### Article

## Развитие метода анализа иерархий (МАИ)

In the article three combined methods of decision-making, based on the Analytic Hierarchy Process (AHP), are offered. Using these methods it is possible to take into account problem situations and to process the views of several experts. These AHP modifications are adapted to the Expert Decision Support System (EDSS). There is comparative analysis of the results of AHP, combined modifications of AHP and EDSS.

To date, a diverse array of expert assessments for quantitative (tangible) and qualitative (intangible) objects based on the analytic hierarchy process (AHP) has been accumulated in various fields of knowledge. The systematization and analysis of the collected data made it possible to put forward the hypothesis that some of the indicators (the eigenvalue of the matrix, the consistency index, and the ration of consistency) can be considered in the form of aggregates of random variables representing an intellectual product and reflecting features of human thinking. The systematization and statistical processing of the results obtained by experts on the basis of the AHP showed that the distribution functions of the values of expert estimates for the consistency indices significantly differ from the similar functions of the generated matrices.

Analytical justification of solutions using a decision support system (DSS) significantly increases the quality of decisions. The existing DSS generally employs 1 or 2 methods of decision-making. It does not always lead to the desired results, as each method is based on certain assumptions and is not universal. The maximum effect can be achieved only insofar a set of decision-making methods is included into the knowledge base of the DSS. The only system that meets these requirements is the Expert Decision Support System (EDSS) developed under supervision of the author. Currently the EDSS includes about 50 decision-making methods. The expansion of the EDSS knowledge base by including new methods will allow for choosing the most suitable method for solving each decision-making task. Enhancing the Decision Table model underlying the system knowledge base allows you to develop the EDSS without complete reworking of the system code. The system knowledge base contains decision rules built on the principle of “if... then...” (if certain conditions of decision making exist, then a definite method of decision-making should be employed). To expand the EDSS knowledge base, ELECTRE collection methods were selected. Their key feature consists in not using the convolution operation of evaluation of the alternatives specified in different scales on individual criteria. This was the reason for selecting the methods of this family. In the article, the algorithms of these methods are adapted for their inclusion in the EDSS. An algorithm for obtaining a criterion-alternative matrix is proposed. It serves as input information for the ELECTRE family methods in cases where there is no objective information for its formation. The results of the study can be used to develop the EDSS, allowing analytical justification of solutions using methods that have not previously been used in the system.

The most expedient way, in our opinion, to generalize the numerous AHP studies is by the formatting of a morphological

table and the corresponding block diagram. On the basis of the table, it is possible to take into account unobvious

variants that can be missed with a simple search. Thus, the AHP, despite its obvious benefits, requires further

research so as it can be successfully applied in the management of supply chains.

The textbook focuses on information support of a decision making process: problem statement, typical stages, approaches to modeling of decision making conditions, as well as consequences of different alternatives selection. The role of expert estimates is examined. Such estimated are used for determining probabilities of problem situation, determining experts' coefficients of competency, forming estimates of the alternatives. The features of group decision making are considered. Special attention is paid to decision support on the strategic management level, in the conditions of changing and hardly predictable environment. Approaches to modeling of problem situations related with possible states of the environment in the future are examined in details. The history, current classification and perspectives of development of modern decision support systems, as well as their role in the integrated management information system are considered. In the textbook different decision support information systems (including ones developed in HSE Department of Business Analytics) are described. Description of the information systems are accomplished by the examples of their practical application. Business cases that are close to real decision making tasks are also examined. The textbook is compliant with current requirements of the Federal standard of higher education and may be useful for students, postgraduates and teachers working in the field of management, decision support and management information systems.

The paper aims to develop a methodology of business process risk evaluation. Research is based on analytic hierarchy process modification, morphological analysis and testing. Practical approval of the methodology is carried out on the business process “Personnel management”.

In the article the results of the Expert Decision Support System (EDSS) development are presented. It is argued that analytical justification of alternatives may be executed relying on different decision making methods in the conditions of risk and uncertainty. Transferring of EDSS to a new technological platform, as well as addition of the interface in English allows to increase the number of users that can work with the system simultaneously.

Let k be a field of characteristic zero, let G be a connected reductive algebraic group over k and let g be its Lie algebra. Let k(G), respectively, k(g), be the field of k- rational functions on G, respectively, g. The conjugation action of G on itself induces the adjoint action of G on g. We investigate the question whether or not the field extensions k(G)/k(G)^G and k(g)/k(g)^G are purely transcendental. We show that the answer is the same for k(G)/k(G)^G and k(g)/k(g)^G, and reduce the problem to the case where G is simple. For simple groups we show that the answer is positive if G is split of type A_n or C_n, and negative for groups of other types, except possibly G_2. A key ingredient in the proof of the negative result is a recent formula for the unramified Brauer group of a homogeneous space with connected stabilizers. As a byproduct of our investigation we give an affirmative answer to a question of Grothendieck about the existence of a rational section of the categorical quotient morphism for the conjugating action of G on itself.

This proceedings publication is a compilation of selected contributions from the “Third International Conference on the Dynamics of Information Systems” which took place at the University of Florida, Gainesville, February 16–18, 2011. The purpose of this conference was to bring together scientists and engineers from industry, government, and academia in order to exchange new discoveries and results in a broad range of topics relevant to the theory and practice of dynamics of information systems. Dynamics of Information Systems: Mathematical Foundation presents state-of-the art research and is intended for graduate students and researchers interested in some of the most recent discoveries in information theory and dynamical systems. Scientists in other disciplines may also benefit from the applications of new developments to their own area of study.

Let G be a connected semisimple algebraic group over an algebraically closed field k. In 1965 Steinberg proved that if G is simply connected, then in G there exists a closed irreducible cross-section of the set of closures of regular conjugacy classes. We prove that in arbitrary G such a cross-section exists if and only if the universal covering isogeny Ĝ → G is bijective; this answers Grothendieck's question cited in the epigraph. In particular, for char k = 0, the converse to Steinberg's theorem holds. The existence of a cross-section in G implies, at least for char k = 0, that the algebra k[G]G of class functions on G is generated by rk G elements. We describe, for arbitrary G, a minimal generating set of k[G]G and that of the representation ring of G and answer two Grothendieck's questions on constructing generating sets of k[G]G. We prove the existence of a rational (i.e., local) section of the quotient morphism for arbitrary G and the existence of a rational cross-section in G (for char k = 0, this has been proved earlier); this answers the other question cited in the epigraph. We also prove that the existence of a rational section is equivalent to the existence of a rational W-equivariant map T- - - >G/T where T is a maximal torus of G and W the Weyl group.