In uncertain information environment, bi-polar preferences can be elicited from experts and processed to be exerted over some weights determination for multiple-agents evaluation. Recently, some weighting methodologies and models in uncertain and preference involved environment with multiple opinions from multiple experts are proposed in some literature. However, in that existing method, when collecting different types of preferences from a single expert, sometimes some subtle cognitive inconsistency may occur. To eliminate such inconsistency, this work elaborately analyzes the possible reasons and proposes some amendment together with a new distinguishable set of formulations for modeling. In addition, we further consider two situations of the weighting models for the problem, with one only considering the situation of single expert with no risk of cognitive inconsistency and the other considering the case of multiple experts wherein some inconsistency might occur. Numerical example and comparison are also presented accordingly.

In traditional group decision making, the inconsistent experts are usually forced to make compromises toward the group opinion to increase the group consensus level. However, the strategy of reaching group consensus via an incentive mechanism encouraging adjustment of preferences is more effective than forcing, which is the aim of this article. Specifically, this article establishes a novel incentive mechanism to support group consensus under dynamic trust relationship. First, the supremum and infimum incentives-based rule driven by trust relationship is defined. Based on the assumption that if incentive conditions are met, then experts will be willing to adjust their preferences, the incentive behavior-driven minimum adjustment consensus model is developed to generate optimal incentive-based recommendation preferences. Thus, the proposed incentive mechanism can effectively reduce the preference adjustment cost and promote group consensus reaching. Third, the updated trust relationships between experts are shown to be strengthen by the proposed incentive-driven preference revision. Consequently, the optimization model based on trust interaction relationship is constructed to obtain the final group preference matrix. Finally, a supplier selection case of high-end medical equipment is provided to illustrate the proposed method and show the rationality and advantages of the proposed methodology with both a sensitivity analysis and a comparison analysis.

Based on subjective possibilistic semantics, an agent's subjective probability mass function is dominated by a qualitative Possibility Mass Function (PossMF), which can also be transformed into a unique consonant mass function. However, the existing transformation method cannot maintain the consistency of combination rules, i.e., fusing PossMFs and consonant mass functions with same information content, respectively, the results no longer maintain the reversible transformation. To address the above issue, a novel belief functions transformation is proposed, which can be interpreted based on both Smets' canonical decomposition and Pichon's canonical decomposition. The proposed method is validated based on consistency of combination rules, the least commitment principle, and its application in the fusion of information. In addition, based on the two canonical decompositions, we extend the transformation to possibilistic belief structure, and offer a new perspective of relationship between possibilistic information and evidential information.

In social network group decision making (SN-GDM), overlapping communities are special community structures that can assist opinion interaction to reach group consensus. However, the specific mechanisms of how overlapping structures facilitate community interaction need to be further explored. In addition, the compromise behavior of decision makers (DMs) is conducive to group consensus, but it is usually fixed at the same value, and then it need further research the characteristic of the dynamics compromise limits. To this end, the overlapping community structures under DMs’ trust network is detected. Then, the effect of community overlap in social networks on community interaction is explored. Meanwhile, a limited compromise function is built based on prospect theory to describe the dynamic compromise behavior of communities. Hence, a dynamic compromise behavior driven bidirectional feedback mechanism with overlapping communities is proposed in the context of SN-GDM, and an illustrative example with comparative analysis is provided to testify the advantages of proposed method. It is proved that overlapping communities can improve the compromise willingness compared to nonoverlapping communities, indicating that overlapping communities can serve as a bridge to facilitate interaction, and the dynamic compromise behavior can more realistically describe the real behavior of DMs. In general terms, the proposed method provides a solution to the consensus reaching issue of SN-GDM from a new perspective. Specifically, it can be applied to real-life application scenarios, such as group recommendation to recommend acceptable solutions for social network group users.

In evaluation and decision making frequently we need to consider and then well model the subjective preferences and behaviors of decision makers. This study firstly defines literally three types of conformities, namely, the Bounded Conformity, Extreme Conformity and Bounded Extreme Conformity. Then, we define two types of value-dependent preference aggregation operators, s-t-A extreme operators and Level dependent Preferences Ordered Weighted Averaging operators, to specifically model Extreme Conformity and Bounded Extreme Conformity, respectively. Some related mathematical properties and simple numerical examples are also presented.

Uncertainties are pervasive in ever-increasing more practical evaluation and decision making environments. Numerical information with uncertainty losses more or less credibility, which makes it possible to use bi-polar preference based weights allocation method to attach differing importance to different information granules in evaluation. However, there lacks effective methodologies and techniques to simultaneously consider various categories of involved bi-polar preferences, not merely the magnitude of main data which ordered weighted averaging aggregation can well handle. This work proposes some types and categories of bi-polar preference possibly involved in preference and uncertain evaluation environment, discusses some methods and techniques to elicit the preference strengths from practical backgrounds, and suggests several techniques to generate corresponding weight vectors for performing bi-polar preference based information fusion. Detailed decision making procedure and numerical example with management background are also presented. This work also presents some practical approaches to apply preferences and uncertainties involved aggregation techniques in decision making.

In real life, multiple attribute decision problems (MADM) can be applied in different areas and numerous related extensions and methodologies have been proposed by researchers. Combining three-way TOPSIS decision ideas with MADM is a feasible and meaningful research direction. In light of this, this paper generalizes the classical TOPSIS method with the help of mean and standard deviation and proposes the so-called modified three-way TOPSIS. First, using a pair of thresholds which is derived by mean and standard deviation, we divide decision alternatives into three segments, and then a preliminary rank results of decision alternatives can be obtained. Furthermore, in each decision region, we use two ranking regulations (one-way TOPSIS or modified two-way TOPSIS method) to rank decision alternatives. A practical example of urban expressway route selection illustrates the feasibility of the proposed method. Finally, we test the feasibility and validity of the modified three-way TOPSIS method by comparing with some existing method.

Sugeno-like operators are binary operations-based generalization of Sugeno integral and are still defined on real valued fuzzy measures. This article discusses the aggregation methods in the situation where both inputs and fuzzy measures are attached with numerical uncertainties. That is, when an input vector and a fuzzy measure are given, each of the entries of the vector and the measure value on each subset can be attached with numerical uncertainty degrees. To fulfill this meaningful aim of performing Sugeno-like aggregation with numerical uncertainties, it needs two parts of work. We first formally define basic uncertain vector and basic uncertain fuzzy measure. Then, we discuss the methods to construct or adjust basic uncertain vector and basic uncertain fuzzy measure in two situations, the general fuzzy measure situation and the probability measure only situation, respectively. With uncertain input and uncertain fuzzy measure, we then accordingly propose the corresponding restricted Sugeno-like operator, for which some different logic restrictions are analyzed. All the proposals also have full consistency for they can immediately degenerates into Sugeno-like operator when the attached uncertainties disappear.

The paper looks at the structure of fuzzy rule-based models from the point of view of a function relating membership grades of inputs with rule outputs. This view in turn is generalized by an approach that produces the output functions of the fuzzy rules using input and output data. In this view, a formulation to compute the output of the model consists of estimating the parameters of the output functions. Essentially, the paper suggests an alternative method for fuzzy modeling based on output functions constructed from level sets and input and output data. The data driven method provides an easy and efficient way to develop and process fuzzy models. Examples of function estimation problems show that the data driven level set method is very effective when compared with alternative modeling techniques.

Easy access to large amounts of data triggers an interest in various data analysis techniques. On multiple occasions, an insightful and meaningful ‘view’ of a phenomenon represented by the data is not easy to spot in the obtained results. The users would like to learn more about essential aspects in the context of their own perception of considered concepts and gain a more human-like and linguistic-based interpretation of the results. Yet, existing techniques cannot work with imprecise linguistic terms and do not provide users with results expressed in a human-friendly format.This paper proposes a simple data analysis technique based on calculating correlation. The method investigates the relationships between linguistic terms defined – individually by the users – on data features. It leads to personalized summarization of data that includes the essential relations. In addition, we use the proposed method on the temporal facet/angle of data allowing us to provide the users with the inside regarding the temporal volatility of relations.

In the creation of better multiple attribute decision-making (MADM) patterns to address the ambiguity in the expanding sophisticated of expert systems, the hypothesis of interval-valued intuitionistic fuzzy sets has proven to be an effective and advantageous technique. We employ Aczel–Alsina operations to remedy the MADM issue, wherein all data supplied by decision-makers is conveyed as interval-valued intuitionistic fuzzy (IVIF) decision matrices with all components described by an IVIF number (IVIFN). This allows us to satisfy much more demands from fuzzy decision-making concerns (IVIFN). In the framework of IVIFNs, we primarily describe several novel Aczel–Alsina operations. On the basis of these operations, we construct several novel IVIF aggregation operators, such as the IVIF Aczel–Alsina weighted averaging operator, the IVIF Aczel–Alsina order weighted averaging operator, and IVIF Aczel–Alsina hybrid averaging operator. We built up several features of such operators. We recommend an MADM technique dependent on the advanced IVIF aggregation operators. To demonstrate the effectiveness of the developed technique, we present an overview of research scientist selection. The experimental results show the viability and benefits of the created strategy by contrasting it with the different strategies. This paper reveals that some existing IVIF aggregation operators are particular instances of the operators induced in this paper.

This paper describes the new intuitionistic fuzzy aggregation operators in consequence of Aczel–Alsina operations that possess certain advantages in cases of solving real life problems. We first present some new operations of intuitionistic fuzzy sets (IFSs), for example, Aczel–Alsina sum, Aczel–Alsina product, and Aczel–Alsina scalar multiplication. At that point, we create some IF aggregation operators, for example, the IF Aczel–Alsina weighted averaging operator, the IF Aczel–Alsina ordered weighted averaging operator and IF Aczel–Alsina hybrid averaging operator. We set up different properties of these operators. It is demonstrated that suggested averaging operators have the properties of idempotency, boundary, monotonicity, and commutativity. Then, we design new techniques dependent on these operators to fix multiattribute decision making issues. We present an example of human resource selection to elaborate on the performance of our proposed approach. The outcome shows the practicality and viability of the new technique. Eventually, an organized comparison between the prevailing techniques and the suggested technique has been given.

The paper attempts to bridge the gap between widely accepted models of biological systems based on the Tsetlin automata acting in random environment and traditional artificial neural networks that consist of the McCalloch and Pitts neurons. Using recently developed algebra with uninorm and absorbing norm aggregators, we consider the neurons as extended Tsetlin automata that implement multi-valued not - xor operator applied to the aggregated inputs and internal states, and then construct the network using these neurons. The inputs of the neurons are specified by the synapses that implement multi-valued joined and and or operations. We demonstrate that for favorable (in the sense of learning) states the suggested neurons act similarly to the traditional neurons, while for unfavorable states they immediately change their activity to the reverse one. Such properties of the neurons both results in the correct activity of the network and demonstrates better correspondence with the logics of natural neural networks.

With some theoretical derivation and practical decision-making problems, in this article, we propose a constructing paradigm for generating fuzzy measures in some certain decision situations, such as in risk evaluation and decision making. The constructing paradigm involves four constructing factors: source measure space, construction function, construction choice, and object/constructed measure space. We use some strict and consistent formulation to reinvent the processes of constructing the well-known belief measure and plausibility measure. In this a, the belief measure and plausibility measure are regarded as the two extremities of some pessimism–optimism preferences, and we further propose two construction choices to construct the families of fuzzy measures to generalize and seamlessly connect them. We also propose three different ways to determine construction functions in some certain decisional situations. Several relevant mathematical properties and numerical examples are also presented.

The consideration of social influence in solving group decision-making (GDM) problems has caused widespread concern in recent years. In most cases, the interaction of experts and the association efforts of alternatives are often not to be ignored in the actual decision-making environment, especially for those situations with incomplete information in the process of opinion changes. In this study, a novel measure-based GDM model is proposed to more comprehensively consider such a factor. Also, a positive cooperation-first principle is proposed as a guide for situations with incomplete information. One of the main advantages of the proposed model is that the estimation of missing fuzzy preference relations is completed according to the modeling results of peer interactions, which makes the preference evolution results more reliable. With the utilization of the fuzzy measure framework, the Shapley function and interaction indicator, originated from the cooperative game theory, are also introduced to extract the interaction features. After several iterations, the global opinion can be eventually obtained via the social influence network technique. Furthermore, it is also demonstrated that the evolution of individual opinions converges to the ultimate collective opinion. A case study of supplier selection with the proposed model is implemented to illustrate its practicality and effectiveness. Related comparisons and discussion regarding related methods are also mentioned.

The increased popularity of Linked Open Data (LOD) and advances in Natural Language Processing techniques have led to the development of Question Answering Systems (QASs) that utilize Knowledge Graphs as data sources. QASs perform well on simple questions providing precise and concise answers. Yet, most of them cannot process answers that contain a large volume of numerical values and are not able to provide users with answers in a human-friendly format. In this paper, we propose a user-defined method for constructing linguistic summarization of multi-feature data. It selects suitable summarizers and quantifiers and works with linguistic constraints imposed on the data. The method relies on definitions of linguistic terms constructed by users using an easy and simple graphical interface. Additionally, we introduce a Context-based User-defined Weighted Averaging (CUWA) operator. It allows determining an average value of data that satisfies multiple constraints that are account for the context defined by the user. We include several illustrative examples.

This study proposes some standard and general forms of induced ordered weighted averaging (GnIOWA) operators where the inductive information is ordered weighted averaging (OWA) weight vectors instead of real numbers. It shows the usefulness of such type of generalized induced OWA in decision-making and evaluation and many other applications. We propose three weights allocation methods that are specifically designed for the proposed GnIOWA operators. For each of the proposed weights allocation methods, a numerical example is also attached accordingly. With the use of convex/concave Regular Increasing Monotone quantifiers, we further discuss some mathematical properties of these weights allocation methods.

The dual probabilistic linguistic (DPL) term sets are considered superior to probabilistic linguistic term sets. Further, the generalized Dombi (GD) operators are pretty flexible with the general parameters during the aggregation process. Besides, the Bonferroni mean (BM) operator has the advantage of considering interrelationships between criteria. In this study, we combine the merits of the GD operator, and BM operator for handling multicriteria group decision-making issues under a DPL setting. The existing research on DPL term sets do not focus on both the subjective and objective weights of decision-experts. As a result, the evaluation results are likely to be distorted. To tackle this situation, in this paper, we utilize the concepts of consistency and similarity between the decision-experts to determine the decision-experts subjective and objective weights, respectively. To calculate the weights of criteria, the grey correlation coefficient of the assessment value of criteria is used to reflect the similarity between the criteria and its reference value. Since the existing aggregation operators fail to capture the interrelations between criteria under DPL setting, so for aggregating criteria values, we propose DPL generalized Dombi BM weighted averaging and geometric aggregation operators. We provide a case study regarding biomass feedstock selection to focus on the applicability of these proposed operators. Furthermore, we investigate the effects of the parameters upon ranking order. We also perform a sensitivity assessment of criteria weights to test the stability of our method. Lastly, we provide a comparison between our approach with various extant methods.

Bi-Choquet Integrals based on bi-capacities are very powerful aggregation operators introduced by Grabisch and Labreuche, and they can generalize Choquet Integrals. Bi-capacities can model a lager range of preferences than capacities. However, there are few existent knowledge about constructing bi-capacities, and there lacks the practical and reasonable orness/andness definitions for bi-capacities to present an effective optimism/pessimism indicator. We present a convenient way to generate bi-capacities. As the main proposal of this study, we propose an orness/andness definition for bi-capacities. Although structurally complex in defining, with the help of one interesting proposition related to bi-Choquet Integral, its reasonability in the way of constructing further proves it can indeed serve as a perfect generalization of orness/andness for capacities.

OWA operators and related aggregation techniques generally focus on input vector with a linear ordering. However, in commonly faced multi-criteria and multi-sources evaluation and decision making, the inputs involved form an evaluation matrix. Considering the fact that the data under evaluation are all with two dimensional meanings, this study explores and proposes four novel preference involved aggregation techniques by using RIM quantifiers and OWA operators. The first two models are both with two steps to carry out the aggregation processes, with one using two times of OWA operators and another considering evaluation matrix as a vector lattice. The last two models come from a whole perspective to direct the aggregation processes, with one arising from a global magnitude view and another based on staggered ordering using two specially defined collections of permutations. Illustrative examples and remarks are also spotted immediately following the proposed models or at suitable positions.

