zhChinese    ruRussian
  AM&CP  » Structure » Staff  » Noghin V.D.

NOGHIN Vladimir D.

[photo]

D.Sc., Professor of Department of Control Theory

Room 223
E-mail: u_n_known@rambler.ru

Specialties

Mathematics, Applied Mathematics, Operations Research

Academic degree

Candidate of Phys.-Math. Sciences (1978)
Doctor of Phys.-Math. Sciences (1996)

Academic titles

Associate Professor (1984)
Professor (1997)
Member of the International Higher Education Academy of Sciences (1997)

Position

2003-PresentProfessor, Department of Control Theory, Faculty of Applied Mathematics-Control Processes, St. Petersburg State University
1993-2003Professor, Department of Mathematics, St. Petersburg State Technical University (former Leningrad Polytechnic Institute)
1980-1993Associate Professor, Leningrad Polytechnic Institute
1974-1980Researcher, Leningrad State University

Education

1971-1974Post-graduate, Leningrad State University
1966-1971Student, Leningrad State University

Honors

State Committee of Public Education prize-winner (1988)
Soros Professor (1999, 2000)

Publications

Total number of publications is over 100

Hobbies

Photography, Poetry

Course for pupils of special school

Numerical Methods

Courses for students (1980-present)

Research activity

Began to analyze multicriteria problems in 1972. During 10 years a number of

  1. necessary and sufficient conditions for Pareto-optimality (efficiency), proper efficiency, and weak efficiency
  2. existence theorems
  3. topological theorems
  4. duality theorems

for linear as well as nonlinear multicriteria problems were obtained. Moreover, an original concept of r-optimality was proposed [8].

As a result, in 1982 the publishing house 'Nauka' was published a monograph 'Pareto-optimal Decisions in Multicriteria Optimization Problems' [1]. In this book numerous mathematical results from all over the world related to the basic concepts of multicriteria optimization problem were collected, systemized, and stated. Moreover, a plenty of original theorems by the authors there were presented. It was the first of such kind monograph in Russia. Authors of hundreds papers (not only in Russian) have been referring to this book. Unfortunately, this monograph was not translated into English.

Since 1980 began to teach students at St. Petersburg State Technical University (former Polytechnic Institute). The text book 'Fundamentals of Optimization Theory' [2] was published in 1986. From one point of view different topics of optimization theory such as linear programming, geometrical programming, dynamic programming, nonlinear programming, multicriteria optimization, calculus of variations, and optimal control were presented in this text book.

In 1994 the problem concerning to maximal number of partially ordered sets was formulated [12]. This problem is unsolved up to now.

Since the beginning of 80th the sphere of main interests is removed to various aspects of the relative importance of criteria. This theme is important both for theory and applications but it was not investigated mathematically.

Many researchers characterize the importance of criteria by means of special positive numbers called importance or weight coefficients. They use these coefficients in generalized criteria such as weighted sum, Tchebycheff metric, etc. These authors do not try to formulate any definition for the coefficients. Not infrequently each author uses his own procedure to obtain the coefficients and different procedures lead to distinct results. Evidently, in order to successfully elicit and apply information on the relative importance of criteria, first of all it is necessary to have a rigorous definition for the assertion 'one criterion is more important than other criterion'.

For example, AHP methodology based on the concept of the relative importance of criteria was suggested by Thomas L. Saaty. But he could not formulate a rigorous definition of the relative importance of criteria. Hence, AHP is only heuristic approach, and we, using it in practice, do not know when AHP 'works properly'.

The first papers in English [11, 13] were devoted to a problem of using some quantitative information on the relative importance of criteria in order to facilitate a decision process (see also Chapter 7 in [2]). These articles did not contain a concept of the relative importance of criteria yet. Really this concept was proposed only in [14] and it was a correction of a definition by V.V. Podinovskii (1978).

In 1997 a multicriteria choice model was introduced [14]. It includes three objects - the set of feasible decisions, vector criterion, and the DM's strict preference relation. There were proposed definitions of the relative importance of criteria for two criteria as well as for two groups of criteria. Thus, the assertion 'one group of criteria is more important than another group' has strictly mathematical meaning now. It is very important that given definitions are very simple. The DM can easily learn them and then express his own preferences in terms of the proposed definitions.

If there is no any information on the relative importance of criteria then the Edgeworth-Pareto principle is available. According to the principle each selected decision must belong to the Pareto set. Since XIX-th century hundreds of researches applied this principle thousands times to solve distinct multicriteria problems. However, there are situations in which the selected (i.e. best) decisions are not Pareto-optimal. Due to three reasonable axioms (see [6, 18, and 19]) all multicriteria choice problems were divided into two classes. For any problem from the first class an application of the Edgeworth-Pareto principle is justified. The second class consists of the problems for which this principle may 'not work'. Thus, the famous Edgeworth-Pareto principle was axiomatically justified [6, 18, and 19].

The main goal of the theory of the relative importance of numerical criteria is to use the quantitative information on the relative importance in order to exclude some Pareto-optimal decisions (i.e. to reduce the Pareto set), and by the same token to facilitate a decision process. As was shown in [6, 18, and 19] in order to use the information on the relative importance of criteria we have to recount 'old' vector criterion (i.e. construct some 'new' vector criterion using very simple formulas) and then to find 'new' Pareto set which will be more narrow then the 'old' one. After that we may make a choice within the 'new' Pareto set or try again to use additional information on the relative importance of criteria etc.

Often we deal with a collection of information on the relative importance. Such information may be inconsistent. A consistent collection of information was defined and necessary and sufficient conditions for the consistency were obtained [6, 13, 14, and 16]. There was established how such information to use.

