Gia Sirbiladze

Weighted fuzzy averages in fuzzy environment: part I. Insufficient expert data and fuzzy averages

Three new versions of the most typical value (MTV)1,2 of the population (generalized weighted averages) are introduced. The first version, WFEVg, is a generalization of the weighted fuzzy expected value (WFEV)3 for any fuzzy measure g on a finite set and it coincides with the WFEV when a sampling probability distribution is used. The second and the third version are respectively the weighted fuzzy expected intervals WFEI and WFEIg which are generalizations of the WFEV, namely, MTVs of the population for a sampling distribution and for any fuzzy measure g on a finite set, respectively, when the fuzzy expected interval (FEI)4 exists but the fuzzy expected value (FEV)4 does not. The construction process is based on the Friedman-Schneider-Kandel (FSK)3 principle and results in the new MTVs called the WFEI and the WFEIg when the combinatorial interval extension of a function5 is used.

Fuzzy Dynamic Programming Problem for Extremal Fuzzy Dynamic System

This work deals with the problems of the Expremal Fuzzy Continuous Dynamic System (EFCDS) optimization problems and briefly discuss the results developed by G. Sirbiladze [31]–[38]. The basic properties of extended extremal fuzzy measure are considered and several variants of their representation are given. In considering extremal fuzzy measures, several transformation theorems are represented for extended lower and upper Sugeno integrals. Values of extended extremal conditional fuzzy measures are defined as a levels of an expert knowledge reflections of EFCDS states in the fuzzy time intervals. The notions of extremal fuzzy time moments and intervals are introduced and their monotone algebraic structures that form the most important part of the fuzzy instrument of modeling extremal fuzzy dynamic systems are discussed. New approaches in modeling of EFCDS are developed. Applying the results of [31] and [32], fuzzy processes with possibilistic uncertainty, the source of which is extremal fuzzy time intervals, are constructed. The dynamics of EFCDS’s is described. Questions of the ergodicity of EFCDS’s are considered. Fuzzy-integral representations of controllable extremal fuzzy processes are given. Sufficient and necessary conditions are presented for the existence of an extremal fuzzy optimal control processes, for which we use R. Bellman’s optimality principle and take into consideration the gain-loss fuzzy process. A separate consideration is given to the case where an extremal fuzzy control process acting on the EFCDS does not depend on an EFCDS state. Applying Bellman’s optimality principle and assuming that the gain-loss process exists for the EFCDS, a variant of the fuzzy integral representation of an optimal control is given for the EFCDS. This variant employs the instrument of extended extremal fuzzy composition measures constructed in [32]. An example of constructing of the EFCDS optimal control is presented.

Modeling of extremal fuzzy dynamic systems

Based on the fuzzy-integral model, methods and algorithms are developed for identifying the “input–output” operator of continuous and stationary discrete extremal fuzzy dynamic systems (EFDS). The EFDS “input–output” operator is restored by means of experimental data with possibilistic uncertainty, the source of which is extremal fuzzy time intervals. The regularization conditions for obtaining quasi-optimal estimates are substantiated by the proved theorems. The corresponding algorithms are provided. The results obtained are illustrated by examples in the case of a finite set of EFDS states.

Using a minimal fuzzy covering in decision-making problems

A new criterion is introduced for minimal fuzzy covering problems, which is a minimal value of the average misbelief contained in the possible alternatives. A bicriteria problem is obtained using this new criterion and the criterion of covering price minimization. The proposed approach is illustrated by an example.

Modelling of extremal fuzzy dynamic systems: Part V. Optimization of continuous controllable extremal fuzzy processes and the choice of decisions

This work deals with the problems of optimization of continuous controllable extremal fuzzy processes that have been presented in parts I, II and III of the current paper. A separate consideration is given to the case where an extremal fuzzy control process acting on an extremal fuzzy dynamic system (EFDS) (i) depends and (ii) does not depend on an EFDS state. Sufficient and necessary conditions for the existence of an optimal control are proved. Applying Bellmans optimality principle and assuming that the gain–loss process exists for an EFDS, a variant of the fuzzy integral representation of an optimal control is given for continuous EFDSs. This variant employs the instrument of extended extremal fuzzy composition measures constructed in parts I and II of this work. Finally, the questions of defining a fuzzy gain relation for an EFDS are considered, taking into account the available expert knowledge on the EFDS subject matter.