Yager's q-rung orthopair fuzzy set is a generalization of fuzzy sets, whose prominent feature is that the qth power sum of the membership and the nonmembership degrees is equal to or less than one, and we call its core, an ordered pair, q-rung orthopair fuzzy number (q-ROFN). More recently, the scholars have constructed the q-rung orthopair fuzzy calculus (q-ROFC), which can effectively deal with continuous q-rung orthopair fuzzy information. Nevertheless, the q-ROFC is only based on the basic operational laws of the q-ROFNs, in fuzzy theory, Archimedean t-norms and t-conorms (ATTs) are a significant class of continuous triangular norms and conorms, which are the generalizations of the intersection and union related to fuzzy sets. Thus, in order to extend the q-ROFC to a wider area, in this article, we systematically discuss the q-rung orthopair fuzzy double integrals (q-ROFDIs) in the frame of ATTs. First, we construct the q-ROFDI in the frame of Archimedean t-conorms in detail, and then provide its concrete value. In addition, we reveal the relationships with respect to two types of q-rung orthopair fuzzy spaces. Based on which, we can easily obtain another types of q-ROFDI. After that, we investigate their fundamental properties in detail so as to comprehend these kinds of q-ROFDIs in-depth. Finally, we point out the essences of these kinds of q-ROFDIs, on the basis of which we provide a practical application to show their effectiveness and elasticity via comparing with the existing methods.

A monotonicity of Interval Sugeno Integrals proposed in “X. Pu, R. Mesiar, R. R. Yager, L. Jin, Interval Sugeno integral with preference,” IEEE Trans. Fuzzy Syst. vol. 28, pp. 597-601, 2020 has been found incorrect with a counter example. This letter presents the counter example and analyzes the reason that causes such nonmonotonicity. In addition, two new related forms of Interval Sugeno Integrals are also proposed, which feature the synchronizations about values and subsets, respectively. The similar monotonicities for the new forms are proved to be correct.

The well-known Choquet integrals are based on given fuzzy measures, providing some sound schemes for the evaluation problems in multicriteria decision making. To further generalize and diversify these interesting aggregation functions, this article proposes the concept of derived fuzzy measures, which can serve as a generalization of normally used fuzzy measures. Based on derived fuzzy measures, we present the detailed integral methods to define the corresponding derived Choquet integrals. With some specific derived fuzzy measures, we find that both Choquet and Sugeno integrals are special cases of derived Choquet integrals. In addition, some relevant properties and propositions with practical explanations are also discussed, for example, the less/more increment properties and amplifying/attenuating properties.

Induced ordered weighted averaging (OWA) operators are important extensions of OWA operators and can suit more different decision environments. However, when practitioners perform induced OWA operators, they frequently do not consider the tied values in inducing information, therein lay the problem of possible non-uniqueness of aggregation results. From a novel perspective that can easily remove this flaw, this study uses a systematical frame to better redefine induced OWA operators, which not only covers all the spirits of the original definition of induced OWA operator but also extends it to some wider understandings and usages. One significant feature related to this frame is that we can express induced OWA operators fully in terms of weighted averaging operators, making the usages and expressions of induced OWA operators much more flexible and stricter. In detail, we propose several new definitions, such as positioning transformation, permutation-based orness, and inducing function based orness. As an example of flexibly formulating complex preferences under the proposed concepts, a novel investigators-consultants-decision maker (ICD) two-stage IOWA evaluation model is clearly proposed and illustrated.

This study first revamps Yager prioritized ordered weighted averaging operators, and condenses them into a conceptual frame with putting aside one realization from Yager's original proposal. Then, based on elicited conceptual frame called Yager prioritized preference conceptual frame, this article proposes three distinct realizations to the conceptual frame with corresponding different instances, in which some evaluation models with weights determination methods are provided. Numerical examples are also presented immediately after every instance. Lastly, this study proposes the concepts of outer monotonic, inner monotonic, and global monotonic weights functions, and discusses some related properties, which are often embodied in preferences of decision makers.

The philosophy of soft sets is founded on the fundamental idea of parameterization, while Pawlak's rough sets put more emphasis on the importance of granulation. As a multivalued extension of soft sets, the newly emerging concept called N-soft sets can provide a finer granular structure with higher distinguishable power. This article offers a fresh insight into rough set theory from the perspective of N-soft sets. We reveal a close connection between N-soft sets and rough structures of various types. First, we show how the corresponding structures of Pawlak's rough sets, tolerance rough sets, and multigranulation rough sets can be derived from a given N-soft set. Conversely, we investigate the representation of these distinct rough structures using the corresponding notions derived from suitable N-soft sets. The applicability of these theoretical results is highlighted with a case study using real data regarding hotel rating. The established two-way correspondences between N-soft sets and diverse rough structures are constructive, which can bridge the gap between seemingly disconnected disciplines and hopefully nourish the development of both rough sets and soft sets.

Sugeno Integral is based on Fuzzy Integral Inference and widely used in applications such as decision making and computational intelligence. When concerned inputs are intervals, directly using Sugeno Integral to respectively aggregate the lower bounds and upper bounds of those intervals has limitations and does not embody fuzzy integral inference. This study analyzes the fuzzy integral inference in interval environment, defines some more suitable orderings on the set of all intervals in [0, 1], i.e., the congruent λ-ordering, and then proposes the Interval Sugeno Integral with preference. The novel aggregation technique proposed in this study proves to better embody fuzzy integral inference when performing Sugeno Integral.

Fuzzy cognitive maps (FCMs) play an important role in high-level reasoning but are limited in their ability to model complex systems with singularities. We are interested in systems that exhibit discontinuous behaviors as one or more of their internal node states approach a threshold. In a new approach to FCM dynamics, we define general classes of aggregation functions which “jump” to a boundary value when any input crosses a threshold, or when all inputs do. The threshold value is a context-dependent parameter which can be readily understood by subject matter experts. Aggregation functions are applied separately to positively and negatively causal antecedents to each node then combined to form the nodal state. This modeling is applied in Computing with Words (CWW) settings, in which link strengths and activation levels are elicited using vocabulary words represented by interval type-2 fuzzy membership functions. We illustrate the behaviors of these novel FCM systems in comparison with their nonsingularity versions.

In this paper, we review the idea of fuzzy measures and look at some properties of these measures. We describe a procedure for aggregating multiple fuzzy measures. We discuss the use of fuzzy measures for modeling information about uncertain variables. We carefully look at the properties of possibility measures. We look at the issue of conjuncting possibility measures.

Analysis of large amounts of data requires considerable efforts in order to extract a useful information or build a model of phenomena represented by the data. On multiple occasions data processing activities can be simpler if some understanding of data is obtained. Further, such activities could be even abandoned when a good insight into data is gained. In this paper, we propose and describe a user-friendly approach for examining datasets via their summarization. It is accomplished using questions equipped with linguistic terms representing imprecise yet easy to grasp concepts by humans. Changes in definitions of the terms done by a user result in different answers and allow to learn more about the data. The approach involves an iPad application that provides a means for an easy way to enter definitions of linguistic terms representing user's perception of imprecise concepts.

Wiener's highly classified wartime work on prediction used the least squares approach and did not involve probabilities. The approach described in this paper, call it SBPD, for short, is probability-based, but is non-traditional. Moreover, SBPD employs some concepts and techniques drawn from fuzzy logic. Underlying SBPD is what may be called a qualitative Prediction Principle: to a considerable degree the Future replicates the Past and the degree increases when the Future is close to Present and Past. Additionally: in similar circumstances (contexts) similar inputs produce similar outputs. The point of departure in SBPD is a time-series of descriptive variables which take values in a finite set V or finite set W depending on whether the variables are prediction variables or decision variables. X plays the role of History. In SBPD, History is examined and searched for contexts which are similar to the present context. Prediction and conclusion are expressed as probability distributions. SBPD has a potential for applications in many directions which go beyond those which appear in the title of the paper. An algorithm which plays an important role in SBPD involves competition of the degree of similarity of two time-series.

Multiple-criteria decision problems involve selecting an alternative that best satisfies a collection of criteria as quantified by a scalar corresponding to an aggregation of the alternatives satisfaction to the individual criteria. A fundamental issue is the formulation of decision maker's aggregation function based upon the decision maker's perceived relationship between the criteria. Here, we allow the decision maker to express their perceived relationship between the criteria in terms of information about the criteria importances by providing a fuzzy measure over the criteria such that the measure of any subset of criteria is its importance. With the aid of the Choquet integral, we use this fuzzy measure of importances to construct an aggregation function. As the Choquet integral requires an ordering of an alternatives individual criteria satisfactions, special handling is required in the case when criteria satisfactions are interval valued rather then scalar. Here we use the golden rule representative value in the case of interval values.

Existing extensions to Yager's ordered weighted averaging (OWA) operators enlarge the application range and to encompass more principles and properties related to OWA aggregation. However, these extensions do not provide a strict and convenient way to model evaluation scenarios with complex or grouped preferences. Based on earlier studies and recent evolutionary changes in OWA operators, we propose formulation paradigms for induced OWA aggregation and a related weight function with self-contained properties that make it possible to model such complex preference-involved evaluation problems in a systematic way. The new formulations have some recursive forms that provide more ways to apply OWA aggregation and deserve further study from a mathematical perspective. In addition, the new proposal generalizes almost all of the well-known extensions to the original OWA operators. We provide an example showing the representative use of such paradigms in decision-making and evaluation problems.

This study proposes a novel concept of Scatter for probability distribution (on [0,1]). The proposed measurement is different from famous Shannon Entropy since it considers [0,1] as a chain instead of a normal set. The measurement works easily and reasonably in practice and conforms to human intuition. Some interesting properties like symmetricity, translation invariance, weak convergence and concavity of this new measurement are also obtained. The measurement also has good potential in more theoretical studies and applications. The novel concept can also be suitably adapted for discrete OWA operators and RIM quantifiers. We then propose a new measurement, the Preference Scatter, with its normalized form, the Normalized Preference Scatter, for OWA weights collections. We analyze its reasonability as a new measurement for OWA weights collections with comparisons to some other measurements like Orness, Normalized Dispersion and Hurwicz Degree of OWA operators. In addition, the corresponding Preference Scatter for RIM quantifiers is defined.

The existing definition of generalized intuitionistic fuzzy soft sets (GIFSSs) is clarified and reformulated as a combination of an IFSS over the universe of discourse and an intuitionistic fuzzy set in the parameter set. With this new perspective, two different types of generalized intuitionistic fuzzy soft subsets and various new operations are developed for GIFSSs. The upper and lower substitutions for both intuitionistic fuzzy sets and GIFSSs are defined as well. Some existing notions and results are improved by virtue of these new concepts. Using the expectation score function, two binary relations are proposed for comparing intuitionistic fuzzy values. An algorithm is designed for coping with multiattribute decision making (MADM) problems with a combined use of the GIFSS, the extended intersection operation, the intuitionistic fuzzy weighted averaging operator, and other related notions. A case study concerning a faculty appointment problem is conducted to illustrate the proposed algorithm. Moreover, a comparative analysis between our approach and other related works is given to demonstrate the effectiveness and advantages of the newly proposed method.

Ordered weighted averaging (OWA) operators, a family of aggregation functions, are widely used in human decision-making schemes to aggregate data inputs of a decision maker's choosing through a process known as OWA aggregation. The weight allocation mechanism of OWA aggregation employs the principle of linear ordering to order data inputs after the input variables have been rearranged. Thus, OWA operators generally cannot be used to aggregate a collection of n inputs obtained from any given convex partially ordered set (poset). This poses a problem since data inputs are often obtained from various convex posets in the real world. To address this problem, this paper proposes methods that practitioners can use in real-world applications to aggregate a collection of n inputs from any given convex poset. The paper also analyzes properties related to the proposed methods, such as monotonicity and weighted OWA aggregation on convex posets.

Given probability information, i.e., a probability measure m with a random variable x on the outcome space N, the expected value of that random variable is commonly used as some valuable evaluation result for further decision making. However, there is no guarantee that the given probability information will be convincing to every decision maker. This is possible because decision makers may question the reliability of that provided probability information and can also be because decision makers often have their own different optimistic/pessimistic preferences. Often, such optimistic/pessimistic preferences can be easily embodied and expressed by some ordered weighted average (OWA) weight functions w. This study first compares and analyzes some simpler methods to melt the given OWA weight functions w with the given probability measure m to generate a new probability measure, pointing out their respective advantages and shortcomings. Then, this study proposes the melting axioms, which will both conform to our intuition and have mathematical reasonability. As the main finding of this study, we then propose the Crescent Method, which will effectively melt the given OWA weight function w with the given probability measure m to generate a final resulted fuzzy measure. Based on that melted fuzzy measure, we perform the Choquet integral of x as the more convincing evaluation result to decision makers with preference w. The study also proposes several interesting mathematical results such as the orness of resulted fuzzy measure will always be equal to the orness of the given OWA weight function w.

This paper puts forward a trust-based framework for building a recommendation mechanism for consensus in group decision making with interval-valued intuitionistic fuzzy information. It first presents an attitudinal trust model where experts assign trust weights to others considering the concept of attitude of the group. This approach allows for the implementation of the group attitude in a continuous scale ranging from a pessimistic attitude to an indifferent attitude. Thus, it can express the continuous trust status, and consequently it generalizes the traditional simplified trust model: `trusting' and `distrusting.' In particular, three typical policies are defined as: `extreme trust policy,' `bounded trust policy,' and `indifferent trust policy.' Second, the attitudinal trust induced recommendation mechanism is established by a reasonable rule: the closer the experts, the higher their trust degree. This can guarantee that the consensus level of the inconsistent expert is increased after adopting the recommended advices. In addition to group consensus, experts envisage to keep their original opinions as much as possible. A harmony degree (HD) is defined to determine the extent of the difference between an original opinion and the corresponding revised opinion after adopting the recommended advices. Combining the HD index and the consensus index, a sensitivity analysis with attitudinal parameter is proposed to verify the rationality of the proposed attitudinal trust recommendation mechanism. In practice, this will facilitate the inconsistent experts to achieve a balance between consensus degree and HD by selecting an appropriate attitudinal parameter.

We review the Dempster-Shafer (D-S) belief structure and discuss the measures of plausibility and belief and other measures that can be associated with this uncertainty modeling structure. We provide a view of the D-S structure as a two-step process, the first is the selection of one of the focal elements and the second is the selection of an object from the chosen focal element. The selection of the focal element is uncertain, the uncertainty is based on a probability distribution. Here, we provide a generalization of this model by allowing a more general type uncertainty, a monotonic set measure, to model the underlying uncertainty associated with the selection of the focal element. We look at form of the plausibility and belief in this more general case. We provide a procedure for uncertain decision making when our uncertainty is modeled using this more general D-S belief structure.

There is no doubt that Internet of Things becomes an important component of future governmental, industrial, commercial and private infrastructures. Interconnected devices, from intelligent ones to simple sensors, will continuously generate enormous amounts of data. It seems impossible to have all this data being transmitted to dedicated processing centres. More and more often, attention is being put on different forms of local, multi-source and multi-level data processing schemas. In this paper, we propose a novel approach to process sensory data in a multi-level fashion. We use elements of Evidence Theory and adopt a newly developed method suitable for satisfying uncertain targets to assess the most adequate state of monitored system/phenomena. We perform this in stages, where observable values are required at the lowest level of processing, while calculations occurring on higher levels use the results of lower level computations. At the same time, levels of belief in the assessed states of a system/phenomena are determined.

We introduce the idea of orthopair membership grades and the related idea of general orthopair fuzzy sets. It is noted that these generalize the intuitionistic and Pythagorean fuzzy sets by allowing the support for and against membership to be almost anywhere in [0, 1] × [0, 1], giving systems modelers great freedom in capturing human knowledge. The aggregation of generalized orthopair fuzzy sets is considered with particular concern for the OWA and Choquet aggregation. The concepts of possibility and certainty as well as plausibility and belief are investigated in this general orthopair environment. We study arithmetic operations on general orthopair fuzzy sets. We show how to obtain associated interval valued fuzzy sets from general orthopair fuzzy sets.

Linguistic summarization provides to express large volumes of quantitative data in easy understandable natural language based forms. While various methods have been recommended for linguistic summarization, there is not any approach for linguistic summarization where possibilistic and probabilistic uncertainties exist together in data. In this study, we establish a tie between Z-number concept and type-I and type-II quantified sentences in order to calculate the truth degree of a linguistic summary covering possibilistic and probabilistic information. The proposed approach employs copulas to obtain a joint probability distribution of the variables included in type-II quantified sentence.

We introduce comfort decision modeling for decision problems in which an alternative is to be selected based on a measure of satisfaction we refer to as comfort. We define comfort as the difference between the payoff received by selecting a particular strategy and the worst payoff that could have been received under the manifestation of the same state-of-nature. We define the effective comfort associated with an alternative as the aggregation of an alternative’s comforts across all possible states-of-nature. We study several methods of aggregating an alternative’s individual comforts across the different states-of-nature, incorporating various types of information about the uncertainty associated with the states-of-nature. We provide a Comfort Decision Model to determine the value of alternatives utilizing attitudinal measures of the decision maker. We demonstrate a process of performing sensitivity of the resulting decision to a measure of the attitude of the decision maker. Lastly, we use an illustration to show the practicability and cogency of the new method.

