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Podinovskii V.V., Noghin V.D. (1982) Pareto-optimal Decisions in Multicriteria Optimization Problems.

Contents
Chapter 1. Basic notions and definitions
Chapter 2. Optimality conditions
Chapter 3. Structure and properties of the set of efficient decisions
Chapter 4. Duality in multiobjective programming

Noghin V.D. (1986) Fundamentals of Optimization Theory.

Contents
Preface
1. Basic mathematical notions
2. Theoretical basis of optimization in finite-dimensional spaces
3. Linear programming
4. Dynamic programming
5. Nonlinear programming
6. Geometrical programming
7. Elements of multicriteria optimization
8. Theoretical basis of optimization in functional spaces
9. Calculus of variations
10. Optimal control
11. Synthesis of optimal controls
Conclusion
References
Index

Noghin V.D. (2002) Decision Making in Multicriteria Environment: a Quantitative Approach.


Noghin V.D. (1976) A New Approach to Reduce a Compromise Set.

Abstract

Let f=(f1,f2,...,fm) be m-dimensional vector-function defined on a set X. An element x*∈X is called r-maximal (or r-optimal) if it is a Pareto-maximal with respect to any r-dimensional vector-function formed from some components of f, r=1,2,...,m .

It easy to see that 1-maximal element is a maximal element for all functions f1,f2,...,fm on X simultaneously. This is an "ideal" solution of multicriteria maximization problem. On the other hand, m-maximal coincides with Pareto-maximal element. If 1<r<m then r-maximal element is Pareto-maximal, and it lies between the 'ideal' and the Pareto-maximal solutions. The less r the 'closer' r-maximal element to the 'ideal' solution.

For the above reason r-maximal element with minimal possible r, for which such element there exists, is suggested as the 'best' compromise solution of the multicriteria problem.

In this paper some existence theorems for r-maximal points are obtained in the case when f is a linear vector function and X is a convex set in finite dimensional space.


Noghin V.D. (1991) Estimation of the Set of Nondominated Solutions.

Abstract
Assuming that the Decision Maker's strict preference relation satisfies some reasonable axioms, an upper and lower estimates for the set of nondominated solutions are constructed.


Noghin V.D. (1994) Upper Estimate for a Fuzzy Set of Nondominated Solutions.

Abstract
Assuming that some reasonable axioms for the Decision Maker's fuzzy strict preference relation hold, an upper estimate for a fuzzy set of nondominated solutions is constructed. For a finite set of feasible alternatives an algorithm to construct the upper estimate is given.


Noghin V.D. (1997) Relative Importance of Criteria: a Quantitative Approach.

Abstract
Multicriteria choice model and mathematical definition for the assertion 'a group of criteria is more important than other group' are introduced. It is shown how quantitative information on the relative importance of criteria allows us to obtain a more precise upper estimate for a set of all non-dominated solutions than the well-known Pareto set.


Noghin V.D. (2000) Completeness Theorems in the Theory of Relative Importance of Criteria.

Abstract
Under appropriate assumptions three completeness theorems are proved. The first theorem shows that using only information on the relative importance of criteria it is possible to construct an approximation for the unknown strict preference relation of a decision maker as close as desired. According to the second theorem in order to find an unknown set of nondominated points for a finite number of feasible decisions it is sufficient to get some information on the relative importance of criteria. The third theorem deals with a convergence to the unknown set of nondominated points.


Noghin V.D., Tolstych I.V. (2000) Using Quantitative Information on the Relative Importance of Criteria for Decision Making.

Abstract
On the basis of a rigorous definition of the relative importance of criteria, two approaches to justify restriction of the Pareto set that use a finite collection of quantitative data on the relative importance of criteria are suggested for the rather wide class of multicriteria choice problems.


Noghin V.D. (2001) What is the Relative Importance of Criteria and How to Use It in MCDM.

Abstract
A lot of multicriteria decision-making methods require the use of weights or importance coefficients. Usually authors of the methods do not define these coefficients. Therefore, their methods are only heuristic. In order to successfully elicit and apply information on the relative importance of criteria, it is necessary to have a rigorous definition for the coefficients. In this paper a definition of the assertion 'a group of criteria is more important that other group' is given. Based on the definition the numerical relative importance coefficients are defined. The main goal of the paper is to demonstrate how to apply these notions in decision making in order to reduce the well-known Pareto set.


Noghin V.D. (2001) A Logical Justification of the Edgeworth-Pareto Principle.

Abstract
Since the nineteenth century the Edgeworth-Pareto principle is an effective tool for solving numerous multicriteria problems. Nevertheless there was no an appropriate mathematical formulation of it. The main objective of the paper is to state and axiomatically justify the Edgeworth-Pareto principle for a certain wide class of multicriteria choice problems.


Noghin V.D. (2003) The Edgeworth-Pareto Principle and the Relative Importance of Criteria in the Case of a Fuzzy Preference Relation.

Abstract
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 (reasonable) axioms, the fuzzy set of selected decisions is always a subset of the Pareto set. Developed by the author in the case of a crisp 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.


Noghin V.D. (2004) A Simplified Variant of the Analytic Hierarchy Processes Based on a Nonlinear Scalarizing Function.

Abstract
Substantial disadvantages of the well-known Analytic Hierarchy Process (AHP) when it is used in the case of both a single and several criteria are outlined. A simpler, more accurate and reliable method for finding the priority vector in single-criterion problems is proposed. For multicriteria choice problems, the use of a minimum function as the scalarizing function is justified.