The above-mentioned results put together the first part of the theory of the relative importance. The second part consists of the completeness theorems (see [6, 15]). According to ones we can construct unknown set of nondominated (i.e. potentially optimal) decisions with any accuracy using only an information on the relative importance of criteria.

As it was shown in [19], all principal results of the theory of the relative importance can be formulated and proved in the case of fuzzy preference relation and/or fuzzy set of feasible decisions.

Recently a simplified variant of AHP based on a nonlinear scalarizing function was proposed [20].

Monographs, text books, handbooks

  1. Podinovskii V.V., Noghin V.D. (1982) Pareto-optimal Decisions in Multicriteria Optimization Problems, Moscow, Nauka, 256 pp. (in Russian).
  2. Noghin V.D. et al. (1986) Fundamentals of Optimization Theory. Moscow, Higher School, 384 p. (in Russian).
  3. Noghin V.D. (1988) Elements of Optimization Theory and Mathematical Economics. St. Petersburg, Polytechnic Institute, 96 p. (in Russian).
  4. Noghin V.D. (1994) Introduction to Mathematical Analysis. St. Petersburg, State Technical University, 68 p. (in Russian).
  5. Noghin V.D., Chystaykov S.V. (1998) Application of Linear Algebra for Decision Making. St. Petersburg, State Technical University, 40 p. (in Russian).
  6. Noghin V.D. (2002) Decision Making in Multicriteria Environment: a Quantitative Approach. Moscow, Fizmatlit, 176 p.
  7. a) Noghin V.D. et al. (2004) System Analysis and Decision Making (handbook). Moscow, Visshaya Shkola, 616 p. (in Russian).
    b) Noghin V.D. Decision making under several criteria. - Saint-Petersburg, UTAS, 2007, 104 p.
  8. Noghin V.D. et. al. Systems theory and systems analysis in management (handbook). Moscow: Finance and Statistics, 2006, 848 pp. (in Russian).

Selected papers

  1. Noghin V.D. (1976) A New Approach to Reduce a Compromise Set. Technicheskaya Kibernetika, No. 5, pp. 10-14 (in Russian).
  2. Noghin V.D. (1977) Duality in Multipurpose Programming. Computational Mathematics and Mathematical Physics, No. 1, pp. 254-258 (in Russian).
  3. Noghin V.D. (1980) Existence Criteria of Decisions in Finite Dimensional Multipurpose Optimization Problem. Vestnik St. Petersburg State University (Ser.: Mathematics, Mechanics, Astronomy), No. 7, pp. 27-32 (in Russian).
  4. Noghin V.D. (1991) Estimation of the Set of Nondominated Solutions. Numerical Functional Analysis and Applications, V. 12, No. 5&6, pp. 507-515.
  5. Noghin V.D. (1994) Problem 214. Discrete Mathematics, 135, p. 394.
  6. Noghin V.D. (1994) Upper Estimate for a Fuzzy Set of Nondominated Solutions. Fuzzy Sets and Systems, V. 67, pp. 303-315.
  7. Noghin V.D. (1997) Relative Importance of Criteria: a Quantitative Approach. J. Multi-Criteria Decision Analysis, V. 6, pp. 355-363.
  8. Noghin V.D. (2000) Completeness Theorems in the Theory of Relative Importance of Criteria. Vestnik St. Petersburg State University (Ser.: Mathematics, Mechanics, Astronomy), 40 (25), pp. 13-18 (in Russian).
  9. Noghin V.D., Tolstych I.V. (2000) Using Quantitative Information on the Relative Importance of Criteria for Decision Making. Computational Mathematics and Mathematical Physics, V. 40, No. 11, pp. 1529-1536.
  10. Noghin V.D. (2001) What is the Relative Importance of Criteria and How to Use It in MCDM. Lecture Notes in Economics and Mathematical Systems, V. 507: Multiple Criteria Decision Making in the New Millennium (Proceedings of the XV International Conference on MCDM, Ankara, Turkey, 2000), pp. 59-68.
  11. Noghin V.D. (2001) A Logical Justification of the Edgeworth-Pareto Principle. Computational Mathematics and Mathematical Physics, V. 42, No. 7, pp. 915-920.
  12. Noghin V.D. (2003) The Edgeworth-Pareto Principle and the Relative Importance of Criteria in the Case of a Fuzzy Preference Relation. Computational Mathematics and Mathematical Physics, V. 43, No. 11, pp. 1604-1612.
  13. Noghin V.D. (2004) A Simplified Variant of the Analytic Hierarchy Processes Based on a Nonlinear Scalarizing Function. Computational Mathematics and Mathematical Physics, V. 44, No. 7, pp. 1194-1202.
  14. Noghin V.D. The Edgeworth-Pareto Principle in Decision Making. Tutorial Presentation for the Russian-Finnish Graduate School Seminar "Dynamic Games and Multicriteria Optimization", Petrozavodsk (Russia), 2006.
  15. Noghin V.D. An Axiomatization of the Generalized Edgeworth-Pareto Principle in Terms of Choice Functions. "Mathematical Social Sciences", 2006, v. 52, No 2, pp. 210-216.
  16. Noghin V.D. The Edgeworth-Pareto principle in terms of a fuzzy choice function. "Computational Mathematics and Mathematical Physics. 2006, Vol. 46, No. 4, pp. 554-562.
  17. Klimova O.N., Noghin V.D. Using inderpendent information on the relative importance of criteria in decision making // Computational Mathematics and Mathematical Physics, 2006, v. 46, 12, P. 2080-2091.

Participation at International Conferences