We introduce the concept of a fuzzy measure and describe the process of combining fuzzy measures to form new measures. We discuss the role of fuzzy measures in modeling uncertain information and its use in modeling granular uncertain information with the aid of measure based belief structures. We turn to the problem of fusing multiple measure based belief structures. First we look at the case when the belief structures being fused have the same focal elements. Then we turn to case where the structures being fused have different focal elements. Finally we compare measure-based fusion with Dempster’s rule.

In this paper, we propose a technique for defining specificity measures in terms of measures of referential success. This research completes the study of the relationships between both families of measures opened in a previous proposal for defining referential success in terms of specificity. Our study allows to define new measures of specificity for any application, and it is particularly relevant in the applied and highly topical area of linguistic description of data by means of data-to-text systems.

Finding the maximum and minimum values of a given collection of real inputs is critical in many decision-making and evaluation models. In often encountered evaluation problems where information is collected based on survey or statistics and embodied by a collection of normalized distributions on lattice, we cannot directly obtain the corresponding concept of “maximum” and “minimum.” To process such situations and related evaluation problem, we first define the distributions poset on a given lattice structure and then propose the concept of lattice dominance points, which can sometimes be seen as a type generalized “maximum” points. Often there may be infinite dominance points to choose from. In order to find some more suitable ones according with our intuition and with mathematical completeness, we propose an optimization method and distribution t-conorm methods to elicit lattice dominances based on given collection of normalized distributions. The distribution and bilinear distribution t-conorms examples are well defined, and some related properties are proposed with strict proofs. In addition, a commonly used evaluation model is proposed with results for comparing differently elicited lattice dominance points.

A multi-criteria decision function allows one to aggregate an alternative's satisfaction to the individual criteria to obtain the alternative's overall satisfaction to a decision problem. We provide a rule-based approach to the construction of these decision functions. Here, we make use of a measure-based representation of the importance associated with subsets of criteria. We then focus on the use of this multi-criteria decision function in the situation where the individual criteria satisfactions are interval valued. To have the capability of implementing the required ordering and comparing operations in the case of interval-valued information, we use the Golden Rule to provide a scalar representative value for these intervals.

In multicriteria decision making, the Choquet integral provides a commonly used method for the aggregation of the individual criteria satisfaction. Information about the criteria importance is expressed via a fuzzy measure on the set of criteria. One requirement in using the Choquet integral is to provide an ordering over the individual criteria satisfactions. When there exists some uncertainty regarding the criteria satisfactions, there typically exists some difficulty in providing this ordering. In an attempt to circumvent this difficulty, we suggest an approximation to the Choquet integral criteria aggregation that does not require an ordering. In this approximation, the criteria satisfactions are aggregated using as weights the Shapley index of the criteria. We provide an assessment of the merit of the approximation using the cardinality indices of the importance measure.

In this paper we propose bounding conditions that characterize different families of specificity measures. The proposed conditions are associated to particular semantics that provide us with different criteria for choosing the most appropriate specificity measure for a certain use. In particular, we distinguish predicate-like measures, mostly useful for providing a fuzzy classification of fuzzy sets between specific and non-specific, and index-like measures, which are more appropriate for providing an order relation between fuzzy sets in terms of their specificity. An intermediate class of measures is also studied, where different bounds are associated to values of the cardinality of fuzzy sets for which specificity is expected to be 0.

A search process becomes an essential component of everyday routine for many users. Users constantly look for new, and more or less relevant items that they require for work or for entertainment. On multiple occasions, they try to find other users who match their `likes' and `dislikes'. Many different methods and approaches have been proposed and developed to address such needs. The Pythagorean Fuzzy Sets have been proposed as a new class of non-standard fuzzy sets. They are related to the idea of Pythagorean membership grades (a, b) that satisfy the requirement a2+ b2≤ 1. The interesting aspect of those types of sets is their ability to express a positive support a - a positive membership grade, and a negative support b - a negative membership grade. In this paper, we propose a method based on the application of Pythagorean fuzzy relations for identifying a degree of matching between users based on their evaluations of items. We use triangular compositions to determine users that match positive evaluations, and users that agree on negative ones. The usage of Pythagorean fuzzy sets allows us to take into consideration both positive and negative aspects of evaluations and find users who like or dislike at least the same items as a given user likes or dislikes. The proposed approach is used to identify users that evaluate movies in a similar way.

Measures of k-specificity are generalizations of Yager's measures of specificity of fuzzy sets or possibility distribution to measure the tranquility when choosing one element, extended to the case when choosing k elements from the universe of discourse. In this paper a new measure of κ-specificity is defined, and an axiomatic definition of measure of crispness is provided. Specificity and entropy has been shown to be inversely related.

Extracting useful and interpretable knowledge from raw data is a crucial issue that has been largely addressed by the data mining community especially. In this paper we provide an interactive data exploration approach that relies on two steps. First, a personalized linguistic summary of the data set concerned is built and displayed as a tag cloud. Then, exploration functionalities are provided on top of the summary to let the user discover interesting properties in the data as frequent/atypical/diversified associations between properties.

The International Journal of Intelligent Systems was created in 1986. Today, the journal is 30 years old. To celebrate this anniversary, this study develops a bibliometric review of all of the papers published in the journal between 1986 and 2015. The results are largely based on the Web of Science Core Collection, which classifies leading bibliographic material by using several indicators including total number of publications and citations, the h-index, cites per paper, and citing articles. The work also uses the VOS viewer software for visualizing the main results through bibliographic coupling and co-citation. The results show a general overview of leading trends that have influenced the journal in terms of highly cited papers, authors, journals, universities and countries.

It is now well recognized that knowledge extracted from rich healthcare data play a vital role for delivery, management and planning of healthcare services. So far, however, there is not much study done on the domain of operational and financial healthcare data since, up to now, a great deal of works are dedicated to clinical/medical healthcare data for the purposes of diagnosis and treatment of diseases. In this paper, an attempt is made, by applying fuzzy linguistic summarization, for the first time to discover knowledge from operational and financial healthcare data. Fuzzy linguistic summarization, in its simplest term, provides natural language based summaries from a dataset in a human consistent way along with a degree of truth attached to each summary. While basically valuable, its benefit can be increased by only generating summaries with a degree of truth above than an indicated threshold value. A genetic algorithm is developed within this context in order to eliminate less promising and useless linguistic summaries. We assess the proposed approach experimentally on a real data and evaluate the generated summaries to gain actionable insights from them.

Uncertainty in spatial geometrical issues is represented using Dempster-Shafer (D-S) theory. Interval approaches are used for D-S uncertainty of spatial locations and the associated arithmetic operations on such intervals described. Categories of uncertainty for points and lines are defined using interval formulations. Based on these, approaches for calculation of geometric areas, line length and line slopes are given. Compatibility of imprecise point locations is discussed and potential aggregations for similar points considered. Finally, topological spatial relationships are described for objects with uncertain boundaries. This will provide a formal framework for the use of a D-S interval approach for uncertainty in spatial geometric issues.

The papers in this special section focus on the use of fuzzy techniques and logic for use in financial modeling and simulation. Computational intelligence has attracted a significant and increasing interest from the financial engineering community, and an emerging interest from analytical economics groups. The bar has been raised with the revision of regulations, and the required compliance and risk management. The new rules should be implemented through new processes and supported by developing new computational tools. Computational systems, capturing sentiments, preferences, behavior, and beliefs, are becoming indispensable in financial applications and desirable in economic analysis. They address problems in the classification of credit worthiness and fraud detection, contribute to the analysis and pricing of financial instruments, and effectively support portfolio optimization and investment analysis. They are instrumental in the design of market mechanisms and contagion mechanisms, and are contributing to the simulation of micro- and macro-economic processes. The armory of fuzzy techniques is capable of addressing challenges encountered in financial engineering and analytical economics. Fuzzy logic can effectively describe and incorporate experts’ intuition, market participants’ preferences, and economic agents’ behavior, thus reaching beyond the capabilities of probabilistic models. The objective of this special issue is to bring together the most recent advances in the design and application of fuzzy approaches to real problems in financial engineering and analytical economics.

We note that orthopair fuzzy subsets are such that that their membership grades are pairs of values, from the unit interval, one indicating the degree of support for membership in the fuzzy set and the other support against membership. We discuss two examples, Atanassov's classic intuitionistic sets and a second kind of intuitionistic set called Pythagorean. We note that for classic intuitionistic sets the sum of the support for and against is bounded by one, while for the second kind, Pythagorean, the sum of the squares of the support for and against is bounded by one. Here we introduce a general class of these sets called q-rung orthopair fuzzy sets in which the sum of the qth power of the support for and the qth power of the support against is bonded by one. We note that as q increases the space of acceptable orthopairs increases and thus gives the user more freedom in expressing their belief about membership grade. We investigate various set operations as well as aggregation operations involving these types of sets.

We introduce the idea of maxitive belief structures, MBS, as a framework for modeling imprecise possibility distributions in a manner analogous to the way Dempster-Shafer belief structures allows the modeling of imprecise probability distributions. We study various properties of these MBS. We consider the problem of obtaining possibility distributions compatible with these belief structures. We consider the problem of fusing multiple MBS and the performance of various arithmetic operations with these objects.

The Semantic Web format of data representation called Resource Description Framework (RDF) is used in Linked Open Data to represent and store any type of information on the Internet. Its intrinsic feature of being a graph-like format provides opportunities for new approaches of analyzing and absorbing information. The extended format of participatory learning based on propositions as pieces of information is an attractive way of integrating new knowledge. Further, it has been recognized that information can have a disjunctive or conjunctive form. This paper uses the RDF-based participatory learning process to absorb conjunctive information characterized by different degrees of consistency. The modification of participatory learning process that allows for integrating such information is presented, and a simple case study is included.

In this paper, a novel approach is introduced to let users extract knowledge from a raw dataset in an intuitive way and using their own vocabulary. The inner structure of a raw data set is first identified using a clustering algorithm, structure on which specificity-driven measures are defined to extract the most informative knowledge. To let domain experts interact with the cluster-based structure and its embedded knowledge, a graphical visualisation is proposed as well as dedicated query operators.

Linked Open Data (LOD) consists of numerous data stores that are highly interconnected. LOD stores use Resource Description Framework (RDF) as a data representation format. A graph-based nature of RDF brings an opportunity to develop new approaches for accumulating data from multiple sources characterized by different levels of confidence in them. Recently, a participatory learning mechanism has been extended to cope with RDF. It is an attractive way of integrating new pieces of information with already known ones. Further, it has been recognized that pieces of information describing entities can have a disjunctive or conjunctive form. This paper uses an RDF-based participatory learning process to aggregate information obtained from multiple data stores. This process provides mechanisms that determine overall certainty in combined data based on levels of confidence in already known pieces of information and new ones. The behavior of such a process used for integrating information equipped with different levels of uncertainty is presented, and a simple case study is included.

We introduce the Sugeno integral and describe how it can be used to provide a weighted mean-like aggregation of a collection of values drawn from the unit interval. We explain that the underlying measure provides information about the weights associated with the arguments. We provide an alternative view of the Sugeno integral that enables us to extend its use to situations in which the arguments in the aggregation are possibility distributions. We look at the application of the Sugeno integral to the formulation of decision functions in the case of multi-criteria decision making. We focus on the situation where there exists some possibilistic uncertainty in our knowledge of criteria satisfactions by an alternative. We provide operational formulations for the calculation of some notable decision functions in the case of possibilistically uncertain criteria satisfactions.

The literature discusses several extensions of fuzzy sets. AIFS, IVFS, HFS, type-2 fuzzy sets are some of them. Interval valued fuzzy sets is one of the extensions where the membership is not a single value but an interval. Atanassov Intuitionistic fuzzy sets, for short AIFS, are defined in terms of two values for each element: membership and non-membership. In this paper we discuss AIFS and their relationship with fuzzy measures. The discussion permits us to define counter AIFS (cIFS) and discretionary AIFS (dIFS). They are extensions of fuzzy sets that are based on fuzzy measures.

We describe the basic properties of the Dempster-Shafer belief structure and introduce the associated measures of plausibility and belief. We look at the role of these structures for providing a model of imprecise probabilistic information. We next consider the problem of calculating the satisfaction of target values by a variable V whose value is expressed by a belief structure. We first look at the simplest case when the target is expressed as subset of the domain of V . We then look at the situation when the target is expressed by more complex uncertain structures. Among those considered are a probability distribution, another belief structure, measure, and possibility distribution. At a formal level this paper involves the extension of the concepts of plausibility and belief associated with D-S structures from being mappings of subsets of the underlying domain of V into unit interval to be mappings of these more complex structures into the unit interval.

Complex fuzzy logic is a new multivalued logic system that has emerged in the last decade. At this time, there are a limited number of known instances of complex fuzzy logic, and only a partial exploration of their properties. There has also been relatively little progress in developing interpretations of complex-valued membership grades. In this paper, we address both problems by examining the recently developed Pythagorean fuzzy sets (a generalization of intuitionistic fuzzy sets). We first characterize two lattices that have been suggested for Pythagorean fuzzy sets and then extend these results to the unit disc of the complex plane. We thereby identify two new complete, distributive lattices over the unit disc, and explore interpretations of them based on fuzzy antonyms and negations.

We discuss the need for obtaining aggregation operations on ordinal-based intuitionistic fuzzy subsets (OBIFS) in order to enable the use of user-friendly fuzzy multicriteria techniques in the mobile “Apps” based environment. We next formally define these OBIFS and look at some of their properties. We describe the formal definition of aggregation of these sets, which is central to multicriteria decision making. We look at some fundamental types of aggregation in this ordinal intuitionistic environment. Particularly notable is our discussion of a new class of ordinal t-norms and t-conorms called stick-type operators.

We introduce the concept of a fuzzy measure, discuss some of their basic properties and look at some notable examples. We introduce a correspondence between a fuzzy measure on X and a fuzzy subset over the power set of X. These fuzzy subsets are required to have special properties on the space 2Xand we refer to these as measure type fuzzy sets. We take advantage of this correspondence between a fuzzy measure and a fuzzy set to suggest the formulation of various set operations on fuzzy measures.

We describe the use of the Choquet integral for finding a mean-like aggregated value of a collection of arguments with respect to a fuzzy measure. We observe the need for ordering the arguments in using this integral. We consider the case where the arguments being aggregated are random variables, probability distributions. In this case, we are faced with the problem of having to order probability distributions. Given the difficulty of obtaining a linear ordering over a collection of probability distributions, we must search for other methods for obtaining a Choquet type aggregation of a collection of probability distributions that does not require a linear ordering; we refer to these as surrogates. Here, we provide one surrogate for calculating the Choquet integral in the case where the objects being aggregated are probability distributions called the probabilistic exceedance method.

We are interested in the formulation of multicriteria decision functions based on the use of a measure over the space of criteria. Specifically, the relationship between the criteria is expressed using a monotonic set measure. We then use the Choquet integral to construct decision functions based on the measure. We look at a number of different decision functions generated from specific classes of measures.

We introduce Jeffrey's rule of conditioning. We explain how it enables us to determine the current probability of an event using a collection of conditional probabilities of the event determined from past experiences and the current probabilities of the conditioning events. We note the importance of the joint probabilities of the event of interest and the conditioning events in obtaining the required conditional probabilities. We investigate the use of copula's to help obtain these required joint probabilities. We then apply our results to a problem of financial decision making in which the success of the stock issue of a new company depends on the quality of management of company. Here, past history tells information about the success of a typical company based on its quality of management and our own observations provides information about the quality of the current companies management.

We discuss the use of the ordered weighted averaging (OWA) operator in multi-criteria decision problems as a means of aggregating the individual criteria satisfactions. We emphasize the need for ordering the arguments, the criteria satisfactions, when using the OWA operator. We consider the situation where the criteria satisfactions have some uncertainty, are finite probability distributions and note the requirement of having to order probability distributions. We introduce the idea of using pairwise stochastic dominance to provide the necessary ordering relationship over the probability distributions. We note that while this approach is appropriate, it is often not possible, since the presence of a stochastic dominance relationship between all pairs of probability distributions is not always the case, the relationship is not complete. To circumvent this problem we introduce an approach called the probabilistic exceedance method (PEM), which allows us to provide a surrogate for the OWA aggregation of probability distributions that doesn't require a linear ordering over the probability distributions. We look at this in both the cases in which the criteria have equal and unequal importances.

OWA operators can be seen as symmetrized weighted arithmetic means, as Choquet integrals with respect to symmetric measures, or as comonotone additive functionals. Following these three different looks on OWAs, we discuss several already known generalizations of OWA operators, including GOWA, IOWA, OMA operators, as well as we propose new types of such generalizations.

