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1

Along with the task of filling the gap in domestic publications about Santorio and his works, which are practically unknown to the Russian medical community, this article aims to discuss the significance of his works for the first scientific revolution of the 17th century. With their research, the authors expand the understanding of this significance and substantiate their own position in assessing the relationship between the contributions of Santorio and Galileo to the initiation of the scientific revolution.

knowledge in the experience of direct communication with nature was obtained through feelings, not reason, and to a large extent<...>and the obvious accuracy of the methods by which he promised to preserve health and direct all therapeutic measures

2

No. 1 [Bulletin of Perm University. Series Mathematics. "Mechanics. Informatics", 2018]

The publication includes original research, review articles, scientific notes relating to all areas indicated in the title of the journal, and above all their current problems and open questions. The journal is of interest to scientists working in these areas, as it provides an opportunity to exchange experiences, as well as for graduate students and students of physics and mathematics specialties at universities. The founder of the journal is the Federal State Budgetary Educational Institution of Higher Professional Education "Perm State National Research University" (formerly the State Educational Institution of Higher Professional Education "Perm state university"), responsible for the publication is the Faculty of Mechanics and Mathematics.

Algorithms for packing k-mers on a square lattice and distributing k-mers across clusters have been developed for the model<...>Horizontal and vertical orientations of k-mers are equally likely. k-mers are uniformly distributed throughout<...>; k – k-mer length; р – specified concentration of k-mers; K – number of tests. k-mers can form<...>side (the direction and origin of the k-measure remain the same); d) if such a k-mer is placed, go to point<...>Algorithm for distributing k-mers across clusters Distribution of k-mers across clusters occurs as follows

Preview: Bulletin of Perm University. Series Mathematics. Mechanics. Informatics No. 1 2018.pdf (0.4 Mb)

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The article is dedicated to the poet, publicist, human rights activist Yuri Timofeevich Galansky and his social activities. The dominant place is occupied by the statements of Y. T. Galansky himself: fragments of his letters, articles, messages to the government and other authorities, as well as his poems.

Well, the “rumors” he mentioned were confirmed: “severe measures” were not long in coming.<...>She will sentence you to the highest punishment that exists for an artist - creative sterility.<...>The fate of Russia largely depends on the nature of the evolution of this party, and on the fate of Russia now<...>In no way do I want to highlight his failure (which is somewhat reminiscent of his youthful short-term breakdown<...>On the eve of the third millennium, they are equally obsolete (in my opinion) as 93 Copyright

4

BIOLOGICAL BASES OF REPRODUCTION OF BEECH FORESTS OF THE CRIMEA ABSTRACT DIS. ... DOCTOR OF BIOLOGICAL SCIENCES

INSTITUTE OF EXPERIMENTAL BOTANY

Study of the reproductive capacity of beech stands as one of the most important conditions the appearance of self-seeding under their canopy showed that beech in Crimea bears fruit relatively weakly. Even the benefits of abundant harvests, which were observed twice during the period from 1957 to 1971, healthy nuts fall no more than 350-400 kg per 1 hectare.

meager:; reserves of fresh water, "and if the problem is with water supply): steppe regions & to a known*|extent<...>*participation. Many prominent scientists and practitioners spoke in favor of the immediate application of measures<...>Among the following: ̂ measures, in the Carpathians the optimum illumination for beech lies within 10-20% (P S.<...>-moisture, the plant does not fully utilize the favorable light regime. Plant response to

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No. 31 [Orthodox community, 1996]

New practical measures were not considered, the only exception being means of struggle.<...>A revolution took place, which should be compared not with Thermidor, but with Brew Mer.<...>Although the announcement is associated with the transfer of some knowledge, it in no way comes down to this.<...>They equally belong to this church people. O. Georgy. Certainly. S. Smirnov.<...>There is a lot more that needs to be done for everything to “work” to its full potential. S. Smirnov.

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No. 11 [Posev, 1961]

However, the government measures that followed after this, mainly in relation to limiting earnings<...>This measure concerns the broad masses of the working people and fundamentally increases their exploitation and<...>The measure was taken, he said, to curb the rapidly growing economic situation.<...>These voices are growing stronger as Germany's armaments grow."<...>Here, at least, everything is clear. There is no dialectical fog...

Preview: Sowing No. 11 1961.pdf (0.5 Mb)

7

Jurisprudence. Part 1 Course of lectures

publishing house LKI

As we move to an essence of a different order.<...>The problem of legal understanding is quite complex.<...>Law is the measure, the scale of human freedom and behavior. 3.<...>Subjective duty is a measure of the proper behavior of a participant in a civil legal relationship.<...>Violation of this prohibition is considered as a basis for the application of penalties.

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No. 1 [Issues of legal regulation in veterinary medicine, 2010]

The journal publishes articles on legal issues in the field of veterinary medicine, agriculture and agro-industrial complex.

Based on the Unified Veterinary Requirements, authorized bodies take measures to prevent the import<...>ARTICLE 8 Each Party has the right to develop and introduce temporary veterinary requirements and measures<...>They were characterized by unclear contours and sharp vacuolization of the cytoplasm, irregular shape of the nucleus, swelling<...>The judge issues a ruling on taking measures to secure the claim (Article 141 of the Code of Civil Procedure of the Russian Federation).<...>The Veterinary Administration continued to take measures to combat contagious animal diseases.

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The emergence of the Russian farming lifestyle

M.: Institute of Sociol. RAS

This book was written based on the results of research in the Krasnodar region, the choice of which as one of the territorial objects for collecting sociological information was largely due to the fact of the rapid development of the modern farming movement there, its noticeable influence on the general course of the evolution of farming throughout Russia.

(in % of the number of respondents) j $er(Workers| measures [Nicky j ;agricultural enterprises 1.<...>(in % of the number of respondents) ’ | 4er|£work №№ I measures j nicknames agricultural p/p i j enterprises I.<...>However, there is progress here: 60% of them are in favor of only this measure.<...>$er-|"Council-(Kres.mery :mery .skie" ;tyav |fer|Not |as a whole) |measures ) -("Soviet.skie", .kres tiane Operating<...>Kree("Soviet (tiane j skie" j ferI: kres1 measures "j tiene I! 2 3! 4 ] 5! 6 I.

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No. 3 [Posev, 1983]

Social and political magazine. Published since November 11, 1945, published by the publishing house of the same name. The motto of the magazine is “God is not in power, but in truth” (Alexander Nevsky). The frequency of the magazine has changed. Initially published as a weekly publication, for some time it was published twice a week, and from the beginning of 1968 (number 1128) the magazine became monthly.

that if they continue to publish their works abroad, action will be taken against them<...>Please take precautions."<...>Not a single Soviet leader, for example, can justify concessions in negotiations through good will and<...>in areas of low fertility and measures of the opposite order in areas of high fertility.<...>sound completely wrong, and if you scrape off the sheen of undigested cliches and vague and indistinct

Preview: Seeding No. 3 1983.pdf (0.6 Mb)

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No. 40 [Orthodox community, 1997]

The magazine “Orthodox Community” was published from 1990 to 2000 by the publishing house of the Moscow Higher Orthodox Christian School (modern name: St. Philaret Orthodox Christian Institute). The editor-in-chief of the magazine is priest Georgy Kochetkov.

