Teimuraz Tsabadze

In the first part of the manual, the traditional methods are given, by means of which it is still possible to solve some practical problems. Processes in the modern world are dynamic, which leads to the need for continuous and very often unexpected changes in previously accepted concepts and methods in social sciences, including management.

Due to these circumstances, it became necessary to develop new approaches in solving planning and management tasks.

The second part is devoted to the latest methods of decision-making developed by the authors using the theory of phase sets.

The manual is intended for undergraduate, master's and doctoral students. It will also be useful for all those interested in decision making.

The main subject of the lecture course is the basics of the theory of fuzzy (phase) sets and relations. The elements of the theory of classical sets and relations are discussed, the definitions of phase sets, their varieties, all basic operations are given, the generalization principle of the founder of phase logic, Lot Fizade, is given. Phase operators are distinguished in detail: triangular norms, triangular conorms, averaging operators. The definition of phase relations and the types of phase relations, their properties and operations on the phase relations meter are given. Theoretical discussions are accompanied by practical examples and exercises, the solutions of which will strengthen the received theoretical knowledge. The publication is intended for undergraduate, master's and doctoral students. It will also be useful to all other interested parties.

 This paper introduces theoretical basement of one approach to group decision-making in fuzzy enviroment. Mathematical apparatus, based on the concepts of coordination index and similarity of finite collections of fuzzy sets, has been successively constructed in metric lattice of fuzzy sets

This paper introduces the generalization of the concept of binary relations reduction. The offered approach allows one-valued restoration of any transitive and antisymmetric relation by its reduction. The theoretical basement of this approach has been successively constructed. The example of practical applying of the reduction has been set. The ways of further generalization of introduced concept are outlined

This paper introduces an approach to group decision-making based on the coordination index and similarity of the finite collections of fuzzy sets. The mathematical apparatus of this approach, based on the concept of maximal coordination, is constructed in the metric lattice of fuzzy sets. A specific kind of fuzzy aggregation operator is proposed. A method for fuzzy aggregation is introduced, thoroughly discussed and its algorithm determined. The practical example of application of the introduced method is offered as well.

This paper introduces the concept of T-reduction of binary fuzzy relations and establishes its properties. It presents an approach to providing the one-valued restoration of any boundary transitive and T-asymmetric binary fuzzy relation by its T-reduction. A theoretical basis for this approach is established, showing that T-reduction is the smallest fuzzy relation whose T-transitive closure coincides with the given binary fuzzy relation and where an (n × n) fuzzy relation on a finite universe with cardinality n can be represented by n − 1 values. This approach opens up an original way for constructing the membership functions of certain binary fuzzy relations. Important examples for the application of this concept are given. 

This paper introduces a simple method for group decision-making, where experts opinions are expressed by trapezoidal fuzzy numbers. The method is based on the metric approach providing an easy procedure for achieving a one-valued consensus. A new approach for determining experts degrees of importance is proposed. A theoretical basis is established, showing that the method is correct under the whole class of metrics on the set of trapezoidal fuzzy numbers. A specific fuzzy aggregation operator is constructed for the realization of the proposed approach. It is shown that despite its simplicity the proposed method preserves such important properties as Agreement Preservation, Order Independence, Linearity property etc. A comparative appraisal of the results obtained by the proposed approach and someds is given as well.

 This paper continues another our work, which is the first of two parts where the approach to the processing of quantitative expert evaluations in the process of group decision-making under uncertainty is considered. In the second part, represented by this paper, an approach is proposed for

the processing of qualitative expert evaluations in the process of group decision-making. The approach is based on the use of triangular fuzzy numbers. In group decision-making the opinions of experts are expressed by linguistic variables like very bad, not very bad, problematic, good, and so on. The technique of conversion of expert quantitative opinions to triangular fuzzy numbers is considered.A simple method of expressing expert opinions by triangular fuzzy numbers is introduced. A new approach to determining the expert degrees of importance is proposed. The proposed methodology is discussed in full detail and its algorithm is described.