We introduce a new structure for fuzzy cognitive maps (FCM) where the traditional fan-in structure involving an inner product followed by a squashing function to describe the causal influences of antecedent nodes to a particular consequent node is replaced with a weighted mean type operator. In this paper, we employ the weighted power mean (WPM). Through appropriate selection of the weights and exponents in the WPM operators, we can both account for the relative importance of different antecedent nodes in the dynamics of a particular node, as well as take a perspective ranging continuously from the most pessimistic (minimum) to the most optimistic (maximum) on the normalized aggregation of antecedents for each node. We consider this FCM structure to be more intuitive than the traditional one, as the nonlinearity involved in the WPM is more scrutable with regard to the aggregation of its inputs. We provide examples of this new FCM structure to illustrate its behavior, including its convergence, and compare it with a traditional FCM architecture on a scenario presented in previous works.

Our concern here is with the issue of the determination of the satisfaction (firing level) of the antecedent condition associated with a variable in a fuzzy systems rule base where this antecedent condition is expressed in terms of a normal fuzzy set, linguistic value. We first consider the case where the input information about the variable is also expressed in terms of a normal fuzzy set. After looking at some approaches for determining this firing level we provide the requirements needed by any formulation for this operation when our input information is a fuzzy set. We next introduce the idea of a measure and show how it can be used to more generally express our knowledge about an uncertain value associated with a variable. We then generalize the requirements for any formulation that can be used to determine the satisfaction (firing level) of the antecedent fuzzy set when the input information about the variable is expressed using a measure. We further provide some examples of formulations. Since a probability distribution is a special case of a measure we are able to determine the firing level of fuzzy rules with probabilistic inputs.

We suggest a transformation to obtain the negation of a probability distribution. We investigate the properties of this negation. Using the Dempster-Shafer theory of evidence, we show of all the possible negations our proposed negation is one having a maximal type entropy.

Our interest here is on comparing Atanassov type intuitionistic membership grades, this is a problem since they are not completely ordered. We suggest the use of an associated scalar value that we refer to as the representative value. We note the formal correspondence between interval valued membership grades and Atanassov type intuitionistic membership grades. Inspired by this we review some ideas on the formulation of representative values for interval valued membership grades. We then use this formal correspondence to help transfer these ideas to Atanassov type intuitionistic membership grades. We note while the construction of a representative value requires the satisfaction of some properties, these properties do not completely constrain its formulation and as such leave room for the inclusion of some subjective aspects in formulating representative values. Here we look at different representative values. A particularly notable example of representative values investigated here is what we refer to as the Golden Rule representative value.

We are interested in developing a framework for determining the degree of satisfaction of some imprecisely stated conditions by some relevant variables whose value itself may be imprecise or uncertain. We particularly focus on the use of fuzzy measures as they can provide a unified representation of various types of uncertainty and imprecision. Starting with set measures, mappings from crisp sets into the unit interval, we first discuss the extension of these to provide measures of fuzzy subsets. We next consider the issue of allowing measures to assign values to other measures. This requires the transformation of the set measure to be a measure on the power set. We then look at some methods for transforming measures to the power set.

Our interest is in the problem of comparing alternatives with uncertain payoffs when the uncertainty is represented using a measure. We first describe various aspects of the use of a measure to represent uncertainty. We recall that probability is a special well-understood example of measure-based uncertainty. We note that stochastic dominance provides a well-established method for comparing alternatives in the case of probabilistic uncertainty. Inspired by this we develop an extension of the use of stochastic dominance for comparing uncertainty profiles to the case where the uncertainty is represented by a measure. We refer to this as measure based stochastic dominance. Do to the fact that in most cases a stochastic dominance relationship does not exist between alternatives this requires us to consider the use of surrogates for measure based stochastic dominance to compare alternatives. Here we investigate a class of surrogates for measure based stochastic dominance that we call Measure Weighted Means (MWM). As we see these MWM are numeric values consistent with measure based stochastic dominance.

In this paper, we investigate the use of weighted averaging aggregation operators as techniques for time series smoothing. We analyze the moving average, exponential smoothing methods, and a new class of smoothing operators based on linearly decaying weights from the perspective of ordered weights averaging to estimate a constant model. We examine two important features associated with the smoothing processes: the average age of the data and the expected variance, both defined in terms of the associated weights. We show that there exists a fundamental conflict between keeping the variance small while using the freshest data. We illustrate the flexibility of the smoothing methods with real datasets; that is, we evaluate the aggregation operators with respect to their minimal attainable variance versus average age. We also examine the efficiency of the smoothed models in time series smoothing, considering real datasets. Good smoothing generally depends upon the underlying method's ability to select appropriate weights to satisfy the criteria of both small variance and recent data.

Recently, a new machine learning approach that is termed as the extreme learning machine (ELM) has been introduced in the literature. This approach is characterized by a unified formulation for regression, binary, and multiclass classification problems, and the related solution is given in an analytical compact form. In this letter, we propose an efficient classification method for hyperspectral images based on this machine learning approach. To address the model selection issue that is associated with the ELM, we develop an automatic-solution-based differential evolution (DE). This simple yet powerful evolutionary optimization algorithm uses cross-validation accuracy as a performance indicator for determining the optimal ELM parameters. Experimental results obtained from four benchmark hyperspectral data sets confirm the attractive properties of the proposed DE-ELM method in terms of classification accuracy and computation time.

Many studies on linguistic summarization have addressed the use of ordinary fuzzy set [type-1 fuzzy set (T1FS)] for modeling words; however, few of them have exploited interval type-2 fuzzy set (IT2FS), although IT2FS is better able to deal with the uncertainty associated with words. The existing studies on linguistic summarization using IT2FS have focused on scalar cardinality based degree of truth. In this paper, for the first time, we propose a probabilistic framework based on the idea of interval mass assignment to evaluate interval type-2 fuzzy linguistic summaries based on the type-II quantified sentences and the semi-fuzzy quantifiers as an alternative method to the scalar cardinality based methods. We implement a real case study on linguistic summarization of time series data of Europe Brent Spot Price, as well as a comparison of the results obtained with our approach and those of the existing approaches.

Evidence resolution has been studied previously for crisp hierarchies and here we develop approaches for fuzzy concept hierarchies. Since fuzzy hierarchies are characterized by decompositions of their domains as opposed to partitions, properties and measures of decompositions are developed. Fuzzy complete and partial evidence resolution is defined and related to the valuation of evidence. Finally a for-and-against analysis is developed for the selection of which concept to choose in the generalization.

We first look at some nonstandard fuzzy sets, intuitionistic, and interval-valued fuzzy sets. We note both these allow a degree of commitment of less then one in assigning membership. We look at the formulation of the negation for these sets and show its expression in terms of the standard complement with respect to the degree of commitment. We then consider the complement operation. We describe its properties and look at alternative definitions of complement operations. We then focus on the Pythagorean complement. Using this complement, we introduce a class of nonstandard Pythagorean fuzzy subsets whose membership grades are pairs, (a, b) satisfying the requirement a2+ b2≤ 1. We introduce a variety of aggregation operations for these Pythagorean fuzzy subsets. We then look at multicriteria decision making in the case where the criteria satisfaction are expressed using Pythagorean membership grades. The issue of having to choose a best alternative in multicriteria decision making leads us to consider the problem of comparing Pythagorean membership grades.

Our objective here is to study the role that fuzzy measures can play in systems reliability modeling. We first introduce some basic ideas from fuzzy measure theory and then introduce some specialized ideas about fuzzy measures, cardinality index, and attitudinal character, which will later be useful in comparing systems reliability. Basic ideas of reliability modeling and the role of the system structure function are discussed. The system's signatures associated with a structure function are introduced and we show their relationship to expected time to system failure. We show how we can model a systems structure function using a fuzzy measure's. A description of the role of a measures cardinality index and attitudinal character in providing tools for comparing systems reliability is provided. We describe how to obtain a structure model of a complex system using aggregation operators and the measure representation of individual components.

In decision-making under uncertainty we have a collection of alternatives from which we must choose one. Associated with each alternative is an uncertainty profile consisting of the set of possible outcomes that can occur if we choose this alternative along with some indication of the uncertainty associated with the outcomes. Since it is difficult to compare these uncertainty profiles one commonly used approach is to obtain a representative value, an aggregation of the information in the uncertainty profile, and use these to compare the alternatives. The method that we use to aggregate the information in the uncertainty profile must reflect aspects of the decision attitude of the decision maker. Here we look at the role that the ordered weighted averaging (OWA) can play in the aggregation process used to obtain these representative values. Since an uncertainty profile can involve a discrete set of possible outcomes or an interval of possible outcomes we provide the OWA operator with the capability to perform aggregations in either of these environments. In addition we provide the OWA operator with the ability to perform aggregations in situations where the arguments being aggregated, the potential outcomes, have an associated uncertainty, probability of occurrence. Particularly notable is the extension the OWA operator to perform aggregation with arguments consisting of a continuum of values with an associated probability density function. We also look at the cases where the uncertainty is modeled with a belief structure as well a possibility distribution.

Our interest here is in multicriteria decision-making when we use a fuzzy measure to capture information about the importances and relationships between the criteria. We describe the use of an integral, such as the Choquet or Sugeno integral, to evaluate the overall satisfaction of each of the available alternatives. We discuss three measures particularly useful for these multicriteria decision problems, the additive, cardinality-based, and possibility measures. We note that the usefulness of these measures is a result of the fact that for each of these, the measure's values for any subset just depends on a small number of parameters. We then consider the situation in which we have some imprecision in these underlying parameters. We show how to represent this imprecision in the underlying parameters using a Dempster-Shafer belief structure. We then consider the evaluation of alternatives under this kind of imprecision using the Choquet, Sugeno, and median type aggregations. As a result of the imprecision in the parameters our overall evaluation for the alternatives, rather than being simple scalar values are imprecise, they are intervals. We discuss some methods for associating a scalar value with an interval. One notable method here is what we refer to as Golden Rule aggregation.

An approach is developed for object similarity to support a focus on the role of extreme values in object matching, which is termed hypermatching. Importance weights are first introduced to the matching and variations formulated for objects that do not share all the same attributes. Extreme attribute values are considered by introducing amplification of attribute importance and several significant cases examined. The objects can both possess the same or different extreme valued attributes. Another case is where one object may have an extreme attribute, which the other object does not have. Then, the analysis of the effect of multiple extreme attributes is presented. Finally for attributes with 0-1 values (features), similarity matching is considered and a closed form solution for the similarity is developed. This provides the basis for the Yager-Petry index, which is then compared with the Jaccard similarity index.

NOx emissions from commercial equipment vehicles can be controlled by the introduction of cleansing agents into output of the equipment. This process requires a monitoring of the output by sensors. Typically, these sensors can be of two types, physical and virtual, each of which can be imperfect in their readings. Physical sensors gradually deteriorate in time while virtual sensors have limit range of reliability. We provide an approach for fusing these imperfect sensor readings using the ordered weighted averaging (OWA) operator. The use of the OWA operator allows users to easily provide their preferred tradeoff between costly introductions of too much cleansing agent versus the possibility of penalization for emission violation. We also provide a methodology for determining the state of the physical sensor. We observe that the system described in this paper has been operationally implemented.

Financial asset volatility modeling and forecasting play a central role in risk management, portfolio selection, and derivative pricing. The increasing availability of market data at intraday frequencies has led to the development of improved volatility measurements such as realized volatility. The literature has shown that simple realized volatility models outperform the popular GARCH and related stochastic volatility models in out-of-sample forecasting. Moreover, gains in performance are achieved by separately considering volatility jump components. This paper suggests a nonlinear approach for realized volatility forecasting with jumps using a simplified evolving fuzzy system based on the concept of data clouds. Such an approach offers an alternative nonparametric form of fuzzy rule antecedents that reflects the real data distribution without requiring any explicit aggregation operations or membership functions, thus providing a more autonomous and efficient algorithm. Empirical results based on the Brazilian stock market index Ibovespa reveal the high potential of the evolving cloud-based fuzzy approach in modeling time-varying realized volatility with jump components, outperforming a traditional benchmark based on a linear regression, as well as alternative evolving fuzzy systems.

The paper addresses a relation between logical reasoning and probability and presents probability-generated aggregators. The obtained aggregators implement probability distributions for specification of generator functions; as it was proven in the paper, such implementation is always possible. In the paper, the relation between neutral element of the probabilistic uninorm and parameters of the underlying probability distribution is demonstrated, and a method for specification of the probabilistic uninorm, and thus—of the probability distribution using t-norm and t-conorm—is constructed. In addition, the obtained probabilistic uninorm and probabilistic absorbing norm or nullnorm are briefly considered as algebraic operations on the open unit interval. In is demonstrated, that, in general, the obtained algebra is nondistributive and depends on the distributions, which are used for generating probabilistic uninorm and absorbing norm. The obtained results bridge several gaps between fuzzy and probabilistic logics and provide a basis both for theoretical studies in the field and for practical techniques of digital/analog schemes synthesis and analysis.

We describe the idea of Pythagorean membership grades and the related idea of Pythagorean fuzzy subsets. We focus on the negation and its relationship to the Pythagorean theorem. We look at the basic set operations for the case of Pythagorean fuzzy subsets. A relationship is shown between Pythagorean membership grades and complex numbers. We specifically show that Pythagorean membership grades are a subclass of complex numbers called Π-i numbers. We investigate operations that are closed under Π-i numbers. We consider the problem of multicriteria decision making with satisfactions expressed as Pythagorean membership grades, Π-i numbers. We look at the use of the geometric mean and ordered weighted geometric operator for aggregating criteria satisfaction.

Our focus is on decision making in uncertain environments. We first introduce the Dempster–Shafer framework to model the uncertainty associated with possible outcomes. We then describe an approach for decision making when our uncertainty is captured using the Dempster–Shafer model and where the payoffs are numeric values. An important part of this approach is the role of the decision attitude as well as the aggregation of the possible payoffs. We then look at the situation where the payoffs, rather than being numbers, are values drawn from an ordinal scale. This requires us to provide appropriate operations for combining payoffs drawn from an ordinal scale.

The concept of moving average is studied. We analyze several extensions by using generalized aggregation operators, obtaining the generalized moving average. The main advantage is that it provides a general framework that includes a wide range of specific cases including the geometric and the quadratic moving average. This analysis is extended by using the generalized ordered weighted averaging (GOWA) and the induced GOWA (IGOWA) operator. Thus, we get the generalized ordered weighted moving average (GOWMA) and the induced GOWMA (IGOWMA) operator. Some of their main properties are studied. We further extend this approach by using distance measures suggesting the concept of distance moving average and generalized distance moving average. We also consider the case with the OWA and the IOWA operator, obtaining the generalized ordered weighted moving averaging distance (GOWMAD) and the induced GOWMAD (IGOWMAD) operator. The paper ends with an application in multi-period decision making.

The web is perceived as a new social platform. Very often, the users look at the web as a place where they can find an individual or group of people with the same or similar interests, or even find new friends. Such situation is reflected in one of the aspects of the web 2.0 called tagging. Tagging is a process of labeling (annotating) digital items-resources-by users. The labels-tags-assigned to those resources reflect users' ways of seeing, categorizing, and perceiving particular items. In general, a single user can label a number of items with a number of different tags. The results of this activity-labeled items and used tags-can be perceived as information characterizing the user. This paper describes an approach for constructing a user signature representing her interests and opinions based on used items and tags. The signature is determined as a fuzzy relation built on two fuzzy sets proposed here: a fuzzy set representing resource attractiveness, and a fuzzy set representing tag popularity. Furthermore, users' signatures are used to determine similarity between users, and potentially give users a method to find new web friends with similar interests and opinions. The paper also describes a process of building different signatures representing a group of users. Signatures of users that are members of the group are aggregated using OWA operator and different linguistic quantifiers to describe the group in a number of ways. A real-world case study illustrating the process of finding similar users and/or groups of users is included.

CIFEr, Computational Intelligence for Financial Engineering and Economics is the major collaboration between the professional engineering and financial communities, and is one of the leading forums for new technologies and applications in the intersection of computational intelligence and financial engineering and economics. Intelligent computational systems have become indispensable in virtually all financial applications, from portfolio selection to proprietary trading to risk management. But nowhere in this definition does it explain the significant role that these three mediums, Computational Intelligence and Financial Engineering and Economics, play in the various financial communities and in our professional lives. CIFEr has been a means of exploration for us and many other professionals who work in a financial arena. We need leading edge Financial Engineers now more then ever to work with "a new" set of "basics" which have evolved recently. Our subprime/liquidity crisis and flash crashes has affected us globally. Program and Algorithmic Trading will continue to grow while changing the way Traders trade and Trade Support supports. The bar has been raised with Regulatory and Compliance and Risk Management. The new rules will not work with the old processes. We need to find new ways to working with the new tools we have developed with Computational Intelligence. We believe CIFEr is one of those forums that will help Financial Engineers get through some perplexing problems at this very difficult time.

