This paper describes a federated learning model capable to process imprecise and missing data. Federation learning is a technique to solve the problem of data governance and privacy by training algorithms without exchanging the data itself. The performance of the proposed method is demonstrated on medical data of breast cancer cases. Results for different data loss scenarios and corresponding measures of classification quality are presented and discussed.

In the paper, we show how the importance of individual ranges of values of attributes describing cases can be determined using the attribute fuzzification process. The importance is determined on the basis of classification capabilities. The described approach is based mainly on fuzzy set theory and the rough set based discretization method. Moreover, an experimental study of the computer-aided classification task is presented.

In this paper, we concentrate on study interval-valued fuzzy relations in decision problems based on preference relations, and a preference structure making up of the strict preference relation, indifference relation, and incomparability relation which may be defined with the use of interval-valued aggregation and interval-valued fuzzy negation function. We analyze the influence of some new types of fusion functions on the effectiveness of the decision process. The studies concern different aggregation classes due to the type of monotonicity/order used.

The article deals with the problem of selecting the most appropriate attributes for a given classification method with the use of inclusion and similarity measures for interval-valued fuzzy sets. These types of measures with uncertainty were introduced using partial or linear order. The article introduces a modified IV-Relief algorithm using the above-mentioned measures. The theoretical considerations were supported by the analysis of the effectiveness of the proposed algorithm on a well-known dataset on breast cancer diagnostics. The proposed methods make it possible to extend the recognized classification methods so that they operate on uncertain data.

The problem of measuring the degree of entropy based on precedence indicator and similarity measures under conditions of uncertainty or imprecision was studied. So we call back to the notion of precedence and similarity measures of interval-valued fuzzy sets (IVFSs) and we construct an entropy measure with uncertainty by applying for IVFSs of different orders. In addition, we discuss the impact of entropy measures reflecting the uncertainty in the decision-making problem that employed these new measures in the problem of missing values.

A new concept of equivalence between intervals and the induced indistinguishability between interval-valued (IV) fuzzy sets are proposed and considered. A new notion of the degree of IV equivalence is presented where partial or linear orders and the width of intervals are involved reflecting uncertainty. Furthermore, construction methods of the considered equivalences are provided and the relation between them and other notions of IV equivalences are studied. Finally, a methodology to apply the proposed equivalences in the field of image processing is shown, along with an illustrative example.

In this contribution the concept of a local contrast of a fuzzy relation with the use of a consensus measure is introduced. A construction method of such local contrast using aggregation functions and fuzzy implications is considered. Other construction methods using similarity measure are also pointed out. Several examples of local contrasts are provided. The usability of introduced local contrast measures is evaluated by applying them in image processing for salient region detection.

In this contribution the new concept of the operator in the interval-valued fuzzy sets, which is a pre-aggregation function, and their application in medical diagnosis is presented. Taking into account the widths of the intervals the type of interval operator measures are proposed in decision making problem. We suggest to apply the new interval-valued operator in decision model of medical diagnosis support.

In this contribution we propose new methodology to compare interval-valued fuzzy cardinal numbers (IVFCN). The new methods are based on interval subsethood measures which take into account widths of the intervals. An application of introduced methodology is presented on an example of decision algorithm for medical diagnosis support.

Fuzzy relations are compared by membership values and as a consequence new types of local properties of fuzzy relations are introduced. In the new properties of fuzzy relations an arbitrary binary relation is involved. Particularly, a binary aggregation function may be used to define these properties. Connections between the new local properties of fuzzy relations are described. Furthermore, preservation of these properties in aggregation process is considered. Finally, notes on applications of the presented local properties in the context of fuzzy multisets and decision making are provided.

In this paper, the problem of measuring the degree of inclusion and similarity measure for interval-valued fuzzy sets is considered. We recall inclusion and similarity measures with uncertainty by using the partial or linear order on intervalvalued fuzzy sets. Moreover, we discuss an influence of inclusion and similarity measures with uncertainty to decision-making algorithm that uses those new measures.