There are also other quite primitive things, for example, sex, etc.<...>Averintsev that the devil needs all confusion, all vagueness inside a person, in thoughts, actions,<...>And a person who ceases to know his own limits, that is, who does not know humility, also dies.<...>For example, today's people often do not say that they communicate, they say that they are in contact.<...>It doesn't matter whether you are happy by conventional standards or not.

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No. 27 [Orthodox community, 1995]

The magazine “Orthodox Community” was published from 1990 to 2000 by the publishing house of the Moscow Higher Orthodox Christian School (modern name: St. Philaret Orthodox Christian Institute). The editor-in-chief of the magazine is priest Georgy Kochetkov.

Here a “measure” is given for each person that is feasible and flexible.<...>If you want the “highest measure” for yourself, first of all, show yourself an example of such an attitude towards others.<...>This is the “highest measure”!<...>For example, a person’s faith said one thing, but his life showed something completely different.<...>This would cause at least bewilderment in the interlocutor.

Preview: Orthodox community No. 27 1995.pdf (0.4 Mb)

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No. 1 [Posev, 1996]

Social and political magazine. Published since November 11, 1945, published by the publishing house of the same name. The motto of the magazine is “God is not in power, but in truth” (Alexander Nevsky). The frequency of the magazine has changed. Initially published as a weekly publication, for some time it was published twice a week, and from the beginning of 1968 (number 1128) the magazine became monthly.

I dare to say that in modern Russia at least two very different groups, <...>But it was riots and strikes that showed that the measure of prisoners’ obedience had been exhausted and, in the hope of reducing<...>The measure of human responsibility must begin from childhood and end only with death.<...>Like his father, he was charged with Article 58-10, 58-11 of the Criminal Code, and detention under arrest was chosen as a preventive measure.<...>The youth movement that appeared in the late fifties in Moscow was formed to a certain extent

Preview: Seeding No. 1 1996.pdf (4.8 Mb)

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No. 6 [Energy security in documents and facts, 2007]

The peculiarity of the publication is that it is informative, scientifically sound, and innovative. Only reliable materials that have scientific and practical value are published. The pages of the magazine cover issues of safety and efficiency of energy in all industries, energy saving, labor protection, personnel training, the latest developments of leading industrial and scientific organizations, trends in the development of alternative energy, regulations and documents.

parties to their implementation; execution technical specifications; deadlines network organization Mero<...>insurance carried out by the policyholder in the previous year to finance preventive measures<...>2007 No. 787 "On financing in 2008 and during the planning period 2009 (2010) of preventive measures<...>For example, in case carboxylic acids their equation is as follows: The essence of the equation is<...>Safety measures during equipment flaw detection.

Preview: Energy security in documents and facts No. 6 2007.pdf (0.2 Mb)

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No. 4 [Posev, 1993]

Social and political magazine. Published since November 11, 1945, published by the publishing house of the same name. The motto of the magazine is “God is not in power, but in truth” (Alexander Nevsky). The frequency of the magazine has changed. Initially published as a weekly publication, for some time it was published twice a week, and from the beginning of 1968 (number 1128) the magazine became monthly.

Russian law, which had previously developed for at least a thousand years (sometimes less or more successfully<...>It has long been known (in Russia at least since the time of Speransky, who wrote about it) that it is necessary<...>For example, in Chelyabinsk 15% of voters voted for the regional mini-Khasbulatov.<...>It all depends on the extent of your needs and scope of possibilities.<...>Not in the least.

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Aviation instruments and information-measuring systems. Book 1 [textbook allowance]

Publishing house SSAU

Book 1. Programs used: Adobe Acrobat. Works of SSAU employees (electronic version)

<...>“Science begins as soon as they begin to measure... Exact science is unthinkable without measure,” said the Russian<...> <...>This error decreases as the speed Vв decreases and in horizontal flight Δ kg = 0.<...>α = 0, which is very difficult to ensure, but to reduce the error from friction forces in the supports to a significant extent

Preview: Aviation instruments and information-measuring systems. Book 1.pdf (1.2 Mb)

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No. 6 [Posev, 1994]

Social and political magazine. Published since November 11, 1945, published by the publishing house of the same name. The motto of the magazine is “God is not in power, but in truth” (Alexander Nevsky). The frequency of the magazine has changed. Initially published as a weekly publication, for some time it was published twice a week, and from the beginning of 1968 (number 1128) the magazine became monthly.

Revived, often through coercive measures, culture is a version of a cheap booth with<...>This is manifested to the highest degree when this denial is implicated in bitterness and lies.<...>We will correct our mistake to the extent that it occurred if we do not give up ours to anyone.<...>After all, there are nations that seem to be “dashing” and “quiet,” at least outwardly.<...>The experience of the pre-revolutionary zemstvo must be fully used.

Preview: Sowing No. 6 1994.pdf (0.5 Mb)

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No. 2 [Posev, 1992]

Social and political magazine. Published since November 11, 1945, published by the publishing house of the same name. The motto of the magazine is “God is not in power, but in truth” (Alexander Nevsky). The frequency of the magazine has changed. Initially published as a weekly publication, for some time it was published twice a week, and from the beginning of 1968 (number 1128) the magazine became monthly.

And x inhabitants have their own territories, but the boundaries of the territories are blurred, unclear, people easily settle intermixed<...>In some ways, the Zhirinovsky-LDP phenomenon can be a classic example.<...>At least in Russia. After all, in Rousseau’s ideology there is no such thing as sin.<...>But this growth was largely inflated.<...>Hence the omissions, misunderstandings, unclear formulas, and internal contradictions.

Preview: Seeding No. 2 1992.pdf (0.3 Mb)

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No. 8 [Legality, 1990]

As you know, over the past decade and a half in Russia, legislation has been actively updated, on some issues - radically, many legal institutions are undergoing significant changes, and new ones are being introduced. During this time, the magazine’s pages published many discussion articles about the place and role of the prosecutor’s office in our society and state, dedicated to judicial reform, the new Criminal Procedure Code, jury trials, investigation reform in the prosecutor’s office, etc. But this was never to the detriment of materials about the exchange experience and comments on legislation, complex issues of law enforcement practice. Essays on highly acclaimed prosecutors are also regularly published. The magazine has an established team of authors, which includes well-known scientists and passionate law enforcement officers from almost all regions of Russia.

Let's consider the problems on such np11Measure.<...>We also accept an exceptional measure of punishment.<...>For crimes against property, for example, such a measure is cost.<...>For example, measures of social impact!<...>The measures provided for in Article 4 are present!