We discuss a new approach to aid in decision making in risky situations. This approach is based on the use of a fuzzy rule based formulation to valuate a decision maker's preferences. This allows us to model the responsible agents decision function, their attitude with respect to different uncertain risky situations. An important aspect of risky decision-making is the modeling of the uncertainty associated with an alternative. We refer to this as an alternative's uncertainty profile. In the real world this kind of information can be ill defined and imprecise. We discuss the role of perception based granular probability distributions as a means of modeling the uncertainty profiles of the alternatives. We shown how this can be used to modeling imprecision in our knowledge about the uncertainty associated with an alternative. We provide the necessary formalisms to allow for the evaluation of an alternative using the rule-based description of preferences and granular description of an alternatives uncertainty profile.

We first recall the concept of Z-numbers introduced by Zadeh. These objects consist of an ordered pair (A, B) of fuzzy numbers. We then use these Z-numbers to provide information about an uncertain variable V in the form of a Z-valuation, which expresses the knowledge that the probability that V is A is equal to B. We show that these Z-valuations essentially induce a possibility distribution over probability distributions associated with V. We provide a simple illustration of a Z-valuation. We show how we can use this representation to make decisions and answer questions. We show how to manipulate and combine multiple Z-valuations. We show the relationship between Z-numbers and linguistic summaries. Finally, we provide for a representation of Z-valuations in terms of Dempster–Shafer belief structures, which makes use of type-2 fuzzy sets. © 2012 Wiley Periodicals, Inc.

Our focus is on the representation of uncertain information using set measures. We first discuss the basic properties of monotonic set measures. We then discuss the appropriateness of their use in modeling uncertain information. We look at some notable types of measures of uncertain information and investigate in considerable detail cardinality-based measures. We look at the Sugeno measure and provide a formulation of the underlying cardinality-based measures. We then look at quasi-additive uncertainty measures. We discuss the entropy and attitudinal character of an uncertainty measure. Finally, we introduce the ideas of the assurance and opportunity of the occurrence of an outcome. © 2012 Wiley Periodicals, Inc.

In this paper, we present a number of approaches using fuzzy set theory to influence decision-making behavior, which is a type of human persuasion. We couch the approach as the process to draw a conclusion “V is P” given “V is F,” where P is a fuzzy subset of F representing some linguistic value for V that corresponds to a perception of the world V is P, which we want a person to accept. We examine several methods to represent the process to engender such influences. These include using a person's predispositions, framing the context of a discussion and generalization techniques that allow issues to be viewed in a more favorable light. Finally, we discuss approaches to the related topic of negotiations. This paper describes these approaches and the required background formalisms that are used in them.

Here, we are interested in the issue of multisource uncertain information fusion. We first consider the problem of fusing multiple possibility distributions. In this case, we particularly investigate the issue of normalization and suggest a generalized approach to the normalization of conflicting possibility distributions. We then look at the fusion of multiple probability distributions. We then suggest a framework to fuse probabilistic and possibilistic uncertainty based on the idea of conditioning the probability distribution with respect to the possibility distribution. We extend our approach to the case where our information about either of the distributions is imprecise, the so called second-order uncertainties.

We describe the use of monotonic set measures, which are sometimes referred to as fuzzy measures, in the representation of uncertain information about a variable. The representation of significant types of uncertain information, such as probabilistic, possibilistic, and others, is discussed. The concept of the dual of a measure is introduced and is used to define the measures of opportunity and assurance associated with the occurrence of an event. We observe that these generalize the ideas of possibility and necessity in possibility theory, as well as the ideas of plausibility and belief in Dempster-Shafer theory. Using the measures of opportunity and assurance, we are able to introduce a concept of entailment between two uncertainty representations, which generalizes Zadeh's idea of entailment between possibility distributions. We show, using the idea of entailment, how to relate possibilistic and probabilistic uncertainty and discuss a potential application in hard/soft information fusion. We, finally, consider the measure representation of joint variables.

Our objective here is to extend the participatory learning paradigm (PLP) to environments in which we are interested in learning information and knowledge expressed in terms of declarative statements. We first recall the basic idea of participatory learning, which stresses the important role of what is already believed in all aspects of the learning process. We then discuss the representation of declarative-type binary knowledge within Zadeh's framework of approximate reasoning. We look at the approximate reasoning inference mechanism and its capability for weighted propositions. We introduce ideas such as consistency, compatibility, and commitment that are needed for our objective. We then provide a version of the PLP that is appropriate for the task of learning declarative knowledge. Central to this is the new updation algorithm that is introduced. We finally look at the dynamic performance of this framework. A particularly notable feature is the unlearning and then learning that takes place when the external environment changes.

We describe multicriteria aggregation and discuss its central role in many modern applications. The concept of aggregation imperative is introduced to indicate the description of how the individual criteria satisfactions should be combined to obtain the overall score. We focus on a particular type of aggregation imperative called prioritized aggregation that is characteristic of situations where lack of satisfaction to criteria denoted as higher priority cannot be compensated by increased satisfaction by those denoted as lower priority. We discuss two approaches to the formulation of this type of aggregation process. One of these uses the prioritized aggregation operator, and the second is based on an integral-type aggregation using a monotonic set measure to convey the prioritized imperative.

We have introduced the linguistic weighted power mean and its computation via a generalization of the Karnick Mendel algorithm. This new family of aggregation operators admits interval type-2 fuzzy membership functions for both its inputs and its weights to account for imprecise knowledge of these quantities. The simplest member of this family, the linguistic weighted average, is the primary instantiation of a "perceptual computing" engine in Mendel and Wu's recent book by this title. This paper presents a comparison of the linguistic weighted power mean and the linguistic weighted average on one of the examples they describe in detail-the "investment judgment advisor". Our results illustrate the flexibility and range of logical inference provided by this very versatile aggregation operator for computing with words applications.

The Internet becomes an enormous source of information containing billions of documents. Users deal with an overwhelming number of alternatives, and continuously make decisions anytime they want to obtain meaningful information. The paper describes an approach for a simple yet effective selection of the most suitable information that fits user's needs. The novelty of the approach is twofold: the concept of lexicographical-like preferences used for a multi-criteria decision-making with elements of fuzziness and OWA operator; a simple fuzzy number based mechanism for estimating user's degrees of acceptance of criterion satisfaction. The lexicographic preferences allow for mimicking user's attitude that some criteria should be satisfied before other criteria are considered. The acceptance of criterion satisfaction levels are defined with a single threshold that represents a boundary value between acceptable and unacceptable values of attributes of alternatives. The paper includes results of a simple case study preformed on a prototype of a web selection system built using the proposed approach.

Our concern here is with the multi-source fusion problem. Two important aspects of this problem are the representation of information provided by the sources and the formulation of the instructions on how to fuse the information provided, which we refer to as the fusion imperative. We investigate the use of a monotonic set measure as a means of representing the fusion imperative. We look at the fusion of various different types of information, precise data, uncertain information such as probabilistic and possibilistic. We also consider the case of imprecise uncertain information such as that represented by a Dempster-Shafer belief structure.

The Bonferroni mean (BM) was originally introduced by Bonferroni and then more recently generalized by Yager. The desirable characteristic of the BM is its capability to capture the interrelationship between input arguments. Nevertheless, it seems that the existing literature only considers the BM for aggregating crisp numbers instead of any other types of arguments. In this paper, we investigate the BM under intuitionistic fuzzy environments. We develop an intuitionistic fuzzy BM (IFBM) and discuss its variety of special cases. Then, we apply the weighted IFBM to multicriteria decision making. Some numerical examples are given to illustrate our results.

In this paper some axiomatic definitions about specificity for interval-valued fuzzy sets are proposed. Some examples of measures of specificity for interval-valued fuzzy sets are showed. It is also defined a extension of the notion of alpha cut for interval-valued fuzzy sets and a generalized similarity for interval-valued fuzzy relations. An axiomatic definition of specificity of interval-valued fuzzy sets under the knowledge of a generalized similarity is proposed.

User's involvement in creating contents of the web is an essential element of Web 2.0 and social software. One example of such involvement is a process called tagging: users annotate digital items, called resources, with labels, called tags. Tags used for annotation represent users' categorization and perception of resources. As the result a network of interconnected resources and tags is created. Such a network is meant to simplify a search process for relevant content. The paper introduced the idea of constructing fuzzy sets based on the network of resources and tags. One of possible ways of building fuzzy representation of tags is proposed and described here. It is shown how those fuzzy sets can be used for search purposes.

We introduce a measure of diversity related to the problem of selecting n objects from a pool of candidates lying in q categories. We introduce the concept of target diversity index (TDI) and describe various agendas for defining it. We look at the problem of trying to select elements to satisfy some desirable criterion that additionally satisfies a requirement of being diverse. We suggest some aggregation methods for combining these multiple criteria. © 2010 Wiley Periodicals, Inc.

In this paper, we develop characterizations of the properties of categories. We introduce an approach to representing imprecise hierarchies based on the idea of fragmentation. In the case of fuzzy categories, measures of how well fuzzy categorization satisfies the concept of a partitioning for such categories are developed. Finally, issues that are involved in the formulation of fuzzy categorization of a domain using approaches based on the use of prototypes are considered.

The power-average (PA) operator and the power-ordered-weighted-average (POWA) operator are the two nonlinear weighted-average aggregation tools whose weighting vectors depend on the input arguments. In this paper, we develop a power-geometric (PG) operator and its weighted form, which are on the basis of the PA operator and the geometric mean, and develop a power-ordered-geometric (POG) operator and a power-ordered-weighted-geometric (POWG) operator, which are on the basis of the POWA operator and the geometric mean, and study some of their properties. We also discuss the relationship between the PA and PG operators and the relationship between the POWA and POWG operators. Then, we extend the PG and POWG operators to uncertain environments, i.e., develop an uncertain PG (UPG) operator and its weighted form, and an uncertain power-ordered-weighted-geometric (UPOWG) operator to aggregate the input arguments taking the form of interval of numerical values. Furthermore, we utilize the weighted PG and POWG operators, respectively, to develop an approach to group decision making based on multiplicative preference relations and utilize the weighted UPG and UPOWG operators, respectively, to develop an approach to group decision making based on uncertain multiplicative preference relations. Finally, we apply both the developed approaches to broadband Internet-service selection.

We describe the basic properties of a norm and introduce the Minkowski norm. We then show that the OWA aggregation operator can be used to provide norms. To enable this we require that the OWA weights satisfy the buoyancy property,$w_{j} \geq w_{k}$for$j$<$k$. We consider a number of different classes of OWA norms. It is shown that the functional generation of the weights of an OWA norm requires the weight generating function have a non-positive second derivative. We discuss the use of the generalized OWA operator to provide norms. Finally we describe the use of OWA operators to induce similarity measures.

We provide a framework for querying of databases. We model a query as a collection of required conditions and an imperative for combining an object's satisfaction to the individual conditions to get its overall satisfaction. We investigate tools that can enrich this process by enabling the inclusion of more human focused considerations. We discuss how to include flexible conditions with the use of fuzzy sets. We describe some more sophisticated techniques for aggregating the satisfactions of the individual conditions based on the inclusion of importances and the use of the OWA operator. We introduce a new method for aggregating the individual satisfactions that can model a lexicographic relation between the individual requirements. In addition to considering more human focused aspects of the query we look at databases in which the information in the database can have some probabilistic or possibilistic uncertainty.

We consider an application of fuzzy logic connectives to statistical regression. We replace the standard least squares, least absolute deviation, and maximum likelihood criteria with an ordered weighted averaging (OWA) function of the residuals. Depending on the choice of the weights, we obtain the standard regression problems, high-breakdown robust methods (least median, least trimmed squares, and trimmed likelihood methods), as well as new formulations. We present various approaches to numerical solution of such regression problems. OWA-based regression is particularly useful in the presence of outliers, and we illustrate the performance of the new methods on several instances of linear regression problems with multiple outliers.

We consider a problem of mediated group decision making where a number of agents provide a preference function over a set of alternatives. Then, using such information, a new agent provides its own preferences, and, after that, a mediation step is applied to aggregate the individual preferences in order to obtain a group-preference function. Finally, the most supported alternative is selected. Two key aspects are that the preference functions of the former agents may or may not have uncertainty, and that the mediation process rewards those agents that are open to other alternatives besides their most preferred ones. The question for the new agent is how to score its alternatives in such a way that its most preferred one gets the biggest group support. We propose to define such scoring or preference function as the solution of a nonlinear optimization problem. The model also takes into account that imprecision could exist in the preference functions. Through extensive simulations (varying the number of agents, alternatives, etc.), we conclude that the proposal is feasible and effective. Additionally, the usefulness of the mediation process rewarding openness is empirically confirmed.

We discuss the idea of fuzzy relationships and their role in modeling weighted social relational networks. The paradigm of computing with words is introduced, and the role that fuzzy sets play in representing linguistic concepts is described. We discuss how these technologies can provide a bridge between a network analyst's linguistic description of social network concepts and the formal model of the network. We then turn to some examples of taking an analyst's network concepts and formally representing them in terms of network properties. We first do this for the concept of clique and then for the idea of node importance. Finally, we introduce the idea of vector-valued nodes and begin developing a technology of social network database theory.

The recent development of Web 2.0 has provided an enormous increase in human interactions across all corners of the earth. One manifestation of this is the growth of computer mediated social networks. Many notable Web 2.0 applications such as Facebook, Myspace and LinkedIn are social networks. Relational networks are becoming an important technology for modeling these types of social networks and the type of collaborative intelligence that arises from these interactions. Our goal here is to enrich the domain of social network modeling by introducing ideas from fuzzy sets and related granular computing technologies to provide a bridge between a human network analyst's linguistic description of social network concepts and the formal model of the network.

We introduce and discuss the participatory learning paradigm. A formal system implementing this type of learning agent is described. We then extend this system so that it can learn from interval-type observations. We further extend this system to the case when the observation is a more general granular object such as a fuzzy set. In the initial stage, while we allowed our observations to be granular, we restricted the learning to be precise values. In the next part, we allow both the observations and learned object to be granular. An important issue that arises when learning granular values relates to the specificity of the learned value. Learned values that are too unspecific can be useless. We suggest methods for controlling the specificity of the values learned.

We introduce the basic features of the ordered weighted averaging (OWA) operator. Particular emphasis is put on the task of obtaining the associated weights. We discuss the maximal entropy OWA (MEOWA) approach to obtaining the weights. This approach is based upon the specification of a parameter characterizing the desired type of aggregation and then the solving of a mathematical programming problem whose objective is to maximize the entropy of the weights subject to this parameter. Here, we provide an alternative way of getting these MEOWA weights based upon the use of a weight-generating function. The introduction of this function allows us to obtain the MEOWA weights for the case in which each argument has a distinct degree of importance. The development of this approach allows us to use the OWA operator in decision making under risk. Here, we are able include probabilistic information as well as decision attitude to construct customized decision functions.

This paper discusses level sets of functions from a set U into a set C with an order structure, for example a poset or a lattice. Such sets are prevalent in fuzzy set theory. We provide some general facts and then consider the particular case when C is the complete lattice of closed intervals in the unit interval [0,1]; that is, when the function is an interval-valued fuzzy set.

Our focus here is to provide a methodology that can be used by a participant in an adversarial decision-making environment to help choose their action. Central to our approach is the use of knowledge and perceptions about one's adversary to obtain an uncertainty profile indicating which action the adversary will take. Once having this uncertainty profile, the problem of deciding which action to take becomes a problem of decision making under uncertainty. Here, we make considerable use of the Dempster--Shafer belief structure as a way of formalizing the uncertainties. © 2008 Wiley Periodicals, Inc.

An introduction to some basic ideas of graph (relational network) theory is first provided. We then discuss some concepts from granular computing in particular the fuzzy set paradigm of computing with words. The natural connection between graph theory and granular computing, particularly fuzzy set theory, is pointed out. This connection is grounded in the fact that these are both set-based technologies. Our objective here is to take a step toward the development of intelligent social network analysis using granular computing. In particular one can start by expressing in a human-focused manner concepts associated with social networks then formalize these concepts using fuzzy sets and then evaluate these concepts with respect to social networks that have been represented using set-based relational network theory. We capture this approach in what we call the paradigm for intelligent social network analysis, PISNA. Using this paradigm, we provide definitions of a number of concepts related to social networks. © 2008 Wiley Periodicals, Inc.

The Internet holds huge amount of documents available for users. Effective utilization of this enormous repository means a need for systems supporting users in a process of finding related documents. An ontology defined in the framework of the Semantic Web (Berners, 2001) allows for specification of concepts, their instances, and relationships existing between concepts. A hierarchy of concepts (Yager, 2000) is a graph-like structure providing a means for representing human-like dependencies. The article proposes an approach for utilization of a hierarchy of concepts to perform categorization of web pages in the Semantic Web. A user provides a hierarchy that can only partially “cover” their domain of interest. The hierarchy is treated as a “seed” representing user’s initial knowledge about the domain. Ontologies are treated as supplementary knowledge bases. They are used to instantiate the hierarchy with concrete information, as well as to enhance it with new concepts initially unknown to a user.