Optimality and computational efficiency are two very desirable but also competitive attributes of optimal feed planning. A well-designed algorithm can vastly increase machining productivity by reducing tool positioning time subject to limits of the machine tool. The nonlinear optimization problem aims to achieve the highest possible feed along the tool path, while limiting the speed of the actuator level, acceleration and Jerk profiles. Methods proposed in the literature either use rather complex nonlinear optimization solvers, such as Sequential Quadratic Programming, use iterative heuristics that extends computation time, or use conventional assumptions that reduce computation time but lead to slower tool motion.The problem of optimal feed-rate planning along a curved tool path for multi-axis CNC machines with a Jerk limit for each axis is addressed. However, the use of Jerk (rate of change of acceleration) into the feed-rate scheduling problem causes generating both, computationally efficient solutions and simultaneously guaranteeing optimality, is a challenging problem. To solve this problem, we propose the approach of modifying a Jerk parameter, through the use of a pair of convex and concave functions, including aggregation functions. In this technique, the suggested algorithm indicates the points at which the increasing convex/concave function and the adequate dual decreasing function modeling the Jerk parameter is used.

In this paper, the issue of measuring the degree of inclusion and equivalence measure for Atanassov intuitionistic fuzzy setting is considered. We propose an application of the inclusion and equivalence measures created by using the partial or linear order on Atanassov intuitionistic fuzzy setting. Moreover, some properties of inclusion and equivalence measures and some correlation between them and aggregation operators are examined and their possible application in image processing is indicated.

Human beings represent and process imperfect information in a natural way, whereas effective approaches are still being looked for in the area of soft computing. Atanassov's intuitionistic fuzzy sets and especially Atanassov's intuitionistic fuzzy relations are are tools that make it possible to model in an effective way imperfect information. In this paper, we discuss the transitivity problem of Atanassov's intuitionistic fuzzy relations. The transitivity property reflects the consistency of a preference relation. Therefore, transitivity is important from the point of view of real problems appearing, e.g., in group decision making, choice, and utility theories widely making use of the preference procedures. We study here in detail transitivity property of Atanassov's intuitionistic fuzzy relations and propose a new algorithm for generating weakly transitivity among the alternatives (options) considered. Next, making use of the new algorithm we propose a procedure of ranking of the alternatives in a group decision-making problem.

In this paper interval-valued fuzzy relations are studied. The problem of selection of alternatives in decision making is examined. In particular, a new definition of transitivity based on the measure of the intensity preference between pairs of alternatives is presented and its apply to solve decision making problem is proposed.

In this paper we study interval-valued fuzzy relations. We consider preference relations, i.e. a triplet consisting of strict preference, indifference and incomparability which are defined with the use of a fuzzy negation. We analyze the preservation of the fuzzy negation based reciprocity property of interval-valued fuzzy relations by aggregation functions and by some basic interval-valued fuzzy relations. We use diverse representa-tions of aggregation functions. We also consider the connection between N-reciprocal relations and transitivity properties. We provide a numerical example where the final alternative is chosen with the use of generalized voting method, where admissible linear orders for intervals are applied.

Based on the definition of admissible order for interval-valued Atanassov intuitionistic fuzzy sets, we study ordered weighted averaging operators in these sets, distinguishing between the weights associated with the membership and those associated with the nonmembership degree, which may differ from the latter. We also study Choquet integrals for aggregating information, which is represented using interval-valued Atanassov intuitionistic fuzzy sets. We conclude with two algorithms to choose the best alternative in a decision-making problem when we use this kind of sets to represent information.

In the paper properties of intuitionistic fuzzy relations are considered and preservation of some properties by operations, including lattice operations, composition and related operators are studied. Properties of intuitionistic fuzzy relations, namely reflexivity, irreflexivity, connectedness, symmetry, antisymmetry, perfect antisymmetry and transitivity are considered. Moreover, the authors study assumptions under which intuitionistic fuzzy preference relations and related operators fulfil these properties.