Preview: Legality No. 8 1990.pdf (0.4 Mb)

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No. 1 [Problems of social hygiene, health care and history of medicine, 2013]

An important component of the entire complex of measures is the presence of doctors of different specialties in AIDS centers<...>This position of the medical faculty forced Louis XVI to take drastic measures.<...>The study of epidemic constitutions and the implementation of effective measures to prevent diseases required<...>This circumstance was largely associated with the reorganization of the maintenance and treatment of mental health<...>Copyright JSC Central Design Bureau BIBKOM & LLC Book-Service Agency 58 MM England and France increasingly

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No. 6 [Problems of social hygiene, health care and history of medicine, 2015]

Founded in 1994. The editor-in-chief of the magazine is Oleg Prokopievich Shchepin - academician of the Russian Academy of Medical Sciences, Doctor of Medical Sciences, professor, scientific director of the National Research Institute of Public Health of the Russian Academy of Medical Sciences. The magazine covers theoretical issues of social hygiene, the main directions of formation of public health and medical and social assistance, issues of economics, scientific organization of labor, sanitary statistics, history of medicine and healthcare. Publishes articles on new forms and methods of work of medical and anti-epidemic health care institutions in organizing health care services for urban and rural populations. The journal publishes materials on methods and results of studying the social conditions of life and health of the population. It reflects the state of health care, issues of organization and operation of medical institutions in foreign countries, and contains articles on the design and equipment of medical institutions. The development of medical science and healthcare is widely covered, important historical dates and the activities of scientific societies are noted, and information about various conferences and meetings is published.

As we moved north, the incidence increased.<...>The results obtained to a certain extent correlate with data previously obtained using the example of the USSR<...>Fischer, the whole variety of measures developed by “breeding hygiene” was grouped into four<...>IN Russian Federation certain measures are being taken to reduce both the number of road accidents and their severity<...>The only measures to prevent them is to prevent road accidents.

Preview: Problems of social hygiene, healthcare and history of medicine No. 6 2015.pdf (0.4 Mb)

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No. 1-2 (38-39) [Yaroslavl Pedagogical Bulletin, 2004]

The scientific journal “Yaroslavl Pedagogical Bulletin” has been published since 1994 and is the first scientific journal in the Yaroslavl region, which publishes articles on various branches of science. The journal is included in the list of leading peer-reviewed scientific journals and publications in which the main scientific results of dissertations are published scientific degree Doctor and Candidate of Sciences. Materials published in the journal are reviewed by members of the editorial board.

...Whoever raises this question should learn that the stomach could not expand sufficiently and<...>The ideas proposed in it by the Empress were not fully accepted by the delegates of the Legislative Commission<...>To some extent, supervision and control were still carried out.<...>Alcohol consumption statistics: are Russians a nation of alcoholics or a “moderately drinking” people? 2.<...>Sobriety movement in Russia // Proceedings of the commission on the issue of alcoholism and measures to combat it.

Preview: Yaroslavl Pedagogical Bulletin No. 1-2 2004 2004.pdf (0.1 Mb)

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Fundamentals of system analysis and management of organizations: theory and practice

M.: DMK Press

The features of formalization and solution of system problems in the management of organizations are considered, practical recommendations are given on the formulation of various system problems, the creation of models based on the use of modern Fuzzy technology approaches, and the solution of problems of analysis and synthesis of systems. The concepts of observation channels and system behavior functions are given. A significant place is occupied by the mathematical foundations of solving system problems. Methods and approaches to solving problems of reconstructive analysis, optimization of goal-oriented systems and other problems of analysis and synthesis of systems are presented. The book includes five topics. The material is presented in the form of theoretical material and practical tasks that allow you to obtain the necessary amount of knowledge in the areas of system analysis and synthesis of organizational management.

<...>A fuzzy confidence measure is a superadditive fuzzy measure.<...>Formalization of fuzzy measures. Fuzzy Sugeno measures (M.<...>fuzzy measures.<...>Fuzzy Sugeno measures are most often used. These measures are called fuzzy gλ-measures.

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The Choquet integral over a fuzzy measure is a generalization of the weighted average aggregation operator and allows one to take into account the phenomenon of interdependence of criteria when aggregating. Thanks to this, it will be possible to more adequately reflect the expert’s knowledge without introducing simplification into the model, which is expressed in the assumption of independence of aggregation criteria. The difficulties of using fuzzy measures and the fuzzy Choquet integral and possible ways to overcome them are considered. Review completed practical applications this relatively new device.

<...>The difficulties of using fuzzy measures and the fuzzy Choquet integral are considered and possible ways overcoming them<...> <...>Although the theory of fuzzy measures and the theory of fuzzy sets were not directly related, they go well together<...>

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M.: Publishing house of MSTU im. N.E. Bauman

Information retrieval based on weighted zone ranking involves assigning a weight to each zone or field in document metadata using machine learning methods. A method for determining weights is considered, in which the fuzzy Choquet integral is used to calculate the weighted zone relevance instead of the weighted average operator. This allows you to take into account possible interdependencies between zone indicators when calculating relevance, which will ultimately increase the accuracy of the ranking.

<...> <...>An alternative to the weighted average operator can be the Choquet integral over a fuzzy measure.<...> <...>Fuzzy measure identification with weighted zone ranking.

26

M.: Publishing house of MSTU im. N.E. Bauman

A grouping of methods and algorithms for information integration is proposed; methods and algorithms for information integration at the decision level are considered. A new multiclassification algorithm FuzzyBoost is presented, which implements the fuzzy boosting method. The FuzzyBoost algorithm provides the construction of a quasi-linear composition and is based on the AdaBoost algorithm, supplemented by the calculation of a fuzzy integral instead of AdaBoost’s own linear aggregation rule at each boosting iteration. Experimental results showed that in the case of a complex class separating surface, the FuzzyBoost algorithm has better generalization ability than the AdaBoost algorithm.

additional information presented in the form of fuzzy measures characterizing the degree of trust or "<...>measures for the corresponding combinations of base classifiers.<...>fuzzy measures ()()mAσμ .<...>measures<...>Calculate the initial data for the subsequent calculation of fuzzy measures +μ and −μ in accordance with their type and property

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M.: Publishing house of MSTU im. N.E. Bauman

The issues of assessing the effectiveness of the implementation of information systems in enterprises are considered. An expanded approach to assessing the effectiveness of implementation is proposed, based on the aggregation of implementation effectiveness indicators. Some indicators have thresholds that they must reach at the end of implementation to be considered successful. The issues of normalization of performance indicators for the implementation of information systems are considered. A generalized indicator of the effectiveness of information systems implementation based on the Choquet integral is proposed. The situation of dependence of indicators is considered, it is noted that taking into account dependencies allows us to build more accurate models for assessing the effectiveness of implementation.

Keywords: information system, implementation efficiency, aggregation operator, fuzzy measure<...> <...>A fuzzy (discrete) measure is a function of a set:  2 0, 1 ,J  where 2J is the set of all subsets<...>The fuzzy (discrete) Choquet integral of the exponents 1, ..., Hg g with respect to the fuzzy measure  is determined by the expression<...>Let's consider methods for identifying a fuzzy measure, where input information can be represented by signs

28

The oil production process is complex and ambiguous, occurs under conditions of uncertainty and requires accurate knowledge of all internal and external factors. However, in many cases it is impossible to obtain complete information. Partial lack of knowledge and vagueness are some of the aspects of uncertainty. Zadeh L. proposed the concept of Z-number based on the reliability of given information. In this paper, we use Z-information for decision making in oil production problems and propose a decision-making framework based on Z-numbers. The method is associated with the construction of a non-additive measure, lower prediction and its use in the Choquet integral to construct a utility function.

