Toward a general theory of fuzzy variables

A general framework for a theory is presented that encompasses both statistical uncertainty, which falls within the province of probability theory, and nonstatistical uncertainty, which relates to the concept of a fuzzy set and possibility theory [L. A. Zadeh, J. Fuzzy Sets1 (1978), 3–28]. The concept of a fuzzy integral is used to define the expected value of a random variable. Properties of the fuzzy expectation are stated and a mean-value theorem for the fuzzy integral is proved. Comparisons between the fuzzy and the Lebesgue integral are presented. After a new concept of dependence is formulated, various convergence concepts are defined and their relationships are studied by using a Chebyshev-like inequality for the fuzzy integral. The possibility of using this theory in Bayesian estimation with fuzzy prior information is explored.     

Possibility and cost in decision analysis

The thesis of this paper is that a practically relevant decision theory must be based on the concept of possibility. As the concept is interpreted here it covers all the obstacles the decision-maker is facing. In many situations the contemplation of possibility is quite as relevant as the usual concentration on utility and probability. There is a traditional economic concept which conforms to an emphasis on obstacles or possibility in decision-making: opportunity cost. The opportunity cost of a decision is the value of the highest valued possible decision which is inhibited as a result of the decision actually taken. However as opportunity cost is usually interpreted decisions are dichotomized as either possible or impossible to perform. It is argued in the paper that this dichotomization is not very realistic. In reality there must be allowed for a continuum of states between the plain impossibility and the complete possibility. This gradual view of possibility could be dealt with if the fuzzy set theory is used. In the paper it is shown how some relevant concepts regarding possibility could be based on fuzziness and how these concepts could be used to analyze practical situations. A method to deal simultaneously with possibility and probability is stated.     

Solving a multiobjective possibilistic problem through compromise programming

Real decision problems usually consider several objectives that have parameters which are often given by the decision maker in an imprecise way. It is possible to handle these kinds of problems through multiple criteria models in terms of possibility theory.Here we propose a method for solving these kinds of models through a fuzzy compromise programming approach.To formulate a fuzzy compromise programming problem from a possibilistic multiobjective linear programming problem the fuzzy ideal solution concept is introduced. This concept is based on soft preference and indifference relationships and on canonical representation of fuzzy numbers by means of their α-cuts. The accuracy between the ideal solution and the objective values is evaluated handling the fuzzy parameters through their expected intervals and a definition of discrepancy between intervals is introduced in our analysis.     

A survey on fuzzy relational equations, part I: classification and solvability

Fuzzy relational equations play an important role in fuzzy set theory and fuzzy logic systems, from both of the theoretical and practical viewpoints. The notion of fuzzy relational equations is associated with the concept of “composition of binary relations.” In this survey paper, fuzzy relational equations are studied in a general lattice-theoretic framework and classified into two basic categories according to the duality between the involved composite operations. Necessary and sufficient conditions for the solvability of fuzzy relational equations are discussed and solution sets are characterized by means of a root or crown system under some specific assumptions.     

Solving the multiobjective possibilistic linear programming problem

In conventional multiobjective decision making problems, the estimation of the parameters of the model is often a problematic task. Normally they are either given by the decision maker (DM), who has imprecise information and/or expresses his considerations subjectively, or by statistical inference from past data and their stability is doubtful. Therefore, it is reasonable to construct a model reflecting imprecise data or ambiguity in terms of fuzzy sets for which a lot of fuzzy approaches to multiobjective programming have been developed. In this paper we propose a method to solve a multiobjective linear programming problem involving fuzzy parameters (FP-MOLP), whose possibility distributions are given by fuzzy numbers, estimated from the information provided by the DM. As the parameters, intervening in the model, are fuzzy the solutions will be also fuzzy. We propose a new Pareto Optimal Solution concept for fuzzy multiobjective programming problems. It is based on the extension principle and the joint possibility distribution of the fuzzy parameters of the problem. The method relies on α-cuts of the fuzzy solution to generate its possibility distributions. These ideas are illustrated with a numerical example.     

