Papers by Vladimir Noghin
It was V.V. Podinovski who began to investigate the relative importance of criteria in 70 th. vi ... more It was V.V. Podinovski who began to investigate the relative importance of criteria in 70 th. vi Formulas, illustrations, and statements have double numeration, where the first number is that of the chapter. The symbol ■ marks the end of proof. The author thanks Irina Tolstykh who played an appreciable role in improvement of representation. Moreover, she made a contribution to a content; namely, Theorem 4.10 belongs to her. Special thanks to the Russian Fund for Basic Research for financial support since 1998. Saint Petersburg Vladimir Noghin vii Contents 3 Relative importance of two groups of criteria 3.1 Main notions and its properties 3.2 Use of information on relative importance 3.3 Three-criteria problem: a geometric illustration 4 Compound information on relative importance 4.1 Use of compound information: simple cases 4.2 Consistency of compound information 4.3 Use of compound information 4.4 Algorithmic approach 5 Completeness of compound information 5.1 Preliminary consideration 5.2 First completeness theorem viii 5.3 Second completeness theorem 6 Decision making methodology based on compound information on relative importance 6.1 How a person makes a choice? 6.2 Successive reduction of the Pareto set 6.3 Combined methods F. Edgeworth and V. Pareto Bibliography Index * This one is a correction of some definition by V.V. Podinovski (1978).
Reducing the pareto set algorithm based on an arbitrary finite set of information “quanta”
Scientific and Technical Information Processing, 2014
In this paper, in the framework of the axiomatic approach developed by the author over the past 3... more In this paper, in the framework of the axiomatic approach developed by the author over the past 3 decades, we assume four axioms of “reasonable” choice, which define a rather wide class of problems of multi-criteria selection. To reduce the Pareto set we use numerical information about the preference relation of a decision maker. We propose a method for narrowing the Pareto set using an arbitrary consistent finite set of such information. The method is based on an algorithm that generates a new set of criteria (with a minimum elements number) with respect to which a new Pareto set gives o more precise upper estimate than the initial Pareto set.
A New Characterization of Cone Proper Efficient Points
Lecture Notes in Control and Information Sciences - Proceedings, 2022
Studies in Systems, Decision and Control, 2017
This chapter considers in brief the aspects of decision-making by humans and then presents the ax... more This chapter considers in brief the aspects of decision-making by humans and then presents the axiomatic approach to Pareto set (domain of compromise) reduction based on information quanta about the DM's preference relation. The corresponding theoretical background can be found in the previous chapters, and here we describe the axiomatic approach without mathematical details, as well as give some recommendations on usage. In addition, possible ways to combine this approach with some multicriteria scalarization methods and some potential extensions are discussed. 8.1 How Do Humans Make Their Decisions? 8.1.1 Mental Components of Decision-Making Process Decision-making process includes three phases, namely, search, choice and implementation of decisions. Decision-making is an act of volition that forms a sequence of actions towards goal attainment by transforming initial information under uncertainty. The main stages of decision-making process are situational analysis using available information and the decision-making procedure itself, i.e., the formation and comparison of alternatives, the choice of an appropriate alternative and the development of an action plan. On the one hand, decision-making may represent a special form of mental activity (e.g., in management) and, on the other, as a stage of thinking in problem solving. This notion has a very wide range of application. Throughout the book, we understand decision-making as a special process of human activity intended for choosing a best alternative (a best action).
Studies in Systems, Decision and Control, 2017
Studies in Systems, Decision and Control, 2017
This chapter introduces and discusses the basic notions of decision-making in a multicriteria env... more This chapter introduces and discusses the basic notions of decision-making in a multicriteria environment, namely, the set of feasible alternatives, the vector criterion and the preference relation of a decision-maker. Here we formulate the multicriteria choice problem. In addition, Chap. 1 defines a pair of fundamentally important notions, the set of nondominated alternatives and the Pareto set, which are vital for the statement and rigorous substantiation of the Edgeworth-Pareto principle. The statement and substantiation of the above principle form the central outcome of Chap. 1. As established below, the Edgeworth-Pareto principle should be applied to solve the multicriteria choice problems from a certain sufficiently large class. This class comprises the problems satisfying two definite requirements (axioms) that express the "reasonable" behavior of a decision-maker. An attempt to use the Edgeworth-Pareto principle beyond the class is risk-bearing, possibly yielding inadequate results.