We introduce a variation of the Mountain Method called the S-Mountain Method, an algorithm for finding focus points from data points based on the similarity of a focus point with the data points. An important aspect of this algorithm is the association of energy levels with the data points and the reduction of these energy levels as we discover focus points. The rate of reduction of the energy level at a data point is related to its similarity with the recently discovered focus point. We provide a general form for this reduction operation using the t-norm operators. We show how this S-Mountain algorithm can be used to discover focus elements that are granular objects such as fuzzy subsets. Finally we extend the S-Mountain method to environments where our similarities are drawn from an ordinal scale. Here we discuss the interesting question of comparing objects consisting of collections of ordinal values.

This paper describes a conceptual and theoretical framework for the resolution of seemingly contradictory evidence for decision-making. Basic to this approach is the use of granulation provided by the categories obtained by ascending concept hierarchies. This process will be driven by the use of a criterion that represents the utility of granular categories to the user's decision making. The definition of complete and partial evidence resolution and their properties are developed, which permits the formulation of the concept of preponderance of evidence for the decision maker. Finally, we show some preliminary results on the concepts of strength and consensus measures to provide metrics of the goodness of the evidence resolution.

In this paper, an overview of the time series smoothing problem is provided and the role of data aggregation is described using a weighted average of the past observations. Two important features associated with this smoothing process are introduced. One is the average age of the data and other is the expected variance. It is noted how both of these are defined in terms of the associated weights. We show the fundamental conflict between trying to keep the variance small while simultaneously using the freshest data. We study the moving average and exponential smoothing methods. Focusing on the aggregation aspect of the data smoothing problem allows us to draw upon the work done with the ordered weighted averaging aggregation operators to suggest new methods for developing smoothing techniques. A new class of smoothing operators based on the use of linearly decaying weights is introduced and is shown to have some better features than either exponential smoothing or the moving average. Other classes of smoothing operators are introduced. We introduce the idea of an operational estimate that involves adjusting a smoothed value by other considerations.

Fuzzy cognitive maps (FCMs) allow experts to express their knowledge by drawing weighted causal digraphs. Experts can pool or fuse their knowledge by adding the underlying FCM causal matrices. This naturally extends the ordered-weighted-averaging (OWA) technique to averaging dynamical systems and can create complex dynamical systems from several simpler ones. Edge quantization allows experts to state their knowledge in the simpler terms of causal increase (1), decrease (−1), or absence (0). We model the expert FCMs as a sequence of random fields to study the small-sample effects of quantizing both the causal edges and the fuzzy-set concept nodes. The averaged quantized random matrices exhibit large-sample convergence to the population means of the unquantized matrices in accordance with the Strong Law of Large Numbers. But the small-sample averages can show substantial diversity of equilibrium attractors (fixed points or limit cycles). We use statistical tests—chi-square tests, Spearman's rank coefficient, the Kolmogorov–Smirnov test, and the fuzzy equality of limit cycle histograms—to show that this small-sample equilibrium diversity increases as the node multivalence or fuzzy-set quantization increases. The appendix presents a new probabilistic convergence theorem that shows that edge quantization or thresholding does not affect FCM combination for large expert sample sizes: the sample mean of quantized expert causal edge values converges with probability one to the population mean causal edge values. © 2007 Wiley Periodicals, Inc. Int J Int Syst 22: 181–202, 2007.

Future automated question answering systems will typically involve the use of local knowledge available on the users' systems as well as knowledge retrieved from the Web. The determination of what information we should seek out on the Web must be directed by its potential value or relevance to our objective in the light of what knowledge is already available. Here we begin to provide a formal quantification of the concept of relevance and related ideas for systems that use fuzzy-set-based representations to provide the underlying semantics. We also introduce the idea of ease of extraction to quantify the ability of extracting relevant information from complex relationships. © 2007 Wiley Periodicals, Inc. Int J Int Syst 22: 385–396, 2007.

We describe a novel hypercube graph representation for labeled graphs with arbitrary edge weights in the interval [0, 1]. This representation admits graphical models for weighted adjacency matrices, which are useful in a number of real world applications wherein the strength of connections between graph nodes is important. It enables us to bring to bear a full arsenal of fuzzy set theoretic measures such as fuzzy subsethood, entropy, completeness, and mutual subsethood to the description of graphs. Our hypercube representation also provides a direct similarity metric between pairs of graphs, which is particularly useful for external comparisons among sets of graphs. The unitary complement of this similarity metric in turn provides a distance metric between two graphs, thus enabling us to perform vector processing operations on graphs, e.g., clustering, change detection, hypothesis testing as to the independence of two graphs, feature extraction for neural network and/or statistical classifiers, and antecedent specification for fuzzy mappings. We derive the probability mass function of this metric for two independent random graphs. The hypercube graph representation finds applications in problems where we are dealing with labeled graphs, e.g., computer networks, social networks, graphical information retrieval, and data fusion problems involving virtual networks of events. Of special interest are labeled graphs with fixed vertices whose edges and their corresponding weights vary over time, as well as graphs that evolve in time by the addition of new vertices and edges.

The ordered weighted averaging operator attempts to provide a user-friendly class of mean type aggregation operators by allowing a user to easily and naturally go from a mental model of a desired type of aggregation to its formal realization. Our objective in this paper is to support this agenda by providing a new methodology to help a user describe the type of aggregation they desire using the idea of a stress function. With this stress function, a user can "stress" which argument values they want to give more weight in the aggregation. An important feature of this stress function is that it is only required to be nonnegative function on the unit interval. This allows a user to completely focus on the issue of where to put the stress in the aggregation without having to consider satisfaction of any other requirements.

In this paper we provide a computational intelligence methodology for risk and disaster management using attitudinal and fuzzy modeling. We provide a paradigm representing the relationship among threats, events, control alternatives and losses. In evaluating a control alternative we look beyond the traditional expected value as a summary measure of expectation and the traditional variance as a measure of dispersion. We introduce a new measure of dispersion which incorporates the decision maker's attitude. We present a fuzzy model to use this attitudinal variance in conjunction with the attitudinal expected value in order to assess the overall value of control alternatives.

We discuss how the ordered weighted averaging (OWA) operator can provide a basis for generating summarizing statistics over large data sets. We further note how different forms of OWA operators can be induced using weight generating functions. We show how these weight generating functions can provide a vehicle with which a data analyst can express desired summarizing statistics. Our purpose here is to begin the development of a mixed graphical-linguistic means of communication to enable analyst to describe to computing machines various sophisticated types of summaries that may be desired.

Considerable concern has arisen regarding the quality of intelligence analysis. This has been, in large part, motivated by the task of determining whether Iraq had weapons of mass destruction. One problem that made this analysis difficult was the uncertainty in much of the information available to the intelligence analysts. In this work, we introduce some tools that can be of use to intelligence analysts for representing and processing uncertain information. We make considerable use of technologies based on fuzzy sets and related disciplines such as approximate reasoning. © 2006 Wiley Periodicals, Inc. Int J Int Syst 21: 523–544, 2006.

Abstract We point out that question‐answering systems differ from other information‐seeking applications, such as search engines, by having a deduction capability, an ability to answer questions by a synthesis of information residing in different parts of its knowledge base. This capability requires appropriate representation of various types of human knowledge, rules for locally manipulating this knowledge, and a framework for providing a global plan for appropriately mobilizing the information in the knowledge to address the question posed. In this article we suggest tools to provide these capabilities. We describe how the fuzzy set–based theory of approximate reasoning can aid in the process of representing knowledge. We discuss how protoforms can be used to aid in deduction and local manipulation of knowledge. The idea of a knowledge tree is introduced to provide a global framework for mobilizing the knowledge base in response to a query. We look at some types of commonsense and default knowledge. This requires us to address the complexity of the nonmonotonicity that these types of knowledge often display. We also briefly discuss the role that Dempster‐Shafer structures can play in representing knowledge.

The concept of measure of specificity of fuzzy sets is extended in this paper to measure of specificity under the knowledge of T-indistinguishabilities. Four axioms of measure of specificity under T-indistinguishabilities are given. An algorithm to compute the inference independent set /spl Tfr/ from a fuzzy set and a T-indistinguishability is used to measure the specificity of fuzzy sets under T-indistinguishabilities satisfying the axioms.

Our focus is on the issue of decision making in risky situations. We discuss the need for using decision functions to aid in capturing the decision maker's preference among these types of uncertain alternatives. The use of fuzzy rule based formulations to model these functions is investigated. We discuss the role of perception based granular probability distributions as a means of modeling the uncertainty profiles of the alternatives. Various properties of this method of describing uncertainty are provided. Tools for evaluating rule based decision functions in the face of perception based uncertainty profiles are presented. Consideration is given to the situation in which our uncertainty profiles are expressed in terms of a cumulative distribution function. We introduce the idea of a perception based granular cumulative distribution and describe its representation in terms of a fuzzy rule-based model

This paper describes a conceptual and theoretical framework to allow better user control over data summarization for knowledge discovery. Basic to the approach is a measure of quality of summarization of data using categories provided by the hierarchical structure of concept ontology. This involves the modeling, using a fuzzy sets approach, of the four criteria implicit in a summarization imperative: minimum coverage, minimum relevance, succinctness, and usefulness. With these criteria modeled, a multicriteria approach is presented, using a decision function aggregating these criteria that provides an overall quality measure to guide the summarization of the data. The development of the theory is first presented for the simple case of a single attribute to clearly delineate the basic issues and approach and then extended to multiple attributes. Finally, approaches to provide a more user-oriented presentation of the summarized data are considered

This work suggests an unsupervised fuzzy clustering algorithm based on the concept of participatory learning introduced by Yager in the nineties. The performance of the algorithm is verified with synthetic data sets and with the well-known Iris data. In both circumstances the participatory learning algorithm determines the expected number of clusters and the corresponding cluster centers successfully. Comparisons with Gustafson-Kessel (GK) and modified fuzzy k-means (MFKM) are included to show the effectiveness of the participatory approach in data clustering

This paper points out some special properties of LexiT orderings for strict t-norms. Among these properties is that the comparison of two n-tuples takes at most n steps for strict t-norms, and that distinct strict t-norms yield distinct orderings

We consider the problem of multicriteria decision making. We indicate how the evaluation of an alternative involves a determination of the degree to which subsets of criteria are satisfied by the alternative. This calculation is based upon an anding of the satisfactions of the individual criteria in the subsets. We consider the possibility of using t-norms other than the Min for the and operation. Using this generalization we develop an extension of the OWA operators, called the TOWA, which involves a mixing of the t-norm with the OWA operator. We extend this generalization to other aggregation techniques, the Choquet and Sugeno integrals. We introduce the concept of the Power of a t-norm to provide an ordering over the t-norm operators. We look at Power of a number of families of t-norm. © 2005 Wiley Periodicals, Inc. Int J Int Syst 20: 453–474, 2005.

An important issue when using the OWA aggregation operators is the determination of weights. One approach is to link the weights to a desired attitudinal character for the aggregation. The ME-OWA operators provide a pioneering example of this approach. Here we first present an alternative approach to generating OWA weights with a desired attitudinal character. We accomplish this by using a family of recursive OWA operators (R-OWA). We then generalize this with a class that allows of OWA aggregation by iteration (It-OWA). Both families are built with the constraint of keeping constant the attitudinal character at any recursion or any iteration step. This is particularly useful in aggregations that sequentially add arguments to the aggregation.

We discuss the participatory learning paradigm and emphasize the all-inclusive role of the currently beliefs in every aspect of the learning process. We describe how the acceptance mechanism determines which observations are used for learning based upon their compatibility with the current beliefs. We point out how this lead to more efficient learning. A prototypical example of a participatory learning system is introduced. We introduce the concept of a reviewer that monitors the performance of the system and gives it some independence from the current beliefs.

We are concerned with the problem of uncertain decision making. The paradigm of decision making using minimization of maximal regret (MMR) is introduced. We discuss a generalization of the MMR method leading to a parameterized family of minimal regret methods. We apply the minimal regret method of decision making to situations in which our uncertainty profile is represented by a Dempster-Shafer belief structure.

We introduce the t-norm and discuss the representation of this operator by additive generators. An approach is suggested for including importances in t-norm aggregation taking advantage of this representation of the t-norm. We look at formulations resulting for a variety of t-norms. We then apply the approach to the t-conorm and look at its effect on a variety of t-conorms. Finally, we turn to the uninorm and suggest an analogous method for including weighting in uninorm aggregations. © 2004 Wiley Periodicals, Inc.

We discuss the OWA and Choquet integral aggregation operators and point out the central role the ordering operation plays in these operators. We extend the capabilities of the Choquet integral aggregation by allowing the ordering to be induced by some values other then those being aggregated. This allows us to consider an induced Choquet Choquet integral aggregation operator. We look at the properties of this operator. We then look at its applications. Among the applications considered are aggregations guided by linguistic and other ordinal structures. We look at the use of induced aggregation in nearest neighbor methods. We also consider the Choquet aggregation of complex objects such as matrices and vectors.

Based on the strong idempotency of aggregation operators, quantitative weights are incorporated into the fusion process. Our general method is compared with some previous specific cases. More details about weighted aggregation based on some penalty function is given. Further, weighted integral based aggregation linked to quantifier-based fuzzy measures is investigated, especially weighted OWA operators. Several examples are included.

We briefly describe the ordered weighted average (OWA) operator. We discuss its role in decision making under uncertainty. We provide an extension of the OWA operator to the case in which our argument is a continuous valued interval rather than a finite set of values. We look at some examples of this type of aggregation. We show how it can be used in some tasks that arise in decision making. We consider the extension of the continuous interval argument OWA operator to the more general case in which the argument values have importance weights. We use this to introduce the idea of an attitudinal-based expected value associated with a continuous random variable.

We discuss the Dempster-Shafer belief structure. We introduce the idea of a cumulative distribution induced by a Dempster-Shafer belief structure. We call these belief-cumulative distribution (B-CDs) functions. We study the properties of these distribution functions and show that they are interval functions. We investigate the possibility of using these distribution functions as a tool for knowledge representation.

We consider the problem of multicriteria decision making (MCDM) in the situation in which there exists a prioritization of criteria. A good example of prioritization among criteria occurs in the case of air travel, where concerns about passenger safety have a higher priority then economic concerns. Tradeoffs between saving on gasoline usage and jeopardizing passenger safety are unacceptable. We show how this prioritization of criteria can be modeled by using importance weights in which the weights associated with the lower priority criteria are related to the satisfaction of the higher priority criteria. We provide some models that allow for the formalization of these prioritized MCDM problems using both the Bellman-Zadeh paradigm for MCDM and the ordered weighted averaging (OWA) operator method.

We discuss Zadeh's paradigm of computing with words and indicate the three important stages. We focus on the retranslation process, selecting a term from our prescribed vocabulary to express information represented using fuzzy sets. A number of criteria of concern in this retranslation process are introduced. Some of these criteria can be seen to correspond to a desire to accurately reflect the given information. Other criteria may correspond to a desire, on the part of the provider of the information, to give a particular perception or "spin." These types of criteria can be of particular importance in many types of information warfare. We discuss some methods for combining these criteria to evaluate potential retranslations.

We focus on situations in which we must decide on what time to take an action. The action is not in question it is the time of action. We call these "time for action decisions," a prototypical example being deciding when to leave on a journey. We point out that during this type of decision process, the decision-maker recognizes two groups of forces acting on him: one that pushes him to act now and other that pushes him act later. We note that the strength of these forces depends on the information available about various uncertainties associated with the situation. It also strongly depends upon the personality of the decision-maker. We observe that as time passes these conflicting forces tend to build up an anxiety in the decision-maker resulting in an action being taken at a time of most intense anxiety. In this paper using the ideas of possibility and necessity measures to enable different interpretations of uncertain information we investigate the temporal profile of the decision-maker's anxiety as a function of their decision attitude. We investigate the role of maximization of anxiety as decision paradigm. One of our goals here is to try to understand role of the nature and the quality information plays in these types of decisions as well as its interaction with anxiety.

This paper describes a new method for automatic generation of OWA operators. It introduces a Takagi-Sugeno type model to link the process of selecting the OWA weights to the data being aggregated. A parameterized and cardinality independent type of OWA weighting vector is obtained through an analytically expression of the OWA operator as a function of the derivatives of an S-curve. These results lead to a context dependent information aggregation method.

We discuss the role of disjunctive aggregation operators in prototype based reasoning systems such as recommender systems. We indicate a need for these aggregation operators to have the additional property of noble reinforcement: Allowing a collection of high valued arguments to reinforce each other to give complete satisfaction while avoiding the situation in which a collection of low valued arguments act in the same way. We describe the difficulty that most t-conorms have in realizing this feature of noble reinforcement. We show that by relaxing the requirement of associativity we can provide disjunctive type aggregation operators that manifest noble reinforcement.

We introduce the ordered weighted averaging (OWA) operator and discuss how it can provide a basis for generating summarizing statistics over large data sets. We further note how different forms of OWA operators can be induced using weight generating functions. We show how these weight generating functions can provide a vehicle with which a data analyst can express desired summarizing statistics. Our goal is to develop an understanding of the relationship between weight generating functions and resulting summarizing statistics.