Fuzzy identification problem for continuous extremal fuzzy dynamic system

This work deals with the problems of the Continuous Extremal Fuzzy Dynamic System (CEFDS) optimization and briefly discusses the results developed by Sirbiladze (Int J Gen Syst 34(2):107–138, 2005a; 34(2):139–167, 2005b; 34(2):169–198, 2005c; 35(4):435–459, 2006a; 35(5):529–554, 2006b; 36(1): 19–58, 2007; New Math Nat Comput 4(1):41–60, 2008a; Mat Zametki, 83(3):439–460, 2008b). The basic properties of extended extremal fuzzy measures and Sugeno’s type integrals are considered and several variants of their representation are given. Values of extended extremal conditional fuzzy measures are defined as a levels of expert knowledge reflections of CEFDS states in the fuzzy time intervals. The notions of extremal fuzzy time moments and intervals are introduced and their monotone algebraic structures that form the most important part of the fuzzy instrument of modeling extremal fuzzy dynamic systems are discussed. A new approach in modeling of CEFDS is developed. Applying the results of Sirbiladze (Int J Gen Syst 34(2) 107–138, 2005a; 34(2):139–167, 2005b), fuzzy processes with possibilistic uncertainty, the source of which are expert knowledge reflections on the states on CEFDS in extremal fuzzy time intervals, are constructed (Sirbiladze in Int J Gen Syst 34(2):169–198, 2005c). The dynamics of CEFDS’s is described. Questions of the ergodicity of CEFDS are considered. A fuzzy-integral representation of a continuous extremal fuzzy process is given. Based on the fuzzy-integral model, a method and an algorithm are developed for identifying the transition operator of CEFDS. The CEFDS transition operator is restored by means of expert data with possibilistic uncertainty, the source of which is expert knowledge reflections on the states of CEFDS in the extremal fuzzy time intervals. The regularization condition for obtaining quasi-optimal estimator of the transition operator is represented by the theorems. The corresponding calculating algorithm is provided. The results obtained are illustrated by an example in the case of a finite set of CEFDS states.

Modeling of extremal fuzzy dynamic systems. Part III. Modeling of extremal and controllable extremal fuzzy processes

New approaches in modeling the so-called extremal fuzzy dynamic systems (EFDSs) are developed. Applying the results of Parts I and II, fuzzy processes with possibilistic uncertainty, the source of which is extremal fuzzy time intervals, are constructed. Fuzzy-integral representations of controllable extremal fuzzy processes (CEFPs) are given. The dynamics of EFDSs is described. Questions as to the ergodicity of EFDSs are considered, using the properties of both extended composition extremal fuzzy measures and -compositions over them (Part II). Sufficient conditions are proved for stationary CEFPs to be ergodic.

Fuzzy-probabilistic aggregations in the discrete covering problem

In this paper, a new criterion is introduced for the discrete covering problem. In this criterion, the a priori information represented by a fuzzy measure and a misbelief distribution on alternatives are condensed by aggregation instruments. Using the Dempster–Shafer belief structure and representations of fuzzy measures through associated probabilities, several variants of a new criterion for the discrete covering problem are constructed based on aggregations by two types of the most typical value, namely, monotone expectation and fuzzy expected value. A bicriterial problem is obtained using one of the variants of the new criterion and the criterion of average price minimisation. The example on the application of a new criterion is presented, where the possibility distribution on the optimal choice of the candidates (alternatives) is represented by expert valuations.

Theory of connectivity and apportionment of representative activity chains in the problem of decision-making concerning earthquake possibility

In this paper a short model to expedite the study of a subjective estimate by means of a qualitative fuzzy technique has been developed, and recommendations for decision-making regarding the three principal elements of earthquakes (time, place and magnitude) have been formulated. Likewise, the problem related to the investigation of the possibility of using Atkins connectivity theory to deal with this subject has been studied. Such a theory gives suitable information for decision-making concerning earthquake possibility in the shape of representative activity chains (precursors), which cannot be obtained by other methods. The most important and general result of this paper is a better understanding of the interaction of geophysical processes. Naturally, the development and application of indiscreet mathematical methods in earthquake prediction require further investigation.

Multistage decision-making fuzzy methodology for optimal investments based on experts’ evaluations

A new methodology of making a decision on an optimal investment in several projects is proposed. The methodology is based on experts’ evaluations and consists of three stages. In the first stage, Kaufmann’s expertons method is used to reduce a possibly large number of applicants for credit. Using the combined expert data, the credit risk level is determined for each project. Only the projects with low risks are selected.