Fuzzy sets as a basis for a theory of possibility

The theory of possibility described in this paper is related to the theory of fuzzy sets by defining the concept of a possibility distribution as a fuzzy restriction which acts as an elastic constraint on the values that may be assigned to a variable. More specifically, if F is a fuzzy subset of a universe of discourse U={u} which is characterized by its membership function μ_F, then a proposition of the form “X is F,” where X is a variable taking values in U, induces a possibility distribution ∏X which equates the possibility of X taking the value u to μF(u)—the compatibility of u with F. In this way, X becomes a fuzzy variable which is associated with the possibility distribution ∏_x in much the same way as a random variable is associated with a probability distribution. In general, a variable may be associated both with a possibility distribution and a probability distribution, with the weak connection between the two expressed as the possibility/probability consistency principle.A thesis advanced in this paper is that the imprecision that is intrinsic in natural languages is, in the main, possibilistic rather than probabilistic in nature. Thus, by employing the concept of a possibility distribution, a proposition, p, in a natural language may be translated into a procedure which computes the probability distribution of a set of attributes which are implied by p. Several types of conditional translation rules are discussed and, in particular, a translation rule for propositions of the form “X is F is α-possible,” where α is a number in the interval [0, 1], is formulated and illustrated by examples.     

On conditional possibility distributions

In a recent paper in Fuzzy Sets and Systems, L.A. Zadeh has defined the concept of a conditional possibility distribution. In the present paper, we show that, in order to be consistent with the notion of noninteraction of fuzzy variables, the expression for conditional possibility distribution must be normalized. A comparison of the properties of conditional possibility and probability distributions is made, and an application to the optimization of a possibilistic finite-state system is outlined.     

Certain types of irregular <Emphasis Type="Italic">m</Emphasis>-polar fuzzy graphs

The notion of an m-polar fuzzy set is a generalization of a bipolar fuzzy set. We apply the concept of m-polar fuzzy sets to graphs. We introduce certain types of irregular m-polar fuzzy graphs and investigate some of their properties. We describe the concepts of types of irregular m-polar fuzzy graphs with several examples. We also present applications of m-polar fuzzy graphs in decision making and social network as examples.     

Probability,fuzziness and borderline cases

An integrated approach to truth-gaps and epistemic uncertainty is described, based on probability distributions defined over a set of three-valued truth models. This combines the explicit representation of borderline cases with both semantic and stochastic uncertainty, in order to define measures of subjective belief in vague propositions. Within this framework we investigate bridges between probability theory and fuzziness in a propositional logic setting. In particular, when the underlying truth model is from Kleene's three-valued logic then we provide a complete characterisation of compositional min–max fuzzy truth degrees. For classical and supervaluationist truth models we find partial bridges, with min and max combination rules only recoverable on a fragment of the language. Across all of these different types of truth valuations, min–max operators are resultant in those cases in which there is only uncertainty about the relative sharpness or vagueness of the interpretation of the language.     

The case for an interval-based representation of linguistic truth

This paper develops an interval-based approach to the concept of linguistic truth. A special-purpose interval logic is defined, and it is argued that, for many applications, this logic provides a potentially useful alternative to the conventional fuzzy logic.The key idea is to interpret the numerical truth value v(p) of a proposition p as a degree of belief in the logical certainty of p, in which case p is regarded as true, for example, if v(p) falls within a certain range, say, the interval [0.7, 1]. This leads to a logic which, although being only a special case of fuzzy logic, appears to be no less linguistically correct and at the same time offers definite advantages in terms of mathematical simplicity and computational speed.It is also shown that this same interval logic can be generalized to a lattice-based logic having the capacity to accommodate propositions p which employ fuzzy predicates of type 2.     

结构的失效可能度及模糊概率计算方法

依据模糊可能性理论,系统地建立含模糊变量时结构的可靠性计算模型。旨在解决模糊结构、模糊-随机结构和模糊状态假设下结构的可靠性计算问题。所建模型可给出模糊结构失效的可能度和模糊-随机结构失效概率的可能性分布。研究表明:对同时含模糊变量和随机变量的混合可靠性计算问题,把失效概率(或可靠度)作为模糊变量,能更客观地反映系统的安全状况。算例分析说明了文中方法的合理性和有效性。     