<...> <...> <...>Let, .nV W   A fuzzy measure with a fuzzy numerical value ((z) is a fuzzy measure) on  is a function<...>Now we can construct a fuzzy measure with a trapezoidal membership function from a fuzzy set

29

Modeling of food product recipes and technologies for their production: theory and practice. allowance

SPb.: GIORD

The book allows students to master information technologies for developing models of food product recipes, methods of mathematical programming of functional and technological properties of multicomponent recipes, including taking into account the interaction of their components; it is written in accordance with the State educational standard.

Fuzzy measures of similarity between a sample and a standard. . . . . . . . . . . . . . . . . . . . 221 Chapter IV.<...>fuzzy measure mPM uncertainty PM.<...>The physical meaning of the introduced membership measure is that it determines a fuzzy measure of connection<...>Let us summarize the calculated values ​​of the fuzzy similarity measure - the multiplicative estimate ρ - in table. 3.2.<...>Give an equation for a fuzzy measure of similarity between the vectors of experimental and control samples. 7.

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The issues of aircraft flight safety management are considered from the standpoint of the theory of highly reliable technical systems with discrete states defined in fuzzy subsets of the original universal set of elements. It is proposed to assess the risks of the occurrence of critical conditions under which aircraft may end up in catastrophic scenarios depending on the combination of hazardous factors.

Here it is proposed to assess the risks of consequences using the concept of risk as a measure of danger<...>Risk is a fuzzy measure of the amount of danger in STS states with an identified threat and hazardous factors (<...>Chance is a fuzzy (predictable) measure of the amount of “luck” in an experience or in the state of a system under conditions<...>a measure of the level of capabilities being studied.<...>For situations with rare events, the following must be assumed: risk is a fuzzy measure of the amount of danger

31

No. 1 [Engineering magazine: science and innovation, 2012]

M.: Publishing house of MSTU im. N.E. Bauman

Alfimtsev ON THE QUESTION OF THE PRACTICAL APPLICATION OF FUZZY MEASURES AND THE CHOQUET INTEGRAL Choquet integral over fuzzy<...>Email: [email protected] Keywords: aggregation operator, fuzzy measure, fuzzy Choquet integral<...>Let's consider the basic concepts used in the theory of fuzzy measures.<...>In the context of the theory of fuzzy measures, the Shapley index for criterion i J∈ with respect to measure ψ is determined by the expression<...>measures of κ-th order or κ-additive fuzzy measure, where the order of κ is less than the number of aggregates

32

No. 3 [Bulletin of Moscow State Technical University named after N.E. Bauman. Series "Instrument Engineering", 2012]

M.: Publishing house of MSTU im. N.E. Bauman

aggregation, Choquet fuzzy integral, Sugeno fuzzy integral, fuzzy measure.<...>fuzzy measures.<...>Fuzzy measures and integrals.<...>A fuzzy measure is called a gλ-fuzzy measure if the condition holds for it: for all Q,P ⊂ Y such<...>using fuzzy measures and integrals.

Preview: Bulletin of MSTU im. N.E. Bauman. Instrument Engineering Series No. 3 2012.pdf (0.1 Mb)

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M.: Publishing house of MSTU im. N.E. Bauman

IN last decades information systems have become widespread. Almost every enterprise in one form or another uses an information system in its work. At the same time, there are a number of unresolved problems associated with the implementation of such systems. One of these problems is the lack of general formal models and methods for assessing the quality of implementation, which would allow making informed management decisions and assessing the true effects of implementing an information system. The article formulates the concept of the quality of implementation of an information system and provides indicators of the quality of implementation. A model for assessing the quality of information system implementation based on aggregation of quality indicators is considered. This model includes aggregation of indicators using the Choquet integral. Using an example from the applied area, it is shown that implementation quality indicators can be interdependent. The Choquet integral, unlike traditional aggregation operators, allows one to take into account possible mutual influences of these indicators.

at least<...> <...> <...> <...>,G G , it will be natural to apply the least squares method to identify fuzzy measures 1 4, ..., 

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An original approach to finding the maximum independent set (maximum clique) in a fuzzy graph is presented. The approach is based on the representation of fuzzy relations by formulas of multi-valued logics И廊. Lukasiewicz and using them to interpret modal relations. A modality like “possibly” is interpreted by a three-digit calculus formula with a truth value of at least 0.5; a modality of the “necessary” type is interpreted by a three-valued calculus formula with a truth value equal to 1. Rules for the calculus of inferences in fuzzy modal systems have been introduced, allowing one to find three-valued equivalents of arbitrary modal formulas.

Key words: graph, maximal independent set, clique, fuzzy clique, fuzzy logic.<...>programming for graphs corresponding to different gradations (levels) of a fuzzy measure.<...>not connected by a fuzzy edge.<...>There are no fuzzy edges.<...>measures

35

Based on the principles of synergetics, innovative approaches to the formation of a classification of pedagogical dimensions as one of the most important elements of the modernization of domestic education are outlined. The classification is based on a system of psychological principles containing the anthropological principle of Konstantin Dmitrievich Ushinsky, the principle of economy of thinking of E. Mach, the principles of self-organized criticality and functional specialization of the cerebral hemispheres. The principles of classification reflect certain properties of human activity, in which two types of logical thinking are distinguished - formal and intuitive, which determine the classification according to the type of logic implemented in the process of measuring the object in question.

Shannon based on the stochastic measure.<...>The meaning of the term “fuzzy” is also fuzzy, but this usually means indeterminacy<...>Examples of implementation of pedagogical measurements based on fractal and fuzzy measures. Example 4.<...>Fuzzy measurements in the learning process.<...>Distinction between fuzzy and stochastic measures.

36

No. 3 [Engineering magazine: science and innovation, 2012]

M.: Publishing house of MSTU im. N.E. Bauman

“Engineering Journal: Science and Innovation” is a scientific and practical publication that publishes original (i.e., not published in other publications) articles containing the results of scientific research in all sections stated in the rubricator. The choice of the electronic form of the publication was determined by the need to quickly introduce the results of scientific research into scientific circulation, which corresponds to the trend to make the results of scientific work paid for by the state into the public domain. This also presupposes that the editors of the journal choose free access to its content.

measure, fuzzy Choquet integral.<...>Fuzzy measures and the Choquet integral.<...>The fuzzy (discrete) Choquet integral from criteria 1, ..., Hs s with respect to the fuzzy measure ψ is determined by the expression<...>Fuzzy measure identification with weighted zone ranking.<...>fuzzy measures ()()mAσμ .

37

The features of oilfield information and possible approaches to classifying the sources of imperfections existing in oil and gas production are considered. The principles of modeling field data using fuzzy numbers are described, leading to the formulation of a wide range of parametric identification problems in the form of multicriteria optimization problems. A formal description of the fuzzy maximum likelihood principle using the averaging aggregation operator for the f-regression problem is given. The conditions for obtaining estimates of model parameters close to the true values ​​are listed. A numerical example demonstrates the correctness of theoretically based conclusions and properties of f-estimates.