Completeness Property of Information Quanta
In this chapter, we justify theoretically the original axiomatic approach to Pareto set reduction... more In this chapter, we justify theoretically the original axiomatic approach to Pareto set reduction based on a finite collection of information quanta. Here the exposition seems most difficult in mathematical terms, but the readers with an insufficient background may skip it without losing the comprehension of further material. The whole essence of the results derived below can be expressed as follows. Information in the form of quanta is complete: for any multicriteria choice problem from a definite (rather wide) class, it is possible to find the unknown set of nondominated vectors (nondominated alternatives) with an arbitrary accuracy only based on such information. Moreover, if the number of feasible vectors is finite, then the set of nondominated vectors can be constructed precisely.
Approximation of convex fuzzy sets
2017 Constructive Nonsmooth Analysis and Related Topics (dedicated to the memory of V.F. Demyanov) (CNSA), 2017
In this paper we introduce the concepts of fuzzy space, the distances between points and sets of ... more In this paper we introduce the concepts of fuzzy space, the distances between points and sets of this space. In addition, a concept of a polyhedral fuzzy set, a convex hull, and a fuzzy direction are proposed. It is proved that a polyhedral fuzzy set is a convex hull of a finite number of fuzzy points and directions. In accordance with the main result of this paper any convex closed fuzzy set with a continuous membership function can be approximated with any degree of accuracy by a fuzzy polytop.
Pareto Set Reduction Using Elementary Collections of Information Quanta
This chapter focuses on the application of some “simple” collections of information quanta. We es... more This chapter focuses on the application of some “simple” collections of information quanta. We establish that some collections can be inconsistent.
Pareto Set Reduction Based on Collections of Information Quanta
Here we further develop the results of the previous chapter on Pareto set reduction using given f... more Here we further develop the results of the previous chapter on Pareto set reduction using given finite collections of mutually dependent information quanta about the DM’s preference relation.
Using quantitative information on the relative importance of criteria for decision making
Computational Mathematics and Mathematical Physics, 2000
On the basis of a rigorous definition of the relative importance of criteria for a wide class of ... more On the basis of a rigorous definition of the relative importance of criteria for a wide class of multicriteria problems, two approaches to justify restriction of the Pareto set that use a finite set of quantitative data on the relative importance of the criteria are suggested for a rather wide class of multicriteria problems.
Pareto Set Reduction Based on General Information Quantum
The notion of an elementary information quantum for two criteria (see Chap. 2) is extended to the... more The notion of an elementary information quantum for two criteria (see Chap. 2) is extended to the general case of two groups of criteria. We study the properties of a general information quantum, demonstrating how it should be used for Pareto set reduction.
Pareto Set Reduction Using Fuzzy Information
In a series of applications-relevant multicriteria choice problems, the available information abo... more In a series of applications-relevant multicriteria choice problems, the available information about the DM’s preference relation can be fuzzy in the sense that it is impossible to define explicitly the preference for one alternative rather than another, since there exist the pros and cons of it.
The Edgeworth-Pareto principle and the relative importance of criteria in the case of a fuzzy preference relation
Computational Mathematics and Mathematical Physics, 2003
The Edgeworth-Pareto principle substantiated previously by the author is extended to the case of ... more The Edgeworth-Pareto principle substantiated previously by the author is extended to the case of a fuzzy preference relation used by a decision-maker. It is shown that, under three certain axioms, the fuzzy set of selected decisions is always a subset of the Pareto set. Developed by the author in the case of a nonfuzzy preference relation, the approach taking into account quantitative information on the relative importance of criteria is extended to a fuzzy preference relation and a fuzzy set of feasible decisions. An illustrative example is given.
