Domination of ordered weighted averaging operators over t-norms

The fusion of transitive fuzzy relations preserving the transitivity is linked to the domination of the involved aggregation operator. The aim of this contribution is to investigate the domination of OWA operators over t-norms whereas the main emphasis is on the domination over the ukasiewicz t-norm. The domination of OWA operators and related operators over continuous Archimedean t-norms will also be discussed.This work was partly supported by network CEEPUS SK-42, COST Action 274 TARSKI and project APVT 20-023402.     

On the extended set of weights of the OWA functions

ABSTRACT

The paper offers an extension and generalization of the definitions of the weighted arithmetic mean WAM, the ordered weighted averaging function OWA and the discrete Choquet integral using not only positive but also negative weights. Negative weights are important in various contexts, including statistics and robust aggregation, and they directly affect the monotonicity of the mentioned functions. The paper offers insights into the extended WAM, OWA and Choquet integral-based averaging functions with respect to their directional monotonicity, and establish the systems of linear inequalities which define the cone of monotonicity of these functions.     

Uncertain induced aggregation operators and its application in tourism management

We develop a new decision making approach for dealing with uncertain information and apply it in tourism management. We use a new aggregation operator that uses the uncertain weighted average (UWA) and the uncertain induced ordered weighted averaging (UIOWA) operator in the same formulation. We call it the uncertain induced ordered weighted averaging - weighted averaging (UIOWAWA) operator. We study some of the main advantages and properties of the new aggregation such as the uncertain arithmetic UIOWA (UA-UIOWA) and the uncertain arithmetic UWA (UAUWA). We study its applicability in a multi-person decision making problem concerning the selection of holiday trips. We see that depending on the particular type of UIOWAWA operator used, the results may lead to different decisions.     

On the dispersion measure of OWA operators

We describe some basic features of the OWA operator. We turn to the problem of determining the weights associated with this operator and particularly the maximal dispersion (entropy) approach. We consider the possibility of using minimization of dispersion. After discussing concerns with both maximization and minimization of dispersion we investigate the possibility of finding an optimal solution intermediate to these extremes. We next consider alternative measures of dispersion. We introduce a fundamental requirement for a measure of dispersion called the Preference for Equal Division. A number of general classes of dispersion measures are provided notable among these are those based on t-norm and t-conorm operators.     

Nonmonotonic OWA operators

 The basic properties of the Ordered Weighted Averaging (OWA) operator are recalled. The role of these operators in the formulation of multi-criteria decision functions, using the concept of quantifier guided aggregation, is discussed. An extended class of OWA operators, one based upon a relaxation of the requirements on the OWA operators, is introduced. This relaxation allows us to consider a new branch of OWA operators, NOMOWA operators, which have negative weights and which exhibit nonmonotonicity. Some special cases of these operators are discussed and then we investigate the role of these nonmonotonic operators in the formulation of multi-criteria decision functions.     

Extremal fuzzy integrals

Fuzzy integral is a monotone idempotent functional on a fuzzy measure. The extremal fuzzy integrals are introduced. Regular fuzzy integrals do not distinguish functions not distinguishable by the underlying fuzzy measures. These integrals include Choquet, Sugeno, Shilkret and several other well known integrals.     

On topological properties of the Choquet weak convergence of capacity functionals of random sets

In view of the recent interests in random sets in information technology, such as models for imprecise data in intelligent systems, morphological analysis in image processing, we present, in this paper, some contributions to the foundation of random set theory, namely, a complete study of topological properties of capacity functionals of random sets, generalizing weak convergence of probability measures. These results are useful for investigating the concept of Choquet weak convergence of capacity functionals leading to tractable criteria for convergence in distribution of random sets. The weak topology is defined on the space of all capacity functionals on R^d. We show that this topological space is separable and metrizable.     

Aggregation functions: Means

This two-part state-of-the-art overview on aggregation theory summarizes the essential information concerning aggregation issues. An overview of aggregation properties is given, including the basic classification on aggregation functions. In this first part, the stress is put on means, i.e., averaging aggregation functions, both with fixed arity (n-ary means) and with multiple arities (extended means).     

Aggregation of infinite sequences

Infinitary aggregation functions acting on sequences and possessing some a priori given properties as additivity, comonotone additivity, symmetry, etc., are investigated. On the other side, we discuss infinitary aggregation functions related to given extended aggregation functions, where special attention is given to triangular norms, triangular conorms and weighted means.     

Decision Making in Reinsurance with Induced OWA Operators and Minkowski Distances

The decision to choose a reinsurance program has many complexities because it is difficult to simultaneously achieve high levels in different optimal criteria including maximum gain, minimum variance, and probability of ruin. This article suggests a new method by which, through membership functions, we can measure the distance of each alternative to an optimal result and aggregate it by using different types of aggregations. In this article, particular attention is given to the induced Minkowski ordered weighted averaging distance operator and the induced Minkowski probabilistic ordered weighted averaging distance operator. The main advantage of these operators is that they include a wide range of special cases. Thus, they can adapt efficiently to the specific needs of the calculation processes. By doing so, the reinsurance system can make better decisions by using different scenarios in the uncertain environment considered.     

Soft Aggregation Methods in Case Based Reasoning

Our goal is to provide some tools, based on soft computing aggregation methods, useful in the two fundamental steps in case base reasoning, matching the target and the cases and fusing the information provided by the relevant cases. To aid in the first step we introduce a methodology for matching the target and cases which uses a hierarchical representation of the target object. We also introduce a method for fusing the information provided by relevant retrieved cases. This approach is based upon the nearest neighbor principle and uses the induced ordered weighted averaging operator as the basic aggregation operator. A procedure for learning the weights is described.     

Decision-making with distance measures and induced aggregation operators

In this paper, we present a new decision-making approach that uses distance measures and induced aggregation operators. We introduce the induced ordered weighted averaging distance (IOWAD) operator. IOWAD is a new aggregation operator that extends the OWA operator by using distance measures and a reordering of arguments that depends on order-inducing variables. The main advantage of IOWAD is that it provides a parameterized family of distance aggregation operators between the maximum and the minimum distance based on a complex reordering process that reflects the complex attitudinal character of the decision-maker. We studied some of IOWAD’s main properties and different particular cases and further generalized IOWAD by using Choquet integrals. We developed an application in a multi-person decision-making problem regarding the selection of investments. We found that the main advantage of this approach is that it is able to provide a more complete picture of the decision-making process, enabling the decision-maker to select the alternative that it is more in accordance with his interests.