Introduction. Required Background in Set Theory. Fuzzy Measures. Extensions. Structural Characteristics for Set Functions. Measurable Functions on Fuzzy Measure Spaces. Fuzzy Integrals. PanIntegrals. Applications. Index.

The concept of natural extension plays an important role in the theory of imprecise probabilities. It is a nonlinear functional on the class of nonnegative finite measurable functions. The difficulty of its calculation restricts applications of the theory. The paper provides a fast algorithm by which a precise value of the natural extension of a given nonnegative finite function with respect to a given lower probability is computed. Such an algorithm makes the natural extension one of the tools for aggregation in information fusion.

Fuzzy Measure Theory

The calculation of natural extensions with respect to lower probabilities

In this paper, we prove a Stolarsky type inequality for pseudo-integrals. More precisely, we show that: Z sup

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Investigation of a stolarsky type inequality for integrals in pseudo-analysis

. We consider generated pseudo-operations of the followingform: x ⊕ y = g (−1) (g(x) + g(y)), x y = g (−1) (g(x)g(y)), where g isa positive strictly monotone generating function and g (−1) is its pseudo-inverse. Using this type of pseudo-operations, the Riemann-Stieltjes typeintegral is introduced and investigated.AMS Mathematics Subject Classiﬁcation (2000): 26A42, 28A25, 28B10Key words and phrases: pseudo-operatins, generating function, Riemann-Stieltjes integral 1. Introduction It is a well known fact that classical Riemann-Stieltjes integral has great ap-plication in several areas of analysis as well as in probability theory and physics(see [3, 4]). The main aim of this paper is to present pseudo-analysis’ coun-terpart of this integral. By means of generalized generated pseudo-operationsand corresponding measure-like set function, pseudo Riemann-Stieltjes integralhas been constructed. This new Riemann type of integral belongs to the areaof pseudo-analysis, i.e., to the theory that combines approaches from many dif-ferent ﬁelds and is capable of supplying solutions that were not achieved bythe classical tools. Some of important results concerning pseudo-analysis, boththeory and application, can be found in [1, 2, 5, 6, 7, 10, 13, 16, 18].Section 2 contains preliminary notions, such as semiring, generalized gen-erated pseudo-operations and measure-like set function. Operations in ques-tion are generalizations of pseudo-operations that are in the core of g-calculus([9, 11, 12, 13, 14]). Set function deﬁned in this section is based on generat-ing function g and, under some additional conditions, has properties of fuzzymeasure ([13, 18]). In the third section is presented the construction of pseudoRiemann-Stieltjes integral. Some of its basic properties are given for case whengenerating function g is strictly monotone and continuous function (we shouldstress that in this case used operations are operations from g-semiring). Specif-ically, for strictly increasing bijection g strong connection between the proposedintegral and Lebesque integral is obtained. Additionally, we give connections

background

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Riemann-Stieltjes type integral based on generated pseudo-operations

Grabisch and Labreuche have recently proposed a generalization of capacities, called the bi-capacities. Recently, a new approach for studying bi-capacities through introducing a notion of ternary-element sets proposed by the author. In this paper, we propose many results such as bipolar Mobius transform, importance index, and interaction index of bi-capacities based on our approach.

The Importance and Interaction Indices of Bi-Capacities Based on Ternary-Element Sets

We investigate the multiple attribute decision making (MADM) problems with hesitant triangular fuzzy information. Firstly, some operational laws of hesitant triangular fuzzy elements and score function of hesitant fuzzy elements are introduced. Then, we have developed the hesitant triangular fuzzy choquet ordered geometric (HTFCOG) operator. The prominent charac- teristic of the operators is that they can not only consider the importance of the elements or their ordered positions, but also reflect the correlation among the elements or their ordered positions. We have applied the HTFCOG operators to multiple attribute decision making for evaluating the customer credit risk of commercial bank with hesitant triangular fuzzy information. Finally an illustrative example for evaluating the customer credit risk of commercial bank has been given to show the developed method.

Fuzzy Measure Theory

Citations

The calculation of natural extensions with respect to lower probabilities

Investigation of a stolarsky type inequality for integrals in pseudo-analysis

Riemann-Stieltjes type integral based on generated pseudo-operations

The Importance and Interaction Indices of Bi-Capacities Based on Ternary-Element Sets

Approach to multiple attribute decision making with hesitant triangular fuzzy information and their application to customer credit risk assessment

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