<...> <...> <...>The fuzzy implication A → B is a measure of the truth of the statement "B is at least as true as<...>the need to pass a straight line through a fuzzy point, complementing the measure of possibility (7).

38

The article proposes a methodology for assessing operational safety sea ​​vessels based on predicting the possibility of dangerous encounters when maneuvering standards are violated in a system of two objects - sea vessels. It is established that the provisions of the risk-oriented approach to the analysis of properties rare events, developed in aviation, are also applicable to maritime transport

In this case, the category “risk” is defined, according to the works of the Institute of Management Problems (IPU) RAS, as a measure<...>a measure of the level of capabilities being studied without using the traditional probabilistic concept.<...>models on fuzzy subsets of objects.<...>Probability is a measure of the randomness of the occurrence of an event; but this measure is non-random and clear, determining<...>Similarly, you can introduce an additional concept in the form of “chance is a fuzzy (predictable) measure of quantity

39

No. 9 [Automation, telemechanization and communications in the oil industry, 2016]

The main advantage of the Choquet integral is the use of a fuzzy measure to evaluate the relationship between<...>fuzzy reliability or fuzzy probability for such a value.<...>The fuzzy measure is calculated based on the given Z information.<...>Let, .nV W   A fuzzy measure with a fuzzy numerical value ((z) is a fuzzy measure) on  is a fuzzy function

Based on an analysis of existing definitions using the concept of a critical set of objects, the authors formulate the concept of “critically important object”

Then the system efficiency indicator is the system damage US(M), (a1) M M, determined by the fuzzy<...>Then, under the accepted restriction on the set M, the fuzzy measure ν(M), and with it the system damage US<...>sets from the family of so-called gν-measures 4 under the restriction a1ϵ M.<...>, when the system efficiency indicator is represented by an integral over a fuzzy measure 5.<...>Fuzzy sets in control and artificial intelligence models.

42

No. 3 [Bulletin of Moscow State Technical University named after N.E. Bauman. Series "Instrument Engineering", 2013]

M.: Publishing house of MSTU im. N.E. Bauman

Issues in the following areas are covered: information science and computer technology; control systems; radio electronics, optics and laser technology; gyroscopic navigation devices; instrumentation technology, biomedical engineering and technology.

fuzzy sets.<...>Learning based on conditional fuzzy measure.<...>Let Gy be a fuzzy measure on Y, Gy is related to Gx by a conditional fuzzy measure σY (∗Ix): GY = .∫ X σY (∗Ix)Gx.<...>The following interpretation of the introduced measures is assumed: Gx evaluates the degree of vagueness of the statement “one<...>The teaching method must meet the mandatory condition: when obtaining information A fuzzy measure

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43

Methods for modeling the probability of events based on the analysis of the “tree” of incidents and events method. instructions

The guidelines provide rules for constructing an incident tree and an event tree, qualitative analysis of tree-type models, quantitative analysis of tree-type diagrams, illustrative tree-type models, testing of methods for qualitative and quantitative analysis of tree-type diagrams, as well as tasks for independent solutions and questions for self-preparation . When developing methodological guidelines, the works of Belov P.G., Gorsky V.G. were used. and other authors.

Despite these safety measures, the possibility of exposure to rolling stock could not be completely ruled out.<...>Names of the initial premises of the incident under consideration and fuzzy measures of possibility P; their appearance<...>Therefore, to determine the extent of the possibility of a critical situation occurring, it is necessary to use<...>This illustrative example indicates a measure of the possibility of injury to the leuise, estimated by the range<...>A measure of the development of society./ M.I. Gvardeytsev. M.: Radio and communications. 1996. – 325 p. 4 Gelfand, B.E.

Preview: Methods for modeling the probability of events based on the analysis of a tree of incidents and events.pdf (0.7 Mb)

44

M.: Publishing house of MSTU im. N.E. Bauman

The problem of choosing means of protecting information from various attacks in an automated system is considered: a mathematical formulation of the problem is performed in the form of a fuzzy mathematical programming problem with Boolean variables. An efficiency indicator has been introduced, determined through an assessment of the average damage prevented when using the selected means of protection, for the calculation of which fuzzy parameters are used. The total cost of the selected means of protection is used as constraints in the problem. An approach to solving this problem is proposed, and an example of a solution is considered.

Gurov THE PROBLEM OF SELECTING INFORMATION PROTECTION MEANS FROM ATTACKS IN AUTOMATED SYSTEMS UNDER FUZZY<...>, fuzzy mathematical programming.<...>measure) to prevent consequences i-th attack using the j-th means of protection, determined according to statistics<...>Let us analyze the features of a fuzzy description of parameters. Fuzzy description of parameters.<...>Problem (3) with fuzzy parameters,ijp ,i N∀ ∈ j M∈ is a fuzzy mathematical programming problem

45

No. 2 [Bulletin of the Astrakhan State Technical University. Series: Management, computing and information science, 2019]

Main headings: Control and modeling of technological processes and technical systems; Computer software and computing technology; Telecommunication systems and network technologies; Management in social and economic systems

The general fuzzy measure is constructed as an additive union of particular measures.<...>Key words: personnel management, goal, criteria, alternative, fuzzy measure, expert group<...>It has been proven that the measure )(.g satisfies all the axioms of the fuzzy measure .<...>Using a fuzzy measure of the value of criteria in multicriteria selection // Automation.<...>Application λ- 47

No. 6 [Automation, telemechanization and communications in the oil industry, 2016]

Development and service measuring instruments, automation, telemechanization and communications, process control systems, information systems, CAD and metrological, mathematical, software

 – T-norm, intersection operator of fuzzy sets or measures, fuzzy logical “AND” (see.<...>Fuzzy likelihood principle Having expression (7) for the similarity measure  M a between the fuzzy point Q<...>The kth fuzzy point with the model will generally lead to a decrease in the similarity measures of other points.<...>The fuzzy implication A → B is The measure concerned the fishing industry of the Far East.<...>These issues have been addressed to one degree or another in a number of issues of TAE.<...>Pesotsky measures, as in the case of V.<...>Cruel measures often yielded positive results.

Preview: Ecumene. Regional studies No. 3 2010.pdf (0.8 Mb)

49

No. 11 [Engineering magazine: science and innovation, 2013]

M.: Publishing house of MSTU im. N.E. Bauman

“Engineering Journal: Science and Innovation” is a scientific and practical publication that publishes original (i.e., not published in other publications) articles containing the results of scientific research in all sections stated in the rubricator. The choice of the electronic form of the publication was determined by the need to quickly introduce the results of scientific research into scientific circulation, which corresponds to the trend to make the results of scientific work paid for by the state into the public domain. This also presupposes that the editors of the journal choose free access to its content.