Fuzzy rules have been shown to be very useful in modeling relationships between variables that have a high degree of uncertainty or ambiguity. A major question in regards to learning fuzzy rule bases is how to handle interactions between rules of overlapping coverage. Structures, such as Yager's (1993, 1998) HPS (hierarchical prioritized structure), have been proposed to answer this question. In this paper, we present our system of learning a hierarchical fuzzy rule base named FURL (for Fuzzy Rule Learner). FURL takes advantage of the properties of HPS to learn hierarchical levels of fuzzy rules. FURL applies machine-learning techniques from theory revision such as credit assignment and repair to fuzzy rules. We also discuss the results of testing FURL on multiple benchmark datasets and finally discuss our results.

We introduce a general framework which we call prototype based reasoning. We explain its role as a technology for knowledge discovery We look at some types of prototype based reasoning systems. First we consider nearest neighbor based systems. We then look at fuzzy rule based models and view these as prototype based reasoning. This unified perspective allows us to extend the capabilities of fuzzy modeling technology in a number of directions.

The web-shopping activity is considered with the aim of improving the quality of the information available to consumers for making their purchasing decisions. A technique called product space map is proposed, which allows a qualitative description of the main features of the considered products. By applying this technique, the products in a considered product line (for example televisions) are first clustered into price categories such as low end, moderate and high end. For each category a set of linguistic summaries outlining the properties shared by the contained products is defined. An example of a linguistic summary is "Most TV's in the high price category provide extremely high resolution".

We focus on the problem of decision-making in the face of uncertainty. The issue of the representation of uncertain information is considered and a number of different frameworks are described: possibilistic, probabilistic, belief structures, and graded possibilistic. We suggest methodologies for decision-making in these different environments. The importance of decision attitude in the construction of decision functions is strongly emphasized. © 2002 Wiley Periodicals, Inc.

We discuss the fuzzy integral. The centrality of the ordering operation, based upon the arguments to be aggregated, is pointed out. We then extend the fuzzy integral aggregation operator by allowing the ordering operation to be based upon values other then those being aggregated. This leads to the induced fuzzy integral aggregation operator. We look at this new operator and study its properties. We show its relationship to a formulation called limited fuzzy integral aggregation. It is shown how this new induced fuzzy integral operator provides a natural framework for the implementation of nearest neighbor rules. Throughout this work, use is made only of the ordinal aspects of the information used. © 2002 Wiley Periodicals, Inc.

We focus on the problem of constructing decision functions to aid in the valuation of alteratives in uncertain decision making. We discuss different types of scales available for representing the payoffs associated with the alternatives. Here we consider the case in which our basic scale is an ordinal case. Here however we augment this ordinal scale by allowing an additional notion. We indicate one special element on the scale which we call the denoted element. We name such a scale a Denoted Ordinal Scale (DOS). We point out that a DOS allows a binary partitioning of the basic ordinal scale which can be associated with differing semantics and used for various purposes. Here we focus on a binary partitioning and use a semantics which provides a classification of payoffs as to whether they are acceptable or not. This allows us to have information such as A is preferred to B but both are acceptable. We show how a DOS with this semantics can be used to construct sophisticated decision functions.

We introduce the fuzzy measure and discuss its use as a unifying structure for modeling knowledge about an uncertain variable. We show that a large class of well-established types of uncertainty representations can be modeled within this framework. A view of the Dempster-Shafer (D-S) belief structure as an uncertainty representation corresponding to a set of possible fuzzy measures is discussed. A methodology for generating this set of fuzzy measures from a belief structure is described. A measure of entropy associated with a fuzzy measure is introduced and its manifestation for different fuzzy measures is described. The problem of uncertain decision making for the case in which the uncertainty represented by a fuzzy measure is considered. The Choquet integral is introduced as providing a generalization of the expected value to this environment.

The basic principle used in the construction of nearest-neighbor models is discussed. The induced ordered weighted averaging (IOWA) operators are shown to provide a useful formal structure for building nearest-neighbor models. A methodology for learning IOWA operator nearest-neighbor models is described. Various types of nearest-neighbor rules are investigated, including those based on a linguistic specification. The situation in which the value of interest lies in an ordinal set is also considered. It is shown that the weighted median provides a useful tool for constructing nearest-neighbor rules in this case.

Fuzzy cognitive maps (FCMs) allow experts to express their knowledge by drawing weighted causal digraphs. Experts can pool or fuse their knowledge by adding the underlying FCM causal matrices. This naturally extends the ordered-weighted-averaging technique to averaging dynamical systems and can create complex dynamical systems from several simpler ones. Edge quantization allows experts to state their real-valued knowledge in the simpler terms of causal increase (1), decrease (-1), or absence (0). We model the expert FCMs as a sequence of random fields to study the small-sample effects of quantizing both the causal edges and fuzzy-set concept nodes. The averaged quantized random matrices exhibit large-sample convergence to the population means of the unquantized matrices in accordance with the strong law of large numbers, but the small-sample averages can show substantial diversity of equilibrium attractors. We use statistical tests and a novel measure based on the fuzzy equality of limit-cycle histograms to show that this small-sample equilibrium diversity increases as the node multivalence or fuzzy-set quantization increases. More realistic probabilistic assumptions can more accurately model the equilibrium behavior of large-scale knowledge fusion.

Dempster–Shafer belief structures provide a useful framework for the representation of information about a variable whose value is uncertain. Important parameters in these structures are the weights associated with the focal elements. These weights, which can be viewed as probabilities, are required to be precisely known. Here we relax this requirement and we consider the situation in which our knowledge of the weights associated with the focal elements is that they lie in some known interval rather then being precisely specified. This relaxation will allow us to more realistically model situations in which the weights cannot be precisely obtained. At a formal level, this situation can be viewed as one in which we have some uncertainty as to what is the actual belief structure, this uncertainty being of the possibilistic type. We introduce the measures of plausibility and belief in this environment. We also look at the issue of combining belief structures for these interval type belief structures. © 2001 John Wiley & Sons, Inc.

In this paper, we focus on the web-shopping activity and on ways to improve the quality of the information available to consumers. We describe a tool, called a product space map, for the presentation of information about a product category which can help consumers in making purchasing decisions. Using this tool we first provide a clustering or segmentation of a product line, that is, 27-inch televisions, into price categories such as low end, moderate and high end. Once having this partitioning we then use the idea of linguistic summaries to describe the properties of each category with respect to relevant features. An example of such a summary is “Most TV's in the high price category provide extremely high resolution.” With the aid of such information it becomes much easier for consumers to understand the product line, see what they are getting for their money, and more easily and confidently locate products that are of particular value for the money. Considerable use is made of fuzzy set technology to provide the ability to describe the information in a way, using linguistic expressions, that is particularly consumer friendly. © 2001 John Wiley & Sons, Inc.

We consider the problem of information fusion, specifically the task of fusing information from two different categories, information which is directly about an object of interest (OBJOIN information) and information about related objects (peer information). We discuss the representation of these different types of information, the first in terms of a possibilistic distribution and the second in terms of a probability distribution. We introduce an approach to information fusion based upon the use of the fuzzy modeling technology. In this approach we represent the fusion function in terms of rules which indicate when to use the different types of information. Particularly notable here is the role of information quality as a guiding factor in the fusion process. © 2001 John Wiley & Sons, Inc.

We look at the issue of obtaining a variance like measure associated with probability distributions over ordinal sets. We call these dissonance measures. We specify some general properties desired in these dissonance measures. The centrality of the cumulative distribution function in formulating the concept of dissonance is pointed out. We introduce some specific examples of measures of dissonance.

We consider the problem of group decision making where the selection process is based upon a group preference function, obtained by an aggregation of the participating agents individual preference functions. We describe some methods for formulating the group preference from the individual preference functions. We note the possibility of the individual agents strategically manipulating the information they provide, so as to further their own goal of getting their most preferred alternative selected by the group. With this in mind, we suggest ways of modifying the formulation of the group decision functions to discourage strategic manipulation by the participating agents.

We introduce the power average to provide an aggregation operator which allows argument values to support each other in the aggregation process. The properties of this operator are described. We discuss the idea of a power median. We introduce some possible formulations for the support function used in the power average. We extend the supported aggregation facility of empowerment to a wider class of mean operators, such as the OWA (ordered weighted averaging) operator and the generalized mean operator.

We discuss the nature of possibilistic uncertainty. Two approaches to the determination of the equivalent value for a possibilistically uncertain variable are introduced. These approaches are then generalized to yield a parameterized class of functions for obtaining the equivalent value of a possibilistic variable. A novel aspect of this class is that it is parameterized by a weak ordering. A discussion to try to understand the semantics of this parameterizing ordering is included. A method is suggested for obtaining the weak ordering directly from a scalar value. We briefly look at the issue of variance in possibilistic uncertainty.

There is much imprecise information in financial environments; for example, in forecasting economic scenarios in order to assess the potential payoff of an investment or to optimize asset allocation. We provide a new methodology which may be used in situations in which the likelihood of a state of nature is given in the form of an interval probability. Our approach incorporates the behavioral disposition of the decision maker.

The distinction between possibilistic and veristic variables is pointed out, the former, such as a persons age, takes exactly one value, while the latter, such as a persons hobbies, takes any number of solutions. We introduce a number of different types of statements involving veristic variables and then show how these can be translated into expressions and constraints, within the framework of the theory of approximate reasoning. We discuss the methods for combining veristic information.

We describe the idea of a possibilistic variable and introduce the problem of obtaining an equivalent value for a possibilistic variable. This value is seen as being analogous to the expected value in probability theory. A number of properties required of the equivalent value are describe. Two approaches for obtaining the equivalent value for a possibilistic variable are introduced. These approaches are then synthesized to yield a generalized approach for obtaining a possibilistic equivalent value. A novel aspect of this approach is that it is parametrized by a weak ordering. We discuss the effect of the parametrizing weak ordering on the process used to calculate the possibilistic equivalent value.

We studied here on some simple examples the interaction between valuation family, parameters and ranking result. The ranking method studied is based upon the idea of associating with a fuzzy number a scalar value, its valuation, and using this valuation to compare and order fuzzy numbers. The valuation method considered was introduced initially by the Yager and Filev. This valuation consists in the integration over α-levels, of the average of each α-cut weighted by a weight distribution function. We finish by introducing a new weight distribution function.

It is often necessary to divide a certain amount of money between n participants, i.e., to assign, to each participant, a certain portionwi≥0 of the whole sum (so that w1+⋯+ wn=1). In some situations, from the fairness requirements, we can uniquely determine these "weights" wi. However, in some other situations, general considerations do not allow us to uniquely determine these weights, we only know the intervals[Formula: see text] of possible fair weights. We show that natural fairness requirements enable us to choose unique weights from these intervals; as a result, we present an algorithm for fair division under interval uncertainty.

Fuzzy measures provides a structure for modeling the knowledge available about variables whose values are unknown and uncertain. A large class of different types of uncertainty can be represented in this framework. In this work, we provide a measure of entropy that can be used to calculate the amount of uncertainty associated with a fuzzy measure.

We deal with the development of tools useful for the construction of multicriteria decision functions that allow for the modeling of the types of complexity that is the hallmark of human intelligence. We first discuss the fuzzy measure, describe its potential for characterizing relationships between multiple criteria, and introduce the class of ordered aggregation functions that can be based on a fuzzy measure. We then focus on the two fuzzy measures associated with the Dempster-Shafer belief structure, plausibility, and belief, and describe the types of ordered aggregation functions obtained using these measures. This leads us to introduce a new class of aggregation functions obtained by allowing a decision maker to provide his decision imperative in terms of components (concepts) that contribute to his overall satisfaction. Each component consists of a value, a subset of criteria and an agenda for combining the criteria in the component. Finally, it is shown how these components can be combined to allow for the representation of hierarchical decision functions.

Presents a scheme for object recognition by classificatory problem solving in the framework of fuzzy sets and possibility theory. The scheme has a particular focus on handling the imperfection problems that are common in application domains where the objects to be recognized (detected and identified) represent undesirable situations, referred to as crises. Crises develop over time, and observations typically increase in number and precision as the crisis develops. Early detection and precise recognition of crises is desired, since it increases the possibility of an effective treatment. The crisis recognition problem is central in several areas of decision support, such as medical diagnosis, financial decision making and early warning systems. The problem is characterized by vague knowledge and observations suffering from several kinds of imperfections, such as missing information, imprecision, uncertainty, unreliability of the source, and mutual (possibly conflicting or reinforcing) observations of the same phenomena. The problem of handling possibly imperfect observations from multiple sources includes the problems of information fusion and multiple-sensor data fusion. The different kinds of imperfection are handled in the framework of fuzzy sets and possibility theory.

We consider here the problem of decision making under uncertainty. We suggest an approach for the construction of decision functions which allow for the inclusion of probabilistic information as well as for the inclusion of information about the decision maker's attitude and preferences. Use is made of the fuzzy modeling technology to construct these functions from specifications provided by the decision maker.

We distinguish between variables having only one solution (possibilistic) and those allowing multiple solutions (veristic). A representation of information contained in statements involving veristic variables is presented. Using this representation we begin to develop a structure for the manipulation of knowledge involving variables that can have multiple solutions. The verity distribution is introduced to capture information about these variables. We show how to combine information involving veristic variables. We study qualified and quantified statements as well as the entailment and extension principles in this framework.

The basic process of Hierarchical Agglomerative (HAG) clustering is described as a merging of clusters based on their proximity. The importance of the selected cluster distance measure in the determination of resulting clusters is pointed out. We note a fundamental distinction between the nearest neighbor cluster distance measure, Min, and the furthest neighbor measure, Max. The first favors the merging of large clusters while the later favors the merging of smaller clusters. We introduce a number of families of intercluster distance measures each of which can be parameterized along a scale characterizing their preference for merging larger or smaller clusters. We then consider the use of this distinction between distance measures as a way of controlling the hierarchical clustering process. Combining this with the ability of fuzzy systems modeling to formalize linguistic specifications, we see the emergence of a tool to add human like intelligence to the clustering process.

We provide an overview of the theory of approximate reasoning and discuss the measurement of information in this reasoning system using specificity. It is then shown how to represent the binary propositional logic in the framework of approximate reasoning. Using the measure of specificity we show how to measure the information contained in the propositions of binary logic. Our measure essentially measures the proportion of possibilities eliminated by the proposition. Next we turn to the possibilistic logic and show how to represent this within the framework of approximate reasoning. We again, using specificity, provide for a measure of information of propositions in this logic. Finally we turn to the issue of quantified statements and show how to represent general quantified statements involving predicates within these two logics. ©1999 John Wiley & Sons, Inc.

Here the Dempster–Shafer belief structure is viewed as providing partial information about the underlying fuzzy measure associated with a uncertain variable. In this perspective there exists many possible fuzzy measures that can be associated with a Dempster–Shafer belief structure. Typically only two of these measures have been made explicit, those being the measure of belief and plausibility. Here we introduce a whole class of fuzzy measures that can be associated with a Dempster–Shafer belief structure. As an aid to choosing between these myriad of possibilities we discuss the entropy of a fuzzy measure. ©1999 John Wiley & Sons, Inc.

The importance as well as the difficulty of the problem of ranking fuzzy numbers is pointed out. Here we consider approaches to the ranking of fuzzy numbers based upon the idea of associating with a fuzzy number a scalar value, its valuation, and using this valuation to compare and order fuzzy numbers. Specifically we focus on expected value type valuations which are based upon the transformation of a fuzzy subset into an associated probability distribution. We develop a number of families of parameterized valuation functions. ©1999 John Wiley & Sons, Inc.

We consider the problem of decision making under uncertainty where the payoffs are measured on an ordinal scale. First the case in which no information is available with respect to the uncertainty is studied. Here decision functions are obtained which reflect the attitude of the decision maker. Next the case where a possibility distribution exists with respect to uncertain variable is considered. Finally we look at the case where both possibilistic and probabilistic uncertainty co-exist.

We discuss the idea of a fuzzy probability assessment, the association a collection of fuzzy probabilities with the outcomes of a random experiment. Fuzzy probability assessments often result from the linguistic specification of probabilities as provided by human experts. The question of consistency of the fuzzy probability assessment is considered. Finally, the problem of decision-making, selecting a best alternative action, in the face of a fuzzy probability assessment is investigated. Here we focus on the issue of obtaining the expected payoff of alternatives in the face of a fuzzy probability assessment. In the course of solving this problem we develop a representation of an effective probability distribution in the face of a fuzzy probability assessment.

The problem of dealing with incomplete information in object-oriented data models (OODMs) is addressed. A method to compute default values for unknown objects' attributes is proposed, based both on the association of typical values with the attributes in the intensional definition of a class and on the application of a prioritized aggregation operator to combine typical values appearing in an inheritance structure. This method can also be applied to refine vague attribute values expressed by means of fuzzy sets interpreted as possibility distributions. A new interpretation of partial inheritance in this context is proposed, introducing the concept of "partial overriding" of typical values.

