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Liu's inference is a process of deriving consequences from uncertain knowledge or evidence via the tool of conditional uncertainty. Using membership functions, this paper derives some expressions of Liu's inference rule for uncertain systems. This paper also discusses Liu's inference rule with multiple antecedents and with multiple if-then rules. [ABSTRACT FROM AUTHOR]

In this paper, attribute reduction in variable precision rough set model is discussed. Several kinds of reducts preserving some of lower approximations, upper approximations, boundary regions and the unpredictable region are discussed. Relations among those kinds of reducts are investigated. As a basis for reduct computation, Boolean function representations of the preservation of lower approximations, upper approximations, boundary regions and the unpredictable region are discussed. Throughout this paper, the great difference between the analysis using variable precision rough sets and the classical rough set analysis is emphasized. [ABSTRACT FROM AUTHOR]

The subject of this paper is a generalized approach to relative scalar cardinalities of fuzzy sets. In the main part of the paper, we discuss basic properties of triangular norm-based relative cardinality such as valuation property, the cartesian product rule and complementary rule. Examples of cardinalities satisfying those properties are also given. [ABSTRACT FROM AUTHOR]

This paper considers a renewal process in which the interarrival times and rewards are characterized as fuzzy variables. A fuzzy elementary renewal theorem shows that the expected number of renewals per unit time is just the expected reciprocal of the interarrival time. Furthermore, the expected reward per unit time is provided by a fuzzy renewal reward theorem. Finally, a numerical example is presented for illustrating the theorems introduced in the paper. [ABSTRACT FROM AUTHOR]

Under an interpretation of the principles of non-contradiction and excluded-middle based on the concept of self-contradiction, this paper mainly deals with the principles' verification in the case of the unit interval of the real line. Such verification is done in the three following cases: (1) The unit interval is totally ordered by the restriction to it of the usual order of the real line, (2) the unit interval is partially ordered by the sharpened order, and (3) the unit interval is under a new particular preorder. The first case is immediately extended to characterize the case of fuzzy sets. [ABSTRACT FROM AUTHOR]

One of the most important issues in the field of fuzzy preference modelling is the construction of a fuzzy strict preference relation and a fuzzy indifference relation from a fuzzy weak preference relation. Here, we focus on a particular class of fuzzy weak preference relations, the so-called fuzzy orderings. The definition of a fuzzy ordering involves a fuzzy equivalence relation and, in this paper, the latter will be considered as the corresponding fuzzy indifference relation. We search for fuzzy strict preference relations compatible with a given fuzzy ordering and its fuzzy indifference relation. In many situations, depending on the t-norm and t-conorm used, this quest results in a unique fuzzy strict preference relation. Our aim is to characterize these fuzzy strict preference relations and to study their transitivity. [ABSTRACT FROM AUTHOR]

Many papers have addressed the task of proposing a set of convenient axioms that a good rule interestingness measure should fulfil. We provide a new study of the principles proposed until now by means of the logic model proposed by Hájek et al.14 In this model association rules can be viewed as general relations of two itemsets quantified by means of a convenient quantifier.28 Moreover, we propose and justify the addition of two new principles to the three proposed by Piatetsky-Shapiro.27 We also use the logic approach for studying the relation between the different classes of quantifiers and these axioms. We define new classes of quantifiers according to the notions of strong and very strong rules, and we present a quantifier based on the certainty factor measure,317 studying its most salient features. [ABSTRACT FROM AUTHOR]

Intuitionistic fuzzy set, was introduced by Atanassov, has been applied to many different fields, such as logic programming, pattern recognition, and decision making, etc. However, so far there has been few investigation on how to transform attribute tested values of alternative into a intuitionistic fuzzy number, and then complete decision making by intuitionistic fuzzy information. In this paper, the original attribute value (objective information) are characterized by crisp number which are given by decision maker. We define the concepts of supporting, opposing and neutral set of alternative respectively, develop an approach for transform attribute values into intuitionistic fuzzy number, and determine the order of alternatives based on the score and the degree of accuracy of the intuitionistic fuzzy number. Finally, a practical example is provided to illustrate the developed method. [ABSTRACT FROM AUTHOR]

There are well established methods for fuzzy clustering especially for the cases where the feature values are numerical of ratio or interval scale. Not so well established are methods to be applied when the feature values are ordinal or nominal. In that case there is no one best method it seems. This paper discusses a method where unknown numeric variables are assigned to the ordinal values. Part of minimizing an objective function for the clustering is to find numeric values for these variables. Thus real numbers of interval scale and even ratio scale for that matter are assigned to the original ordinal values. The method uses the same objective function as used in fuzzy c-means clustering but both the membership function and the ordinal to real mapping are determined by gradient descent. Since the ordinal to real mapping is not known it cannot be verified for its legitimacy. However the ordinal to real mapping that is found is best in terms of the clustering produced. Simulations show the method to be quite effective. [ABSTRACT FROM AUTHOR]