A fuzzy measure is a function of a set: 2 J  , where 2J is the set of all subsets of the set)<...>Unlike the weighting coefficients in the weighted average operator, the fuzzy measure expresses the relative<...>The Choquet integral over a fuzzy measure has the form      1 () () (1) 1 , ..., : , H H h h h h C g g g A<...>An alternative to the weighted average operator is the fuzzy discrete Choquet integral over the fuzzy measure [<...>The fuzzy (discrete) Choquet integral of the exponents 1, ..., Hg g with respect to the fuzzy measure  is determined by the expression

The experience of existing work allows us to draw the following conclusions about the possibility of using these methods to study railway embankments.

For the PGZ method:

> confident study of the structural features of the upper part of railway embankments to a depth of 1-10 m (depending on humidity, soil salinity) or to the roof of loamy soils, which are an absorbing medium for electromagnetic waves;

> continuous inspection of railway embankments;

> cost reduction by reducing the volume of mining and drilling work, reducing the time to obtain the final result of survey work, no need to interrupt train traffic;

> increasing the safety of rolling stock movement through non-destructive inspection techniques;

> reduction of errors when analyzing the causes of deformations and, accordingly, in making design decisions. For example, embankment subsidence, possible

dead after overhaul, due to the lack of information about the shape of the roof of loamy soils.

For the EDS method:

> prompt determination of the depth of the roof of loamy soils;

> obtaining physical and mechanical properties of soils in field conditions;

> use of the obtained results to adjust the data of the PGZ method;

> study of the embankment to a depth of 15m, which is limited by installation capabilities.

The last of these arguments does not apply to soils containing more than 10% coarse inclusions.

The disadvantage of both methods is their limited use in depth and strong dependence on the composition of the soil. In this regard, it is necessary to use these methods in combination with shallow seismic and electrical prospecting, which will increase the depth of research to tens of meters.

The article was accepted for publication on June 29, 2006.

S.A. Sakulin

Visualization of the aggregation operator based on the Choquet integral over an odd measure of 2nd order

Aggregation of numerical criteria is a method of combining them into one numerical criterion (the result of aggregation) to express the combined effect of these criteria. Aggregation is used in fuzzy inference and recognition, multicriteria decision-making problems. An aggregation operator is often called one that has certain specified

properties of the operator ACC:i -", where N

Number of criteria. Some of these properties are constant and correspond to the selected type of aggregation operator. The remaining properties are set by the expert based on his vision of the criteria aggregation process. The properties specified by the expert are expressed using the parameters of the aggregation operator, while the constant properties of the operator do not depend on the values ​​of these parameters.

There is currently no general formal approach to constructing aggregation operators based on expert knowledge; work is underway in this direction. To formally define the aggregation operator, sets of fundamental conditions are proposed. It should be noted that these sets of conditions are not compatible with each other. A set of less stringent conditions is proposed, in accordance with which

The aggregation operator AGG of criteria gH is defined as follows: Definition 1 The aggregation operator AGG is a function i -> satisfying the following conditions:

Identity in the case of unary: if H = 1, AGG = gH;

Boundary conditions:

AGG = 0; AGG[ 1,..., l] = l;

Non-decreasing: gH)<{g[ g"H)^>

AGG.

We will stick to this definition. All additional conditions imposed on the aggregation operator will be added to those listed and correspond to the preferences of the expert.

Criteria are independent if the effect on the result of aggregation caused by a change in each of them (with fixed values ​​of the other criteria) does not depend on the values ​​of the other criteria.

riev , Otherwise the criteria are dependent. In general, the criteria are also dependent.

To reflect expert knowledge about the dependencies between criteria, the concepts of fuzzy measure and fuzzy integral are used.

Definition 2 A fuzzy (discrete) measure is

function y/: 27 -> , where 2") is the set of all subsets of the set of criterion indices Y - (1,..., H), which satisfies the conditions: y/(0) = O, = £>сЯ =><^(Я)

We will omit the curly braces and write /, I] instead of (/), (/,у) respectively. Instead of

For the sake of brevity, we will also use “criterion I” to denote “criterion with index / e 3”.

In general, a fuzzy measure is not additive, or

y/(p)l-y/(B~)Fu/f^B) where D Vs/; £>nB = 0. The value of the measure u/f) can be interpreted as the “weight” or “importance” of subset O of the set of criteria Y.

Let yс(7-(r" and y)). Then the criteria / and y interact positively (or, following the terms of game theory, tend to cooperate) if the local contribution of the criterion y" to any subset of criteria,

u/f and / and y) - u/f and 0 > y/(O and y) -u/f)- (1) Criteria / and y are independent if equality occurs

u/f and I and y) -u/f and 0 = y) -^f). (2)

Criteria / and y interact negatively (or, following the terms of game theory, have a tendency opposite to cooperation) if the local contribution of criterion y to any subset of criteria containing

criterion I, less than the local contribution of criterion y to the same subset, where criterion r is excluded: u/f ugiD-^fi 0<у/(£Юу)-у/(£>)" (3) Migo^Y and Bopes1a proposed the following definition of the index of interaction between criteria I and y:

„ (Y-|L|-2)!|1)|!G. (4)

I PI L, 1 and y) - q, (B and |) - y (D and L + y(£>)]

This index is interpreted as a weighted average of the total impact produced by the criteria / and y, placed together, in all

considered combinations, When the index /(?",./) is positive (negative), the relationship between criteria I and y is called positive (negative).

The index of interaction among subset criteria was introduced in 1997 as a natural generalization of the special case when |2?| = 2:

Correlation is the best known and most intuitive of the relationships between criteria. Two criteria r, y and y are positively correlated if the expert can observe a positive correlation between the contributions to the aggregation result associated with the criteria r and y, respectively.

A positive correlation between the criteria will then be expressed by the inequality y/(y)< УЧО + УО) С учётом других комбинаций, если критерии I и у положительно коррелированны, то локальный вклад критерия у в любую комбинацию критериев, содержащую критерий I, строго меньше, чем локальный вклад критерия у в той же самой комбинации, где критерий I исключён, то есть справедливо неравенство (3).

Now suppose that the criteria / and y are negatively correlated, then y/(r, y) > y/(r) + y (y), taking into account other combinations, inequality (1) is satisfied. If the criteria / and y are not correlated,

equality (2) is true.

Another type of dependence is substitution (interdependence) of criteria. Let us consider again the criteria r and y. Suppose the expert believes that satisfying only one criterion produces almost the same effect as satisfying both.

Here, the importance of a pair of criteria y is close to the importance of each of them separately, even in the presence of other criteria. In this case, we observe that the criteria / and y are almost replaceable or interchangeable. In this case, just as in the case of positive correlation of criteria, inequality (3) is satisfied.

Conversely, an expert may require that satisfying only one criterion may produce a very weak effect compared to satisfying both. Then we can talk about their interdependence, modeled by the fuzzy measure y/ so that

inequality (1).

Note that, unlike the phenomenon of criterion correlation, substitution and interdependence between criteria cannot be detected through statistical observations. They only represent the expert's opinion about the relationship between the importance of the criteria, regardless of the contributions of these criteria to the aggregation result,

Preferential dependence of criteria and its opposite - preferential independence - are well known in utility theory. Suppose

that the expert’s preferences on the set of implementations of criteria A are known and expressed by a relation of non-strict order. Let us denote by g£) the implementation of criteria gi, where /e/), and let us denote by gJ_D the implementation of criteria g¡, where ge3-V.