We briefly describe the Ordered Weighted Averaging (OWA) operator and discuss a methodology for learning the associated weighting vector from observational data. We then introduce a more general type of OWA operator called the Induced Ordered Weighted Averaging (IOWA) Operator. These operators take as their argument pairs, called OWA pairs, in which one component is used to induce an ordering over the second components which are then aggregated. A number of different aggregation situations have been shown to be representable in this framework. We then show how this tool can be used to represent different types of aggregation models.

The problem of fusing opinions provided by multiple independent experts is pervasive to many decision making situations. This problem is of particular concern in financial engineering where one is often faced with drawing upon multiple experts to forecast market related events. The authors describe a new approach to information fusion based upon an extension of the ordered weighted averaging (OWA) operators. Central to this approach is a weighting associated with each expert dependent upon their recent performance.

In this work, we focus on the OWA operator. We view this operator as the inner product of two vectors: a weighting vector containing the weights associated with the aggregation, and a second vector containing the arguments to be aggregated. We note the centrality to this operator of the process used to index the elements in the argument vector, and suggest some new mechanisms for ordering the arguments. In the second part, we study the situation in which the weighting vector is context dependent. Finally, we consider the case in which the arguments to be aggregated are drawn from an ordinal scale; the solution to this problem leads us to consider random weighting vectors. © 1998 John Wiley & Sons, Inc.

Textbooks often claim that quantum mechanics explained the periodic system: namely, the actual configuration of electronic orbits that is responsible for the element's chemical properties can be described as the one that minimizes the total energy, and the energy of each configuration can be computed by using quantum mechanics. However, a careful analysis of this explanation reveals that, in addition to the basic equations of quantum mechanics, we need some heuristic rules that do not directly follow from quantum physics. One reason why additional heuristics are necessary is that the corresponding numerical equations are extremely difficult to solve, and as we move to atoms with larger and larger atomic numbers Z, they become even more difficult. Moreover, as Z grows, we must take relativistic effects into consideration, and this means going from partial differential equations to even more mathematically difficult operator equations. In this paper, we show that if instead of the (often impossible) numerical optimization, we consider the (available) ordinal information, we can then explain the observed periodic system.

We investigate the problem of fusing collections of documents provided by multiple information retrieval systems. A parameterized approach is suggested in which a parameter determines how the documents in the individual collections are interleaved to form a fused list of documents. We then suggest a mechanism for learning this parameter.

Ordered weighted averaging (OWA) operators are described and are shown to provide a general class of parameterized aggregation operators that include the max, min, and average. We suggest an approach to the inclusion of importances in the OWA operator aggregation technique. The methodology suggested involves a transformation of the scores to be aggregated by their respective importances. We show how we can construct transformation functions with the aid of the fuzzy modeling technology.

A review of the basic ideas of fuzzy systems modeling is provided. We introduce a hierarchical-type fuzzy systems model called a hierarchical prioritized structure (HPS) and review its structure, operation, and the interlevel aggregation algorithm. We then turn to the issue of constructing the HPS. Consideration is first given to the case in which rules are provided by an expert. Detailed consideration is given to the problem of completing incomplete priorities by use of the principle of maximal buoyancy. A mathematical programming method is introduced for the implementation of this approach. The issue of tuning hierarchical models is addressed. We next introduce a dynamic approach to the formulation of an HPS directly from data that enables us to continually update our model as more observations become available. This approach allows a system's builder to start with a default model and include exceptions.

We introduce the concept of upward reinforcement in aggregation as one in which a collection of high scores can reinforce or corroborate each other to give an even higher score than any of the individual arguments. The concept of downward reinforcement is also introduced as one in which low scores reinforce each other. Our concern is with full reinforcement aggregation operators, those exhibiting both upward and downward reinforcement. It is shown that the t-norm and t-conorm operators are not full reinforcement operators. A class of operators called fixed identity MICA operators are shown to exhibit the property of full reinforcement. We present some families of these operators. We use the fuzzy system modeling technique to provide further examples of these operators.

Genetic Algorithms (GA) are a useful and convenient tool to find the solution in combinatorial optimal problems, and widely used in the various engineering fields. Here we apply GA to identify both of the composite operations and fuzzy relations under that operation at the same time from the given input-output system data. There exist many composite operations and associated fuzzy relations, which satisfy the same input-output system data. Then, it is supposed that many composite operations and fuzzy relations, which satisfy the original data, are generated when we apply GA to this problems. Tne authors propose a method to identify the fuzzy system from these composite operations and fuzzy relations, generated by GA, by an unweighted pair-group method using arithmetic average (UPGMA) which was developed to make a taxonomic tree of the expression in molecular biology.

The increasing use of computers for transactions and communication have created mountains of data that contain potentially valuable knowledge. To search for this knowledge we have to develop a new generation of tools, which have the ability of flexible querying and intelligent searching. In this paper we will introduce an extension of a fuzzy query language called Summary SQL which can be used for knowledge discovery and data mining. We show how it can be used to search for fuzzy rules.

We consider the problem of fusion of multiple information sources, particularly in environments when the sensor observations are imprecise. The concept of a combinability relationship is introduced to enable the inclusion in the fusion process of information about the appropriateness of fusing different elements from the observation space. This idea allows for the use of an expert knowledge base, containing information about the domain of the particular problem, in the fusion process and leads to a more intelligent aggregation. We show that if we use a combinability relationship that only allows fusion of identical elements then the only idempotent fusion of any collection of fuzzy observations is their intersection. Using the idea of the fuzzy measure we considered situations in which we allow a partial collection of the observations determine the fused value. © 1997 John Wiley & Sons, Inc.

The concept of typicality is studied and the relationship between a typical value and the mode is explored. It is suggested that a typical value should be a fuzzy subset. A measure of typicality associated with any fuzzy subset is introduced. We provide an algorithm for finding the best typical value which is of the form of an interval. © 1997 John Wiley & Sons, Inc.

We describe a new paradigm for the inclusion of advertisements on the WWW. This paradigm takes advantage of the internet's great ability for instantaneous online processing of information in real time. A methodology is described for the use of intelligent agents to help in the determination of the appropriateness of displaying a given advertisement to a visitor to a site using very specific information about potential customers. Use is made of fuzzy systems modeling for the construction of these agents. © 1997 John Wiley & Sons, Inc.

An exhaustive study of uninorm operators is established. These operators are generalizations of t-norms and t-conorms allowing the neutral element lying anywhere in the unit interval. It is shown that uninorms can be built up from t-norms and t-conorms by a construction similar to ordinal sums. De Morgan classes of uninorms are also described. Representability of uninorms is characterized and a general representation theorem is proved. Finally, pseudo-continuous uninorms are defined and completely classified.

We discuss the applicability of fuzzy logic multi criteria ranking techniques and genetic algorithms in solving problems concerning financial risk management. Fuzzy logic techniques are useful in soliciting information on user perceptions of risk factors. However, since people are notoriously inaccurate and unreliable in reporting their preferences, we also employ a genetic algorithm to help validate user supplied data. The genetic algorithm helps clarify how and when user preferences effect the perceived desirability of a particular outcome. The genetic algorithm also helps tune the parameters of fuzzy multiple criteria decision models.

The problem of selecting an investment option in the face of uncertainty with respect to the payoff is considered. Methods for the representation of uncertainty based on the theory of fuzzy sets and the Dempster-Shafer belief structure are described. Approaches for comparing alternatives under various kinds of uncertainty are discussed.

We consider multicriteria aggregation problems where, rather than requiring all the criteria be satisfied, we need only satisfy some portion of the criteria. The proportion of the critera required is specified in terms of a linguistic quantifier such as most. We use a fuzzy set representation of these linguistic quantifiers to obtain decision functions in the form of OWA aggregations. A methodology is suggested for including importances associated with the individual criteria. A procedure for determining the measure of “orness” directly from the quantifier is suggested. We introduce an extension of the OWA operators which involves the use of triangular norms. © 1996 John Wiley & Sons, Inc.

In this work we consider a fuzzy set based approach to the issue of discovery in databases (database mining). The concept of linguistic summaries is described and shown to be a user friendly way to present information contained in a database. We discuss methods for measuring the amount of information provided by a linguistic summary. The issue of conjecturing, how to decide on which summaries may be informative, is discussed. We suggest two approaches to help us focus on relevant summaries. The first method, called the template method, makes use of linguistic concepts related to the domain of the attributes involved in the summaries. The second approach uses the mountain clustering method to help focus our summaries. © 1996 John Wiley & Sons, Inc.

This paper examines the problem of selecting an alternative in situations in which there exists uncertainty in our knowledge of the state of the world. We show how the ordered weighted averaging aggregation operators provide a unifying approach to selecting alternatives under uncertainty. In particular, we see how these operators provide a type of probability associated with our degree of optimism. We also show how the Dempster-Shafer belief structure provides a general framework for representing the information a decision maker has regarding relevant events. We then propose a methodology for decision making under uncertainty, integrating the ordered weighted averaging aggregation operators and the Dempster-Shafer belief structure. The proposed methodology is applied to a real world case involving risk management at one of the nation’s largest banks.

The basic paradigm for decision making under uncertainty is introduced. A methodology is suggested for the calculation of the variance associated with each of the alternatives in the case when the uncertainty is not necessarily of a probabilistic nature.

We introduce an automated procedure for extracting information from knowledge bases that contain fuzzy production rules. The knowledge bases considered here are modeled using the high-level fuzzy Petri nets proposed by the authors in the past. Extensions to the high-level fuzzy Petri net model are given to include the representation of partial sources of information. The case of rules with more than one variable in the consequent is also discussed. A reasoning algorithm based on the high-level fuzzy Petri net model is presented. The algorithm consists of the extraction of a subnet and an evaluation process. In the evaluation process, several fuzzy inference methods can be applied. The proposed algorithm is similar to another procedure suggested by Yager (1983), with advantages concerning the knowledge-base searching when gathering the relevant information to answer a particular kind of query.

The focus of this work is to provide a procedure for aggregating prioritized belief structures. Motivated by the ideas of nonmonotonic logics an alternative to the normalization step used in Dempster's rule when faced with conflicting belief structures is suggested. We show how this procedure allows us to make inferences in inheritance networks where the knowledge is in the form of a belief structure.

We introduce and define the concept of mean aggregation of a collection of n numbers. We point out that the lack of associativity of this operation compounds the problem of the extending mean of n numbers to n+1 numbers. The closely related concepts of self identity and the centering property are introduced as one imperative for extending mean aggregation operators. The problem of weighted mean aggregation is studied. A new concept of prioritized mean aggregation is then introduced. We next show that the technique of selecting an element based upon the performance of a random experiment can be considered as a mean aggregation operation.

Multiple decision problems appear in many applications of financial decision making. We suggest the use of quantifier guided aggregations to model these problems. Use is made of the fuzzy set concept of a linguistic quantifier and ordered weighted averaging aggregation technique.

The focus of this article is on the issue of information aggregation. We introduce two new aggregation operators, called MOM and MAM operators, which are, respectively, generalized and and or operators. We describe their relationship to the multivalued logic triangular norm operators and show how they generalize these operators by weakening the associativity property. We provide a duality theorem between these new operators. We present some special classes of these operators. We extend these operators to allow for weighted aggregations, which enable us to include importances. We introduce some families of these weighted MOM and MAM operators. We show how the typical neural aggregation is a special class of these weighted MOM and MAM operators. This generalization allows us to consider neural network and fuzzy logic methods in the same framework. © 1995 John Wiley & Sons, Inc.

We consider a general characterization of a measure of ambiguity suggested by Fishburn and investigate the appropriateness of this characterization. It is shown that a number of concepts related to uncertainty and taken from fuzzy logic-measures of fuzziness, measures of specificity, and measures of possibility/certainty interval-satisfy this characterization. In addition, a measure from classical logic indicating the certainty of our knowledge of the truth or falsity of a proposition in the face of a collection of premises also satisfies the characterization of Fishburn. © 1995 John Wiley & Sons, Inc.

The problem of decision making under doubt is described. the concept of immediate probabilities is introduced. It is seen as a modification of typical probabilistic knowledge with information about the payoffs, mediated through dispositional information (optimism/pessimism), resulting in a modified formulation of an agents perception of the probabilities in effect in the current decision. We use the Dempster rule of combination to help obtain an expression for these probabilities. We show how immediate probabilities allows us to explain the Allais paradox. A number of properties of these probabilities are described. the strategic use of these probabilities are explored as a means for effecting other people's decisions. © 1995 John Wiley & Sons, Inc.

Abstract: We discuss the MICA operators and show they provide a very general framework for modeling aggregation operators. We present a generic class of MICA operators that can be adapted to various type of situations through the tuning of parameters. These parameters can be set by a human expert or automatically learned from data through a genetic algorithm.

Discusses some basic ideas from the Dempster-Shafer theory of evidence. The authors describe the concept of fuzzy systems modeling used in fuzzy logic control. The authors use the Dempster-Shafer framework to provide a machinery for including randomness in the fuzzy systems modeling process. The authors show how to represent additive noise in this combined framework.>

We discuss the idea of a linguistic quantifier and fuzzy set representations of these objects. We describe two formalisms for evaluating the truth of linguistically quantified propositions such as Most winter days are cold. the first approach is based upon a probabilistic interpretation and the second is based upon a logical interpretation, and uses a generalization of the “and” and “or” operations via OWA operators. We suggest an application of these quantified statements for the representation of the quotient operator in fuzzy relational data bases. © 1994 John Wiley & Sons, Inc.

This note investigates how various ideas of "expectedness" can be captured in the framework of possibility theory. Particularly, we are interested in trying to introduce estimates of the kind of lack of surprise expressed by people when saying "I would not be surprised that…" before an event takes place, or by saying "I knew it" after its realization. In possibility theory, a possibility distribution is supposed to model the relative levels of possibility of mutually exclusive alternatives in a set, or equivalently, the alternatives are assumed to be rank-ordered according to their level of possibility to take place. Four basic set-functions associated with a possibility distribution, including standard possibility and necessity measures, are discussed from the point of view of what they estimate when applied to potential events. Extensions of these estimates based on the notions of Q-projection or OWA operators are proposed when only significant parts of the possibility distribution are retained in the evaluation. The case of partially-known possibility distributions is also considered. Some potential applications are outlined.

We discuss the issue of weighted Min and Max aggregation. We provide for a formalization of the process of importance weighted transformation. The transformation of zero importance items to the identity element of the aggregation is emphasized. Median type aggregation is discussed and we describe its properties. We suggest a new technique for weighted median aggregation. We show its usefulness as an alternative the weighted average when we are using an ordinal scale.

We develop a simple and effective approach for approximate estimation of the cluster centers on the basis of the concept of a mountain function. We call the procedure the mountain method. It can be useful for obtaining the initial values of the clusters that are required by more complex cluster algorithms. It also can be used as a stand alone simple approximate clustering technique. The method is based upon a griding on the space, the construction of a mountain function from the data and then a destruction of the mountains to obtain the cluster centers.>

We suggest a new approach to the construction of fuzzy logic controllers based upon the selection of systems parameters. We first show that the standard fuzzy logic controllers have four basic operations which determine the nature of their functioning, the aggregation process used to combine individual antecedent firing levels to give a rule firing level, the determination of rule output based on antecedent firing level, the aggregation of individual rule outputs to find combined output, and the defuzzification process. The next show how we can parameterize these operations using S-OWA operators. These parameterized models give us a new class of flexible structured fuzzy logic controllers (FS-FLC). We look at the structure and performance of these controllers. We then suggest that one can improve the structure of fuzzy logic controllers by learning the values of the parameters introduced.>

We discuss the participatory learning model originally introduced by Yager [IEEE Trans. Syst. Man Cybern. SMC-20, 1229–1234 (1990)]. We analyze the learning mechanism as a stable control strategy. We show how the learning mechanism used in participatory learning can be expressed in the form of a fuzzy rule base. We use this rule base formulation to provide new learning rules. We modify the Widrow-Hoff rule to include a participatory learning mechanism. © 1993 John Wiley & Sons, Inc.

We discuss the structure of nonmonotonic reasoning. We suggest that an important aspect of nonmonotonic reasoning systems is the ordering relationship (preference structure) on the propositions in the knowledge. We suggest that this ordering relationship must be a weak order, a complete and transitive relationship. We provide a comprehensive discussion of some central issues in preference theory. In most real applications of nonmonotonic knowledge bases the ordering relationship available is often not complete. We suggest a mechanism for completing these relationships. The basic imperative used in providing this completion is to do it in such a manner as to add as few unjustified inferences as necessary. To formally accomplish this task we introduce a measure of buoyancy associated with a weak order and suggest that the preferred completion is the one with the maximal buoyancy.