We consider approaches to making decisions for control problems in nonstandard situations in the absence of previous experience and incomplete knowledge of the considered problem. In such cases we usually cannot do without expert evaluations that lead to the process of group decisionmaking, and it becomes necessary to solve a problem of alternatives aggregation. It has been proposed to solve such problems by means of fuzzy sets. The material of the paper is divided into two parts. In Part 1, an approach is proposed for the processing of quantitative expert evaluations that are used in group decision-making. The approach is based on the coordination index and the similarity of finite collections of fuzzy sets and takes into account the specific character of the fuzzy aggregation operator. The approach is discussed in full detail and its algorithm is presented. An example of the application of the proposed method is given. 

This paper introduces one approach to group decisionmaking, where experts’ opinions are expressed by fuzzy sets. It is meant that there are several groups of experts and, therefore, several finite collections of fuzzy sets turn out. Then a quite simple approach for aggregation of obtained finite collections of fuzzy sets into resulting one is proposed and its algorithm is determined.

Keywords—Group decision-making, Finite collection of fuzzy sets, Fuzzy aggregation operator, Regulation, Representative.   

 This paper introduces one method of reaching consensus among several groups of experts. The method is based on the use of fuzzy numbers. It is meant that opinions of experts are expressed by fuzzy triangular numbers and, therefore, several finite collections of fuzzy triangular numbers are obtained. A method for aggregation of the obtained several finite collections of fuzzy sets into the resulting one is proposed. A new approach is introduced for determining degrees of experts’ importance depending on the closeness of experts’ estimates to the representative of a finite collection of all triangular fuzzy estimates. The specific fuzzy aggregation operator is offered. The proposed method is thoroughly discussed and its algorithm is presented.

KEYWORDS: Consensus, Finite collection of fuzzy triangular numbers, Metric lattice, Regulation, Representative, Fuzzy aggregation operator, 

 This paper introduces an attempt to establish natural gas tariffs based on experts’ fuzzy estimations on the example of Georgia. We consider the existent situation in tariffication processes. We discuss the necessity to use new approaches for implementation of these processes. We introduce a new approach for determination of natural gas tariffs based on fuzzy logic and its algorithm. At the end we show the example of practical application of the proposed method

Key-Words: - Natural gas tariff, Parameter, Fuzzy set, Fuzzy aggregation, Algorithm. 

The purpose of this chapter is to introduce a new approach for an assessment of the credit risks. The initial part of the chapter is to briefly discuss the existing models of assessment of the credit risks and justify the need for a new approach. Since a new approach is created for conditions of uncertainty, we cannot do without fuzzy mathematics. The proposed approach is based on group decision-making, where experts’ opinions are expressed by trapezoidal fuzzy numbers. The theoretical basis of the offered approach is laid out in the metric space of trapezoidal fuzzy numbers.The new approach is introduced and discussed, and two realization algorithms are given.The toy example of application of the introduced approach is offered as well.

The purpose of this paper is to develop a new approach for an assessment of the credit risk to corporate borrowers. There are different models for borrowers' risk assessment. These models are divided into two groups: statistical and theoretical. When assessing the credit risk for corporate borrowers, statistical model is unacceptable due to the lack of sufficiently large history of defaults. At the same time, we cannot use some theoretical models due to the lack of stock exchange. In those cases, when studying a particular borrower given that statistical base does not exist, the decision-making process is always of expert nature. The paper describes a new approach that may be used in group decision-making. An example of the application of the proposed approach is given.

This paper is intended to present a new approach to the hierarchical structuring of various systems. An approach is thoroughly discussed and its theoretical justification is given, in particular, a theorem is proved that gives formalisms for constructing hierarchical structuring. The paper includes a detailed practical example of application of the introduced method. Further ways of development of the presented work are also outlined.