A generalized probability mixture density governs an additive fuzzy system. The fuzzy system’s if-then rules correspond to the mixed probability densities. An additive fuzzy system computes an output by adding its fired rules and then averaging the result. The mixture’s convex structure yields Bayes theorems that give the probability of which rules fired or which combined fuzzy systems fired fo...

In this paper Hermite—Hadamard type inequalities for m-convex and (α,m)-convex functions for fuzzy integrals are given. Some examples are also given to illustrate the results.

In the case of real-valued inputs, averaging aggregation functions have been studied extensively with results arising in fields including probability and statistics, fuzzy decision-making, and various sciences. Although much of the behavior of aggregation functions when combining standard fuzzy membership values is well established, extensions to interval-valued fuzzy sets, hesitant fuzzy sets,...

Ordered fuzzy numbers (OFN) invented by the second author and his two coworkers in 2002 make possible to utilize the fuzzy arithmetic and to construct the Abelian group of fuzzy numbers and then an ordered ring. The definition of OFN uses the extension of the parametric representation of convex fuzzy numbers. Fuzzy implication is proposed with the help of algebraic operations and a lattice stru...

Abstract In this paper, we introduce the non-convex interval-valued functions for fuzzy-interval-valued functions, which are called "Equation missing"-convex by means of fuzzy order relation. This relation is defined level-wise through Kulisch–Miranker given on interval space. By using -->-convexity concept, present fuzzy-interval Hermite–Hadamard inequali...

The lattice Lu of upper semicontinuous convex normal functions with convolution ordering arises in studies of type-2 fuzzy sets. In 2002, Kawaguchi and Miyakoshi [Extended t-norms as logical connectives of fuzzy truth values, Multiple-Valued Logic 8(1) (2002) 53–69] showed that this lattice is a complete Heyting algebra. Later, Harding et al. [Lattices of convex, normal functions, Fuzzy Sets an...

Abstract: Classical cost efficiency (CE) measurement models require exact and accurate knowledge of the input and output values for each decision making unit (DMU). However, the observed values of the input and output data in real-world problems are often imprecise or vague. In recent years, fuzzy data envelopment analysis (DEA) has been successfully used to deal with imprecise or vague data in...