Aggregation of information is of primary importance in the construction of knowledge based systems in various domains, ranging from medicine, economics, and engineering to decision-making processes, artificial intelligence, robotics, and machine learning. This book gives a broad introduction into the topic of aggregation functions, and provides a concise account of the properties and the main classes of such functions, including classical means, medians, ordered weighted averaging functions, Choquet and Sugeno integrals, triangular norms, conorms and copulas, uninorms, nullnorms, and symmetric sums. It also presents some state-of-the-art techniques, many graphical illustrations and new interpolatory aggregation functions. A particular attention is paid to identification and construction of aggregation functions from application specific requirements and empirical data. This book provides scientists, IT specialists and system architects with a self-contained easy-to-use guide, as well as examples of computer code and a software package. It will facilitate construction of decision support, expert, recommender, control and many other intelligent systems.

Aggregation Functions: A Guide for Practitioners

We note that orthopair fuzzy subsets are such that that their membership grades are pairs of values, from the unit interval, one indicating the degree of support for membership in the fuzzy set and the other support against membership. We discuss two examples, Atanassov's classic intuitionistic sets and a second kind of intuitionistic set called Pythagorean. We note that for classic intuitionistic sets the sum of the support for and against is bounded by one, while for the second kind, Pythagorean, the sum of the squares of the support for and against is bounded by one. Here we introduce a general class of these sets called q-rung orthopair fuzzy sets in which the sum of the ${\rm{q}}$ th power of the support for and the ${\rm{q}}$ th power of the support against is bonded by one. We note that as q increases the space of acceptable orthopairs increases and thus gives the user more freedom in expressing their belief about membership grade. We investigate various set operations as well as aggregation operations involving these types of sets.

Generalized Orthopair Fuzzy Sets

Advances in scientific computing have made modelling and simulation an important part of the decision-making process in engineering, science, and public policy. This book provides a comprehensive and systematic development of the basic concepts, principles, and procedures for verification and validation of models and simulations. The emphasis is placed on models that are described by partial differential and integral equations and the simulations that result from their numerical solution. The methods described can be applied to a wide range of technical fields, from the physical sciences, engineering and technology and industry, through to environmental regulations and safety, product and plant safety, financial investing, and governmental regulations. This book will be genuinely welcomed by researchers, practitioners, and decision makers in a broad range of fields, who seek to improve the credibility and reliability of simulation results. It will also be appropriate either for university courses or for independent study.

/pdf/verification-and-validation-in-scientific-computing-262w5ti0ia.pdf

Verification and Validation in Scientific Computing

We describe the idea of Pythagorean membership grades and the related idea of Pythagorean fuzzy subsets. We focus on the negation and its relationship to the Pythagorean theorem. We look at the basic set operations for the case of Pythagorean fuzzy subsets. A relationship is shown between Pythagorean membership grades and complex numbers. We specifically show that Pythagorean membership grades are a subclass of complex numbers called Π-i numbers. We investigate operations that are closed under Π-i numbers. We consider the problem of multicriteria decision making with satisfactions expressed as Pythagorean membership grades, Π-i numbers. We look at the use of the geometric mean and ordered weighted geometric operator for aggregating criteria satisfaction.

Pythagorean Membership Grades, Complex Numbers, and Decision Making

There is given a short overview of the monograph rdquoAggregation Functionsrdquo (M. Grabisch, J. L. Marichal, R. Mesiar, E. Pap), Cambridge University Press (in press).

”Aggregation Functions”, Cambridge University Press

Some properties of overlap and grouping functions and their application to image thresholding

Fuzzy implication functions are one of the main operations in fuzzy logic. They generalize the classical implication, which takes values in the set {0,1}, to fuzzy logic, where the truth values belong to the unit interval [0,1]. These functions are not only fundamental for fuzzy logic systems, fuzzy control, approximate reasoning and expert systems, but they also play a significant role in mathematical fuzzy logic, in fuzzy mathematical morphology and image processing, in defining fuzzy subsethood measures and in solving fuzzy relational equations. This volume collects 8 research papers on fuzzy implication functions. Three articles focus on the construction methods, on different ways of generating new classes and on the common properties of implications and their dependencies. Two articles discuss implications defined on lattices, in particular implication functions in interval-valued fuzzy set theories. One paper summarizes the sufficient and necessary conditions of solutions for one distributivity equation of implication. The following paper analyzes compositions based on a binary operation * and discusses the dependencies between the algebraic properties of this operation and the induced sup-* composition. The last article discusses some open problems related to fuzzy implications, which have either been completely solved or those for which partial answers are known. These papers aim to present todays state-of-the-art in this area.

Advances in Fuzzy Implication Functions

A review of the relationships between implication, negation and aggregation functions from the point of view of material implication

http://library.utia.cas.cz/separaty/2011/E/mesiar-a%20class%20of%20aggregation%20functions%20encompassing%20two-dimensional%20owa%20operators.pdf

Ana Pradera

Papers

Aggregation Functions: A Guide for Practitioners

Some properties of overlap and grouping functions and their application to image thresholding

Advances in Fuzzy Implication Functions

A review of the relationships between implication, negation and aggregation functions from the point of view of material implication

A class of aggregation functions encompassing two-dimensional OWA operators