Media theory is a new branch of discrete applied mathematics originally developed in mid-nineties to deal with stochastic evolution of preference relations in political science and mathematical psychology. The theory focuses on a particular semigroup of ‘messages’ acting as transformations of a set of ‘states’, called a ‘medium’, whose axioms are both strong and natural. The term ‘medium’ stems from a particular application in which the transformations formalize the effects, on an individual, of ‘tokens’ of information delivered by the environment—that is, the ‘medium’. However, many different types of examples can be found, ranging from learning spaces to hypercube computers, suggesting that this concept is ubiquitous. The paper presents very basic concepts and results of media theory and is aimed at a wide body of researchers in discrete applied mathematics. [ABSTRACT FROM AUTHOR]

This paper is to compute a Nash equilibrium in a fuzzy environment, which is represented by a fuzzy approximate Nash equilibrium in a space of discrete mixed strategies. For discrete mixed strategies, the relationship between the discrete degree and the approximate degree is discussed. Based on the fuzzy regret degree, a genetic algorithm for computing a fuzzy Nash equilibrium is given. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the concept of interval-valued nested set on the universal set X, propose two representation theorems and equivalent classification theorem of interval-valued fuzzy set. These works can be used in setting up the basic theory of interval-valued fuzzy set. [ABSTRACT FROM AUTHOR]

Inference problems are one of the main research topics in the artificial intellect field. So far there have been various inference systems, some of them have been applied in fuzzy control according to their feature. In 1989, the concept of truth-valued flow inference was introduced by Wang1, and its mathematical theory of truth-valued flow inference was set up by Wang2 in 1995. In this paper, aimed at the real meaning of the truth-valued in the truth-valued flow inference, we introduce the concepts of the inner product truth-valued and inner product truth-valued flow inference, and analyze some inference algorithms of fuzzy control at present in detail. We reveal that the fuzzy inference algorithms in fuzzy control at present are all regarded as inner product truth-valued flow inference algorithm. Finally, inner product truth-valued flow inference approach is generalized in the multiple inputs and single output. [ABSTRACT FROM AUTHOR]

This paper explores Heilpern's notions of fuzzy mapping and the fixed point theorem for fuzzy mappings. The fixed point theorem for fuzzy mappings as introduced by Heilpern has been generalized and some characterizations are done in this context. [ABSTRACT FROM AUTHOR]

In this paper, by using Sengupta and Pal's method of comparison of interval numbers and a new set of arithmetic operations for interval numbers, we propose a theory for the study of arithmetic operations on interval numbers. [ABSTRACT FROM AUTHOR]

Tandem queueing models play an important role in many real world systems such as computer systems, production lines, and service systems. This paper proposes a procedure to construct the membership functions of the performance measures in tandem queueing systems, in that the arrival rate and service rates are fuzzy numbers. The basic idea is to transform a fuzzy tandem queue to a family of crisp tandem queues by applying the α -cut approach. Then on the basis of α -cut representation and the extension principle, a pair of mathematical programs is formulated to describe this family of crisp tandem queues, via which the membership functions of the performance measures are derived. Two numerical examples are solved successfully to demonstrate the validity of the proposed approach. Since the performance measures are expressed by membership functions rather than by crisp values, the fuzziness of input information is completely conserved. Thus the proposed approach for fuzzy systems can represent the system more accurately, and more information is provided for designing queueing systems. The successful extension of tandem queues to fuzzy environments permits tandem queueing models to have wider applications. [ABSTRACT FROM AUTHOR]

Traditionally the beta distribution has been used due to its ease for computing the mean of tasks, times, cash-flows, etc. because, in this case, the expected value is a weighted average of optimistic, most likely and pessimistic values. One of the critiques of PERT has been the difficulty of introducing the level of confidence that the expert has in his own estimate of the modal value. The Belief in Fuzzy Probability Estimations of Time (BIFPET) model uses human judgement instead of stochastic assumptions to determine the duration of a project. Thus, each supervisor responsible of an activity specifies the three values with their respective probabilities. The manager accepts these probabilities or may extend them within a certain range, according to his belief in these likelihoods or the degree of control exercised by himself in achieving the lower bound of the expected value. However, this approach can be implemented in the PERT methodology. Thus, in this paper, we introduce the confidence that the manager has in the most likely value supplied by the supervisor in order to determine the specific beta distribution to model the variable, showing the advantages of this intuitive procedure which moreover can be easily implemented. [ABSTRACT FROM AUTHOR]