Belief and probability: A general theory of probability cores

This paper considers varieties of probabilism capable of distilling paradox-free qualitative doxastic notions (e.g., full belief, expectation, and plain belief) from a notion of probability taken as a primitive. We show that core systems, collections of nested propositions expressible in the underlying algebra, can play a crucial role in these derivations. We demonstrate how the notion of a probability core can be naturally generalized to high probability, giving rise to what we call a high probability core, a notion that when formulated in terms of classical monadic probability coincides with the notion of stability proposed by Hannes Leitgeb [32]. Our work continues by one of us in collaboration with Rohit Parikh [7]. In turn, the latter work was inspired by the seminal work of Bas van Fraassen [46]. We argue that the adoption of dyadic probability as a primitive (as articulated by van Fraassen [46]) admits a smoother connection with the standard theory of probability cores as well as a better model in which to situate doxastic notions like full belief. We also illustrate how the basic structure underlying a system of cores naturally leads to alternative probabilistic acceptance rules, like the so-called ratio rule initially proposed by Isaac Levi [34].Core systems in their various guises are ubiquitous in many areas of formal epistemology (e.g., belief revision, the semantics of conditionals, modal logic, etc.). We argue that core systems can also play a natural and important role in Bayesian epistemology and decision theory. In fact, the final part of the article shows that probabilistic core systems are naturally derivable from basic decision-theoretic axioms which incorporate only qualitative aspects of core systems; that the qualitative aspects of core systems alone can be naturally integrated in the articulation of coherence of primitive conditional probability; and that the guiding idea behind the primary qualitative features of a core system gives rise to the formulation of lexicographic decision rules.     

Bilattices for deductions in multi-valued logic

In this exploratory paper we propose a framework for the deduction apparatus of multi-valued logics based on the idea that a deduction apparatus has to be a tool to manage information on truth values and not directly truth values of the formulas. This is obtained by embedding the algebraic structure V defined by the set of truth values into a bilattice B. The intended interpretation is that the elements of B are pieces of information on the elements of V. The resulting formalisms are particularized in the framework of fuzzy logic programming. Since we see fuzzy control as a chapter of multi-valued logic programming, this suggests a new and powerful approach to fuzzy control based on positive and negative conditions.     

Relating and extending semantical approaches to possibilistic reasoning

Two main semantical approaches to possibilistic reasoning with classical propositions have been proposed in the literature. Namely, Dubois-Prade's approach known as possibilistic logic, whose semantics is based on a preference ordering in the set of possible worlds, and Ruspini's approach that we redefine and call similarity logic, which relies on the notion of similarity or resemblance between worlds. In this article we put into relation both approaches, and it is shown that the monotonic fragment of possibilistic logic can be semantically embedded into similarity logic. Furthermore, to extend possibilistic reasoning to deal with fuzzy propositions, a semantical reasoning framework, called fuzzy truth-valued logic, is also introduced and proved to capture the semantics of both possibilistic and similarity logics.     

Fuzzy multi-objective programming for supplier selection and risk modeling: A possibility approach

Selection of supply chain partners is an important decision involving multiple criteria and risk factors. This paper proposes a fuzzy multi-objective programming model to decide on supplier selection taking risk factors into consideration. We model a supply chain consisting of three levels and use simulated historical quantitative and qualitative data. We propose a possibility approach to solve the fuzzy multi-objective programming model. Possibility multi-objective programming models are obtained by applying possibility measures of fuzzy events into fuzzy multi-objective programming models. Results indicate when qualitative criteria are considered in supplier selection, the probability of a certain supplier being selected is affected.     

基于犹豫模糊二元语义的多属性决策方法

深入研究了犹豫模糊二元语义多属性决策问题。首先利用幂均算子给出了犹豫模糊二元语义集的均值函数,并基于均匀分布概率准则和二元语义的距离测度提出了犹豫模糊二元语义集两两比较的可能度公式,进一步给出了可能度排序公式的性质。针对属性值为犹豫模糊二元语义集的多属性决策问题,提出了一种基于熵权的多属性决策方法。最后结合实际问题,验证了该方法的有效性和可行性。     

Possibility distributions: A unified representation of usual direct-probability-based parameter estimation methods

The paper presents a possibility theory based formulation of one-parameter estimation that unifies some usual direct probability formulations. Point and confidence interval estimation are expressed in a single theoretical formulation and incorporated into estimators of a generic form: a possibility distribution. New relationships between continuous possibility distribution and probability concepts are established. The notion of specificity ordering of a possibility distribution, corresponding to fuzzy subsets inclusion, is then used for comparing the efficiency of different estimators for the case of data points coming from a symmetric probability distribution. The usefulness of the approach is illustrated on common mean and median estimators from identical independent data sample of different size and of different common symmetric continuous probability distributions.     

A fuzzy approach to cooperative n-person games

The object of this paper is to provide a systematic treatment of bargaining procedures as a basis for negotiation. An innovative fuzzy logic approach to analyze n-person cooperative games is developed. A couple of indices, the Good Deal Index and the Counterpart Convenience Index are proposed to characterize the heuristic of bargaining and to provide a solution concept. The indices are examined theoretically and experimentally by analyzing three case studies. The results verify the validity of the approach.