Definition 3 A subset of criteria B a3 is said to be preferably independent of the subset J - D if and only if, for each pair of criteria implementations, from

(%D>£J-D)t.(%"D,%J-D) for some implementation follows The Alya of all realities

lizations g/_¿), where denotes the preference relation (non-strict order) on A. Otherwise, the subset of criteria B c: 3 is preferentially dependent on the subset 3 - /),

Fuzzy Choquet integral (SIocie!), introduced in 1974 Bidepo based on non-additive Choquet measures, is used as an aggregation operator that allows you to reflect the expert’s knowledge about the dependencies between criteria by selecting the values ​​of the corresponding parameters. Its use for constructing aggregation operators of dependent criteria is discussed in. In particular, preferential criterion independence, modeled using the Choquet integral, is considered in .

Definition 4 Fuzzy (discrete) Choquet integral of criteria g1,..., gн with respect to a fuzzy measure

y/ e ^ is determined by the expression

where (*) means a permutation of indices in Y such that - - X(H)» 4n) = ((A),..., (I)) and

The Choquet integral has the following properties

Satisfaction of boundary SYN(0,..., 0) = 0, SYD1,..., 1) = 1;

Non-decreasing:

Idempotency:

I, = £2 = = FROM, =

From these properties it follows that the Choquet integral corresponds to our accepted definition of the aggregation operator. For reflection during aggregation, expert

If you have sufficient knowledge about the dependencies between the criteria, you need to specify a fuzzy measure y/.

A fuzzy measure can be represented in a unique way so that = ^ a(B), where

Ss/; a(O) is a set function on 3, which in combinatorics is called the Möbius function with respect to y/ and is expressed by the formula:

af) = £ (-1)Х%(£>), where in с 3. Not every

a set of 2 coefficients π(ξ>) can represent a fuzzy measure y/, the boundary conditions and the monotonicity condition must be satisfied:

a(0) = 0; ] >(£>) = 1;

The fuzzy measure y/ is additive if y/φ) + y/(B) = \1/(ωB), where D1)n5 = 0. In this case, to specify it, you need to set the values ​​of the weights: y/(H). In the general case, it is necessary

you can set 2 weight values ​​corresponding to

2 subsets of set 3.

It is obvious that even with a relatively small

number of criteria Н = \з\ the expert is not able to give

so much information. In addition, the meaning of the u/f values ​​is not always clear to the expert. In many cases, an expert is able to judge the importance of individual criteria or pairs of criteria, but not the importance of subsets of criteria consisting of a larger number of them. And conversely, if a fuzzy measure is given, the expert is not able to judge its values ​​in terms of his own subject area,

In order to overcome the problem of formalizing expert knowledge with a large number of values

weights (2i), bgaYzsb proposed the concept of fuzzy conditions: measures £. th ORDER £< |У| = Я . Суть этой концепции заключается в том, что для упрощения задания нечётких мер из рассмотрения исключаются зависимости между более чем к - критериями.

Let us consider the 2nd order case, which, in accordance with the above considerations, is the most interesting from a practical point of view.

really, only

N + Сgn=Н+-

2!(I -2)! 2 coefficients are necessary in this case to determine the value of the fuzzy measure, namely:

1/(0 = a(i), i€ J; y/(ij) = ail) + a(j) + ci(ij), (i,j)œ3. The remaining coefficients are then:

Note that the second order case is equivalent to accepting that the interaction index I(B) is equal to

zero for subsets consisting of at least three elements. In this case, the Choquet integral will take the form:

Index of interaction between criteria / and y: I(i,j) = a(ij), (/,y")eY. Note also that a(i) e [OD] for all y e J, I(i, j) e [-1,1] for all (r,y) e Y . Finally, in this context, conditions (6) for the coefficients a(0), a(i), a(i,j), ((i, j)ej), defining the fuzzy measure, take the form:

a(0) = 0; 2>(0+ X *G0 = 1

a(i) > 0 Vi e J (9)

a(i) + £ a(ij) > 0, Vi e J, Vi) with У - (/)

Let us return to the previously discussed dependencies between the criteria for the case of a 2nd order model.

Let Z)c;(/-(iuу")), then based on (11) we

we can write expressions for the 2nd order fuzzy measure of the corresponding subsets:

y(B)=^a(p) + X

/>s=Z) (p,q)c,D p&D

J^a(p) + £ «(/>

pv-D 1p.<})£й peD p*D

If criteria i and y are positively correlated, inequality (3) is satisfied; Substituting expressions (10), (11), (12), (13) into it, we obtain:

^a(pL + ai) + a(d)<^а(рЛ+а(Л ^ «G0< 0.(14)

Therefore, to reflect the positive correlation of criteria i and y in the case of a second-order model, it is sufficient to set the interaction index I(ij) = a(ij)< 0, не принимая во внимание остальные критерии и зависимости.

In the case of a negative correlation between criteria i and y, the index of their interaction will be set to I(ij) > 0, which, similarly to (14), will reflect inequality (1),

If the criteria are not correlated, then the following expression is valid:

X a(PJ") + a(L + = Z +aU) =>

The case of replacement of criteria \ and ) is characterized by inequality (3), and interdependence (1) respectively. We will assume that if the expert believes that the criteria / and y are replaceable (interdependent), he will not simultaneously take into account their positive or negative correlation in the model. Indeed, a positive (negative) correlation of criteria is identified on the basis of the expert’s statistical observations, while substitution (interaction) is nothing more than his opinion regarding the need to satisfy these criteria, which has greater priority when choosing the value of the aggregation result.

Now we come to a difficult problem: how to express the preferential dependence or independence of criteria using a fuzzy measure. With the beginning of the use of fuzzy measures and integrals to construct aggregation operators, it was understood that the non-additivity of a fuzzy measure should allow modeling the preferential dependence of criteria. However, an apparatus has not yet been developed that allows this to be done strictly formally, and the phenomenon of preferential dependence of criteria itself has been poorly studied. MigoM and Zidepo proved the following theorem:

Theorem 1 Let gl9...i be a set of criteria. Let us denote gJ_(i) the implementation of criteria gj, where y e 3 - (/). Here gt is called an integral criterion if 3 gi,g"¡ are such that

Let us limit the set of aggregation operators by operators based on the Choquet integral, i.e. gя) = Cffw(gl,..., 8н). That-

where, if we have at least three integral criteria, then the following statements are equivalent:

1. criteria gl,..., gн mutually preferable

independent;

2. the fuzzy measure y/ is additive.

Thus, we will reflect the preferred dependence (independence) of the criteria using the 2nd order Choquet integral using a fuzzy measure based on the interaction indices of the criteria (correlation and substitution), as well as partial order on the set of implementations of criteria A (training sample).

Currently, applications of the Choquet integral as an aggregation operator in some practical applications are known. In particular, a system for selecting the optimal software interface is considered, a speech recognition system is described, and a description of a navigation system for pedestrians using the Choquet integral is given.

