Induced aggregation operators in decision making with the Dempster-Shafer belief structure

We study the induced aggregation operators. The analysis begins with a revision of some basic concepts such as the induced ordered weighted averaging (IOWA) operator and the induced ordered weighted geometric (IOWG) operator. We then analyze the problem of decision making with Dempster-Shafer theory of evidence. We suggest the use of induced aggregation operators in decision making with Dempster-Shafer theory. We focus on the aggregation step and examine some of its main properties, including the distinction between descending and ascending orders and different families of induced operators. Finally, we present an illustrative example in which the results obtained using different types of aggregation operators can be seen.aggregation operators, dempster-shafer belief structure, uncertainty, iowa operator, decision making

Construction of aggregation operators with noble reinforcement

This paper examines disjunctive aggregation operators used in various recommender systems. A specific requirement in these systems is the property of noble reinforcement: allowing a collection of high-valued arguments to reinforce each other while avoiding reinforcement of low-valued arguments. We present a new construction of Lipschitz-continuous aggregation operators with noble reinforcement property and its refinements. <br /

Aggregation on bipolar scales

The paper addresses the problem of extending aggregation operators typically defined on $[0,1]$ to the symmetric interval $[-1,1]$, where the ``0'' value plays a particular role (neutral value). We distinguish the cases where aggregation operators are associative or not. In the former case, the ``0'' value may play the role of neutral or absorbant element, leading to pseudo-addition and pseudo-multiplication. We address also in this category the special case of minimum and maximum defined on some finite ordinal scale. In the latter case, we find that a general class of extended operators can be defined using an interpolation approach, supposing the value of the aggregation to be known for ternary vectors.bipolar scale; bi-capacity; aggregation

Pointwise construction of Lippschitz aggregation operators with specific properties

This paper describes an approach to pointwise construction of general aggregation operators, based on monotone Lipschitz approximation. The aggregation operators are constructed from a set of desired values at certain points, or from empirically collected data. It establishes tight upper and lower bounds on Lipschitz aggregation operators with a number of different properties, as well as the optimal aggregation operator, consistent with the given values. We consider conjunctive, disjunctive and idempotent n-ary aggregation operators; p-stable aggregation operators; various choices of the neutral element and annihilator; diagonal, opposite diagonal and marginal sections; bipolar and double aggregation operators. In all cases we provide either explicit formulas or deterministic numerical procedures to determine the bounds. The findings of this paper are useful for construction of aggregation operators with specified properties, especially using interpolation schemata.<br /

Homogeneous aggregation operators

summary:Recently, the utilization of invariant aggregation operators, i.e., aggregation operators not depending on a given scale of measurement was found as a very current theme. One type of invariantness of aggregation operators is the homogeneity what means that an aggregation operator is invariant with respect to multiplication by a constant. We present here a complete characterization of homogeneous aggregation operators. We discuss a relationship between homogeneity, kernel property and shift-invariance of aggregation operators. Several examples are included

Identification of general and double aggregation operators using monotone smoothing

Aggregation operators model various operations on fuzzy sets, such as conjunction, disjunction and averaging. Recently double aggregation operators have been introduced; they model multistep aggregation process. The choice of aggregation operators depends on the particular problem, and can be done by fitting the operator to empirical data. We examine fitting general aggregation operators by using a new method of monotone Lipschitz smoothing. We study various boundary conditions and constraints which determine specific types of aggregation.<br /

A copula-based approach to aggregation functions

This paper presents the role of copula functions in the theory of aggregation operators and an axiomatic characterization of Archimedean aggregation functions. In this context we are focusing our attention about several properties of aggregation functions, like supermodularity and Schur-concavity.Aggregation functions, supermodularity, Schur-concavity, copula, Archimedean copulae

Aggregation operators on partially ordered sets and their categorical foundations

summary:In spite of increasing studies and investigations in the field of aggregation operators, there are two fundamental problems remaining unsolved: aggregation of $L$-fuzzy set-theoretic notions and their justification. In order to solve these problems, we will formulate aggregation operators and their special types on partially ordered sets with universal bounds, and introduce their categories. Furthermore, we will show that there exists a strong connection between the category of aggregation operators on partially ordered sets with universal bounds (Agop) and the category of partially ordered groupoids with universal bounds (Pogpu). Moreover, the subcategories of Agop consisting of associative aggregation operators, symmetric and associative aggregation operators and associative aggregation operators with neutral elements are, respectively, isomorphic to the subcategories of Pogpu formed by partially ordered semigroups, commutative partially ordered semigroups and partially ordered monoids in the sense of Birkhoff. As a justification of the present notions and results, some relevant examples for aggregations operators on partially ordered sets are given. Particularly, aggregation process in probabilistic metric spaces is also considered