On Multicriteria Choice Based on Type-2 Fuzzy Preference Relation: an Axiomatic Approach
2021 IEEE International Conference on Fuzzy Systems (FUZZ-IEEE), 2021
A multicriteria choice problem is considered. The setting of this problem includes three objects,... more A multicriteria choice problem is considered. The setting of this problem includes three objects, namely, a set of feasible alternatives, a numerical vector criterion, and a decision maker's binary strict preference relation. The Edgeworth — Pareto principle is a fundamental instrument to solve multi-criteria problems. Previously, the validity of this principle was established in the case of a crisp as well as a type-1 fuzzy preference relation. We assume that the preference relation is a type-2 fuzzy relation. Under two reasonable axioms the Edgeworth—Pareto principle is established. In accordance with the first axiom, an alternative not chosen in a pair should not be selected from the whole set of feasible alternatives. The second axiom is the Pareto axiom, which provides greater preference for those alternatives that have larger (smaller) values of one or more criteria.
What is the Relative Importance of Criteria and how to Use it in MCDM
Lecture Notes in Economics and Mathematical Systems, 2001
A lot of multiple criteria decision-making methods require the use of weights or importance of co... more A lot of multiple criteria decision-making methods require the use of weights or importance of coefficients. Usually authors of the methods do not mathematically define these coefficients. Therefore, their methods are only heuristic. In order to successfully elicit and apply the relative importance of criteria, it is necessary to have a rigorous definition for the coefficients. In this paper a mathematical definition of the assertion ‘a group of criteria A is more important than a group of criteria B with two sets of positive parameters‘ is given. Based on the definition the numerical relative importance coefficients are mathematically defined. The main objective of the paper is to demonstrate how to apply those notions in decision making in order to restrict the well-known Pareto set.
Studies in Systems, Decision and Control, 2018
The series "Studies in Systems, Decision and Control" (SSDC) covers both new developments and adv... more The series "Studies in Systems, Decision and Control" (SSDC) covers both new developments and advances, as well as the state of the art, in the various areas of broadly perceived systems, decision making and control-quickly, up to date and with a high quality. The intent is to cover the theory, applications, and perspectives on the state of the art and future developments relevant to systems, decision making, control, complex processes and related areas, as embedded in the fields of engineering, computer science, physics, economics, social and life sciences, as well as the paradigms and methodologies behind them. The series contains monographs, textbooks, lecture notes and edited volumes in systems, decision making and control spanning the areas of Cyber
Scientific and Technical Information Processing, 2018
The multicriteria choice problem with a fuzzy preference relation is considered. This problem inv... more The multicriteria choice problem with a fuzzy preference relation is considered. This problem involves a set of feasible alternatives, a numerical vector criterion, and a fuzzy preference relation of a decision-maker (DM). The concepts of fuzzy vector space, a polyhedral fuzzy set, and the distance between convex fuzzy sets and cones are used. To reduce the Pareto set, ultimate possibilities of using information about the fuzzy preference relation in the form of its quanta are studied. For a sufficiently wide class of choice problems, it is proved that an originally unknown fuzzy set of nondominated elements can be arbitrarily accurately approximated using a finite set of fuzzy information quanta.
Computational Mathematics and Mathematical Physics, 2017
A multicriteria choice problem involving a decision-maker's binary preference relation is conside... more A multicriteria choice problem involving a decision-maker's binary preference relation is considered. Several two-stage methods are proposed for its solution. First, the Pareto set is reduced by applying an axiomatic approach. Then the problem is scalarized on the resulting set by using the Chebyshev or Euclidean metric. The methods proposed are substantiated with the help of well-known and new techniques for characterizing weakly efficient and proper efficient points. Illustrative examples are given.
Scientific and Technical Information Processing, 2010
This paper considers a multicriteria choice model that includes multiple choices, a numerical vec... more This paper considers a multicriteria choice model that includes multiple choices, a numerical vec tor criterion, and an asymmetric preference relationship of the decision maker (DM). The solution of the multicriteria choice problem is a set of selected choices. Certain "rational" axioms are assumed to be true. Provided that the DM's preference relationship is known only partially, one cannot find a set of selected choices; nevertheless, an upper bound estimate can be obtained from the available information. The infor mation about a DM's preferences is considered to be of the point multiple type. This means that the DM is ready to take losses in some finite groups of criteria for the sake winning in some other group. The information about the DM's preferences can be used to develop a new vector criterion and build a restriction of the Pareto set that produces a more precise estimate of the unknown set of the selected variants than the original Pareto set.
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Papers by Vladimir Noghin