On the Bayes formula for fuzzy probability measures

The fuzzy P-measure is presented here as the unique useful fuzzy probability measure satisfying the Bayes formula.     

On distorted probabilities and m-separable fuzzy measures

Fuzzy measures are used in conjunction with fuzzy integrals for aggregation. Their role in the aggregation is to permit the user to express the importance of the information sources (either criteria or experts). Due to the fact that fuzzy measures are set functions, the definition of such measures requires the definition of 2ⁿ parameters, where n is the number of information sources. To make the definition easier, several families of fuzzy measures have been defined in the literature.In this paper m-separable fuzzy measures are introduced. We present some results on this type of measures and we relate them to some of the previous existing ones. We study generating functions for m-separable fuzzy measures and some properties related to these generating functions.     

On measures of fuzziness of solutions of fuzzy relation equations with generalized connectives

By using a general class of fuzzy connectives of Yager [Fuzzy Sets and Systems4 (1980), 235–242], Pedrycz [Fuzzy relational equations with generalized convectives and their applications, Fuzzy Sets and Systems10 (1983), 185–201] has shown that the classical fuzzy relation equations of Sanchez [in “Fuzzy Automata and Decision Processes” (M. M. Gupta, G. N. Saridis, and B. R. Gaines, Eds.), pp. 221–234, North-Holland, Amsterdam, 1977] can be considered as a particular case of a more extensive class of fuzzy equations. For such types of equations, in this paper the solutions having the greatest energy measure and the smallest possible entropy measure of fuzziness are characterized.     

Correspondence between fuzzy measures and classical measures

In this paper we study the question whether, given a fuzzy measure (as defined in [3] and [4]). there exists a classical measure such that the fuzzy measure of a measurable fuzzy set μ equals the classical measure of the area below the membership function of μ. The results are that in the case of finite additivity there is a one-to-one correspondence between classical measures and fuzzy measures, whereas in the case of countable additivity this result only holds for generated fuzzy σ-algebras. Finally, some connections of that problem with the existence of an extension of a fuzzy measure defined on an arbitrary fuzzy σ-algebra σ to the generated fuzzy σ-algebra $\text{σ}$ are discussed.     

A discrete location model with fuzzy accessibility measures

In crisply defined discrete location problems, a number of facilities are to be located at specific points within an area, according to precisely quantified criteria. However in many location problems, especially those associated with social policies, non-crisply defined criteria are used such as, how ‘near’ or ‘accessible’ a facility is, or how ‘important’ certain issues are, etc. In these cases a fuzzy sets approach is more appropriate.This paper presents an application of the set partitioning (set covering with equality constraints) type of integer programming formulation to a discrete location problem with fuzzy accessibility criteria. The solution method suggested uses the symmetry of the objectives and the constraints introduced by Bellman and Zadeh.     

Fuzzy measures for profit maximization with fuzzy parameters

Profit maximization is an important issue to the firms that pursue the largest economic profit possible. This paper extends the situation from the deterministic to uncertain, where the coefficients are represented by fuzzy numbers. Intuitively, when the problem has fuzzy parameters, the derived profit value should be a fuzzy number as well. The extension principle is utilized to develop a pair of two-level mathematical programs to calculate the upper and lower bounds of the profit value at α-cuts. Following the duality theorem and a variable separation technique, the two-level mathematical programs are transformed into a class of one-level signomial geometric programs to solve. An example is given to illustrate the idea proposed in this paper.     

Additive fuzzy measures and integrals I

A point of view concerning “fuzzy measures” is explained. To this end, a new concept of “disjointness” for fuzzy is introduced and studied. Also, a concept of an “additive class of fuzzy sets” is defined to be a class of fuzzy sets closed under some “additive operations.” The fuzzy measures are defined to be sum-preserving real functions over such additive classes. Some basic properties of the fuzzy measures are derived. In contrast with other homonymous concepts studied in literature, our fuzzy measures lead to an additive fuzzy integral (see the part II of the paper).     