Wider use of this tool is hampered by its poor intuitive understanding by many.

practical specialists. To overcome this circumstance, you can use the visualization mechanism by associating the Choquet integral with some well-known physical object.

The author proposes a method for visualizing the construction of an aggregation operator based on the 2nd order Choquet integral. This method is based on the idea of ​​the metaphor of balance. This idea is to establish a correspondence between a real object, for which a natural intuitive representation is well developed, and a mathematical object - the aggregation operator. Such a real object is a lever, which is fixed at the fulcrum by a spring with a constant stiffness coefficient equal to one (Fig. 1). Weights are installed on the lever that correspond to the importance or “weights” of the criteria. We consider a family of aggregation operators that can be built based on the balance metaphor. The Choquet integral is not included in this family. To build a mechanism for visualizing the 2nd order Choquet integral based on the balance metaphor, we modify the balance metaphor.

In order to be able to take into account the interaction of criteria in the case of a 2nd order model, it is necessary to reflect in the balance metaphor the influence of criteria interaction indices /(//) on the aggregation result. The range of values ​​of these indices is the interval [-

Based on this range of values, we select the interval [-1,1] for the lever scale. We will choose 0 as a neutral element on the scale of the lever (or the place of its attachment). In the non-negative area of ​​the lever scale we will postpone the values ​​of the criteria ^ gп, assigning weights to them. In the negative area of ​​the lever scale we will postpone the values

mt(£.,£.), associated with the weights |/((/)|, if 1(y)< 0. В случае, если индекс взаимодействия критериев /((/)>0, to criterion weight

we will add the value

In Fig. Figure 1 shows the construction of balance described above for the case of two criteria whose interaction index 7(1,2) is negative. Let us write, in accordance with Newton’s second law, the balance equation for the case shown in Fig. 1,

Obviously, increasing the number of criteria will not lead to changes in the structure of the balance sheet; let us write the corresponding equation:

This expression is equivalent to the second-order Choquet integral,

Let us now consider qualitatively the modeling of dependencies between criteria using the proposed visualization mechanism and the corresponding aggregation operator. In accordance with the aggregation scale (Fig. 1), we will call the moment of rotation of the lever directed counterclockwise negative, and directed clockwise positive.

In the case of a positive correlation of criteria or their replacement, we will display their negative interaction, modeled by inequality (3), when constructing a balance.

In the negative area of ​​the lever scale,

the load will be located |/(?)")| at a distance from the zero mark.

Rice. 1. Visualization of the Choquet integral based on the balance metaphor

The lever will be affected by a negative torque due to the values ​​of I(ij)<0 и

min(g.,g-y). At the same time, the total positive

torque due to loads y/(i) and

y/(j)i located at distances g. and g. from

zero mark, will be partially compensated by the negative moment I(ij) mm(g;,gy).

In the case of a negative correlation between criteria i and j or their interdependence, the index of their interaction will be set to /(r>) > 0, which will reflect inequality (1). The lever will be affected by a positive torque due to the values ​​of I(ij) >0 and

mm(gi ,gj). In this case, the total positive moment of rotation due to the loads located at distances g. and g. from the zero mark, will be enhanced by the positive moment /(//) min(gi9gj).

If the criteria are not correlated, and are not substitutable or interdependent, then I(ij) = 0 and we can observe the aggregation of independent criteria. In this case, the position of the lever will be determined by the action of positive moments

Si V(i) and gj yf(J).

In accordance with Theorem 1, in the case of preferential independence of the criteria, the position of the lever will also be determined only by the action of positive moments g. y/(g) and g. y/(j).

The proposed visualization method will allow developers of practical applications to have an intuitive vision of constructing aggregation operators based on the 2nd order Choquet integral. The use of this method will also facilitate the task of training an expert to formalize knowledge in his subject area through the relatively new apparatus of fuzzy measures and integrals.

Bibliography

1. Grabisch M., Orlovski S., Yager R. Fuzzy Aggregation of numerical Preferences, In R, Slowinski, editor, Fuzzy Sets in Decision Analysis, Operations Research and Statistics, Kluwer Academic, 1998, 43 p.

2. Belenky A.G. Selection of scales and aggregation operators when constructing fuzzy intelligent information management systems. -M.: MPEI, 1999. 50 p.

3. Ovchinnikov, S., On Robust Aggregation Procedures, Aggregation Operators for Fusion under Fuzziness. Bouchon-Meunier B. (eds.), 1998, pp. 3-10.

4. Mayor, G. and Trillas E., On the representation of some Aggregation functions, Proceeding of ISMVL, 1986, pp. 111-114.

5. Mesiar R. and KomornOkova M., Aggregation Operators, Proceeding of the XI Conference on applied Mathematics PRIM" 96, Herceg D., Surla K. (eds.), Institute of Mathematics, Novi Sad, 1997, pp. 193- 211.

6. Moulin E. Cooperative decision making: Axioms and models. -M.:Mir, 1991, - 464 p.

7. M. Sugeno, Theory of fuzzy integrals and its applications, Ph.D. Thesis, Tokyo Institute of Technology, Tokyo, 1974, 237 p.

8. M. Grabisch, k-order additive discrete fuzzy measures and their representation, Fuzzy Sets & Systems 92, 1997, pp. 167-189.

9. T. Murofushi and S. Soneda, Techniques for reading fuzzy measures (III): interaction index, in: 9th Fuzzy System Symposium, Sapporo, Japan, May 1993, pp. 693-696.

10. P. Wakker. A behavioral foundation for fuzzy measures. Fuzzy sets & Systems, 37, 1990, pp. 327-350.

11. G. Choquet. Theory of capacities. Annales de I"lnstitut Fourier, 5, 1953, pp. 131-295.

12. T. Murofushi, M. Sugeno Non-additivity of fuzzy mesures representing preferential dependence, 2nd Int. Conf. On Fuzzy Systems and Newral Networks, lizuka, Japan, July, 1992, pp. 617-620.

13. Stanley P. Enumerative combinatorics, - M.: Mir, 1990. -440 p.

14. M. Sicilia, E. Garsia, T. Calvo An Inquiry-Based Method for Choquet Integral-Based Aggregation of Interface Usability Parameters RepDblica Checa Kybemetica, 39(5), 2003, pp. 601-614.

15. T. Pham, M. Wagner, Similarity normalization for speaker verification by fuzzy fusion, The Journal of the Pattern Recognition Society 33, 2000, pp. 309-315.

16. Y. Akasaka and T. Onisawa, Pedestrian Navigation Reflecting Individual Preference for Route Selection -Evaluation on Fitness of Individual Preference Model-, Journal of Japan Society for Fuzzy Theory and Intelligent Informatics, Vol. 18, No. 6, 2006, pp. 900-910.

17. M. Detyniecki and B. Bouchon-Meunier, Building an Aggregation Operator with a Balance, Proceedings of the International Conference on Information Processing and Management of Uncertainty in Knowledge-Based Systems, Madrid, Spain, July 2000, pp. 686-692.

The article was accepted for publication on March 21, 2007

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