Interval-valued fuzzy sets constitute an extension of fuzzy sets which give an interval approximating the "real" (but unknown) membership degree. Interval-valued fuzzy sets are equivalent to intuitionistic fuzzy sets in the sense of Atanassov which give both a membership degree and a non-membership degree, whose sum must be smaller than or equal to 1. Both are equivalent to L-fuzzy sets w.r.t. a special lattice L*. In fuzzy set theory, an important class of triangular norms is the class of those that satisfy the residuation principle. In a previous paper5 we gave a construction for t-norms on L* satisfying the residuation principle which are not t-representable. In this paper we investigate the Smets-Magrez axioms and some other properties for the residual implicator generated by such t-norms. [ABSTRACT FROM AUTHOR]

A new hierarchical uncertainty model for combining different evidence about a system of statistically independent random variable is studied in the paper. It is assumed that the first-order level of the model is represented by sets of lower and upper previsions (expectations) of random variables and the second-order level is represented by sets of lower and upper probabilities which can be viewed as confidence weights for interval-valued expectations of the first-order level. The model is rather general and allows us to compute probability bounds and "average" bounds for previsions of a function of random variables. A numerical example illustrates this model. [ABSTRACT FROM AUTHOR]

A fuzzy variable is a function from a possibility space to the set of real numbers, while a bifuzzy variable is a function from a possibility space to the set of fuzzy variables. In this paper, a concept of chance distribution is originally presented for bifuzzy variable, and the linearity of expected value operator of bifuzzy variable is proved. Furthermore, bifuzzy simulations are designed and illustrated by some numerical experiments. [ABSTRACT FROM AUTHOR]

Intuitionistic fuzzy sets are a generalization of ordinary fuzzy sets which are characterized by a membership function and a non-membership function. In this paper we consider the problem of ranking a set of intuitionistic fuzzy numbers. We adopt a statistical viewpoint and interpret each intuitionistic fuzzy number as an ensemble of ordinary fuzzy numbers. This enables us to define a fuzzy rank and a characteristic vagueness factor for each intuitionistic fuzzy number. We show the reasonablesness of the results obtained by examining several test cases. [ABSTRACT FROM AUTHOR]

Rough set theory is a relatively new mathematical tool for use in computer applications in circumstances which are characterized by vagueness and uncertainty. In this paper, we introduce the concepts of information entropy, rough entropy and knowledge granulation in rough set theory, and establish the relationships among those concepts. These results will be very helpful for understanding the essence of concept approximation and establishing granular computing in rough set theory. [ABSTRACT FROM AUTHOR]

Standard methods for solving influence diagrams consist in stepwise elimination of variables, and along with elimination of a variable a set of new potentials over new domains is calculated. It is well known that these methods tend to produce unnecessarily large domains resulting in excessive consumption of time and memory. The lazy evaluation method represents only a partial solution to the problem. In this paper we extend any potential with two graphs over its domain representing the dependencies of variables. When a node A is eliminated, all necessary structural information for establishing the minimal sets of domains for potentials is contained in these graphs. We push lazy evaluation a step further to avoid performing unnecessary multiplications and subsequent division with equivalent potentials. [ABSTRACT FROM AUTHOR]

In this paper we study conditional independence structures arising from conditional probabilities and lower conditional probabilities. Such models are based on notions of stochastic independence apt to manage also those situations where zero evaluations on possible events are present: this is particularly crucial for lower probability. The "graphoid" properties of such models are investigated, and the representation problem of conditional independence structures is dealt with by generalizing the well-known classic separation criteria for undirected and directed acyclic graphs. Our graphical models describe the independence statements and the possible logical dependencies among the random variables. [ABSTRACT FROM AUTHOR]

In this paper, two new formulas of fuzzy entropy induced by distances between two intuitionistic fuzzy sets are given. These entropy measures can be computed easily and give reliable results. Some examples are illustrated for the comparison with Burillo and Bustince (1996) and Szmidt and Kacprzyk (2001). [ABSTRACT FROM AUTHOR]

In belief functions, there is a total measure of uncertainty that quantify the lack of knowledge and verifies a set of important properties. It is based on two measures: maximum of entropy and non-specificity. In this paper, we prove that the maximum of entropy verifies the same set of properties in a more general theory as credal sets and we present an algorithm that finds the probability distribution of maximum entropy for another interesting type of credal sets as probability intervals. [ABSTRACT FROM AUTHOR]

This section is maintained by Vladik Kreinovich. Please send your abstracts (or copies of papers that you want to see reviewed here) to vladik@utep.edu, or by regular mail to: V. Kreinovich, Department of Computer Science, University of Texas, El Paso, TX 79968, USA. [ABSTRACT FROM AUTHOR]