Using fuzzy measures and habitual domains to analyze the public attitude and apply to the gas taxi policy

Public acceptance and support are the crucial keys for implementing public policies successfully. Thus, the understanding of public acceptance or rejection towards the policy, as well as the important attributes of concern, could be very helpful to implementing the policy. However, most conventional attitude models could not approximate people's subjective evaluation process exactly by virtue of the additivity and independence assumptions. Additionally, people's decision behavior is deeply affected by their existing habits. Since habitual domains exist in the decision process, if the government can change or extend people's habitual thinking in favor of the public policy, the policy will receive satisfactory acceptance. Therefore, this study uses the habitual domain theory to analyze the public's attitude towards public policies. Furthermore, general fuzzy measures and fuzzy integrals, which require only boundary conditions and monotonicity, are also applied to develop a public attitude analysis model. An empirical study on the compress natural gas (CNG) taxi policy in Taipei City is conducted to show the applicability of the proposed model. The empirical results indicate that there are significant differences between the public's concern and governmental publicity, and some valuable strategies are suggested to the government.     

Optimal production inventory policy for defective items with fuzzy time period

In this paper, an optimal production inventory model with fuzzy time period and fuzzy inventory costs for defective items is formulated and solved under fuzzy space constraint. Here, the rate of production is assumed to be a function of time and considered as a control variable. Also the demand is linearly stock dependent. The defective rate is taken as random, the inventory holding cost and production cost are imprecise. The fuzzy parameters are converted to crisp ones using credibility measure theory. The different items have the different imprecise time periods and the minimization of cost for each item leads to a multi-objective optimization problem. The model is under the single management house and desired inventory level and product cost for each item are prescribed. The multi-objective problem is reduced to a single objective problem using Global Criteria Method (GCM) and solved with the help of Fuzzy Riemann Integral (FRI) method, Kuhn–Tucker condition and Generalised Reduced Gradient (GRG) technique. In optimum results including production functions and corresponding optimum costs for the different models are obtained and then are presented in tabular forms.     

Fuzzy integrals and fuzzy measures with their values in complete lattices

Real-valued fuzzy measures and their associated fuzzy integrals were proposed by Sugeno. In this paper we consider fuzzy measures valued in complete lattices and their associated upper and lower fuzzy integrals. Moreover we define a class of functionals, called fuzzy integrals. In the setting of the complete lattices which are both Brouwerian and dually Brouwerian, the upper and lower fuzzy integrals are fuzzy integrals; while in that of the completely distributive complete lattices every fuzzy integral is both lower and upper fuzzy integral with respect to a fuzzy measure.     

Men and measures: capturing knowledge requirements in firms through qualitative system modelling

Knowledge Management (KM) is an issue of great and increasing importance in most if not all areas of managerial endeavour. In this paper, we are concerned with the particular practical difficulty within KM of mapping knowledge in a managed system. This is an important practical issue because without a view of the terrain of explicit and tacit knowledge in the managed system, we have little prospect of planning our managerial interaction. Few if any practical methods exist which reflect the strongly systemic nature of business organizations. We begin by establishing our position with regard to the numerous definitions and perspectives of knowledge in managed systems, and indeed in regard to the disagreements that rack KM over the nature of knowledge itself, where it lies and the role of humans as creators, users and guardians of that knowledge. We relate the nature of system knowledge to well-known taxonomies of knowing what, knowing how, knowing why, knowing who together with the integrated from of knowing in the managed system as a whole. The method presented, Systems Based KM (or SBKM), is based on a non-positivist qualitative method deriving from System Dynamics and it is presented through the medium of a case study of a professional firm.     

Distinction between several subsets of fuzzy measures

In this paper, we place several fuzzy measure subsets in relation one with the other. The subsets under study are those corresponding to the definitions of probability measure. Sugeno's g_λ-measure, Shafer's belief function and Zadeh's possibility measure. We study the intersection of these subsets and we show the particular role of Dirac's measures in this comparison. We limit ourself to the case of mappins whose domain is the collection of all subsets of a finite set.Finally, the obtained partial results are summarized in only one figure which shoul clarify the specificity of each of the above definitions.     

Additive fuzzy measures and integrals,III

This paper is the third in a series of works dealing with a class of fuzzy measures and with their corresponding fuzzy integrals. Its aim is to present some important properties of the additive fuzzy integrals introduced in (Butnariu, J. Math. Anal. Appl., in press; J. Math. Anal. Appl. 117 (1986), 385–410) (e.g., the Lebesgue-Beppo-Levi theorem, etc.). These properties are used to explain some applications of the additive fuzzy integrals in proving a Radon-Nikodym representation theorem for additive fuzzy measures and in constructing solutions for fuzzy games.