Methods of tropical optimization in rating alternatives based on pairwise comparisons

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Abstract

We consider unconstrained and constrained optimization problems in the framework of the tropical (idempotent) mathematics, which focuses on the theory and applications of semirings with idempotent addition. The problems are to minimize or maximize functions defined on vectors over idempotent semifields (semirings with multiplicative inverses). We examine several problems of rating alternatives from pairwise comparisons, including problems with constraints on the final scores of alternatives, and multi-criteria rating problems. We reduce the problems to the log-Chebyshev approximation of pairwise comparison matrices by reciprocal matrices of unit rank, and then represent the approximation problems as optimization problems in the tropical mathematics setting. To solve these problems, methods of tropical optimization are used to provide direct, complete solutions in a compact vector form. We discuss the applicability of the results to real-world problems, and provide numerical examples.
Original languageEnglish
Pages38
Publication statusPublished - 2016
EventAnnual International Conference of the German Operations Research Society: International Conference on Operations Research – Analytical Decision Making - Helmut-Schmidt-Universität / Universität der Bundeswehr Hamburg, Hamburg
Duration: 30 Aug 20162 Sep 2016
http://or2016.de/

Conference

ConferenceAnnual International Conference of the German Operations Research Society
Abbreviated titleOR 2016
CountryGermany
CityHamburg
Period30/08/162/09/16
Internet address

Fingerprint

Pairwise Comparisons
Optimization
Alternatives
Idempotent
Semiring
Multiplicative inverse
Semifield
Chebyshev Approximation
Unconstrained Optimization
Approximation Problem
Multi-criteria
Constrained Optimization Problem
Maximise
Optimization Problem
Minimise
Numerical Examples
Unit

Scopus subject areas

  • Decision Sciences (miscellaneous)
  • Control and Optimization
  • Algebra and Number Theory

Cite this

Кривулин, Н. К. (2016). Methods of tropical optimization in rating alternatives based on pairwise comparisons. 38. Abstract from Annual International Conference of the German Operations Research Society, Hamburg, .
Кривулин, Николай Кимович. / Methods of tropical optimization in rating alternatives based on pairwise comparisons. Abstract from Annual International Conference of the German Operations Research Society, Hamburg, .
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Methods of tropical optimization in rating alternatives based on pairwise comparisons. / Кривулин, Николай Кимович.

2016. 38 Abstract from Annual International Conference of the German Operations Research Society, Hamburg, .

Research output

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T1 - Methods of tropical optimization in rating alternatives based on pairwise comparisons

AU - Кривулин, Николай Кимович

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PY - 2016

Y1 - 2016

N2 - We consider unconstrained and constrained optimization problems in the framework of the tropical (idempotent) mathematics, which focuses on the theory and applications of semirings with idempotent addition. The problems are to minimize or maximize functions defined on vectors over idempotent semifields (semirings with multiplicative inverses). We examine several problems of rating alternatives from pairwise comparisons, including problems with constraints on the final scores of alternatives, and multi-criteria rating problems. We reduce the problems to the log-Chebyshev approximation of pairwise comparison matrices by reciprocal matrices of unit rank, and then represent the approximation problems as optimization problems in the tropical mathematics setting. To solve these problems, methods of tropical optimization are used to provide direct, complete solutions in a compact vector form. We discuss the applicability of the results to real-world problems, and provide numerical examples.

AB - We consider unconstrained and constrained optimization problems in the framework of the tropical (idempotent) mathematics, which focuses on the theory and applications of semirings with idempotent addition. The problems are to minimize or maximize functions defined on vectors over idempotent semifields (semirings with multiplicative inverses). We examine several problems of rating alternatives from pairwise comparisons, including problems with constraints on the final scores of alternatives, and multi-criteria rating problems. We reduce the problems to the log-Chebyshev approximation of pairwise comparison matrices by reciprocal matrices of unit rank, and then represent the approximation problems as optimization problems in the tropical mathematics setting. To solve these problems, methods of tropical optimization are used to provide direct, complete solutions in a compact vector form. We discuss the applicability of the results to real-world problems, and provide numerical examples.

M3 - Abstract

SP - 38

ER -

Кривулин НК. Methods of tropical optimization in rating alternatives based on pairwise comparisons. 2016. Abstract from Annual International Conference of the German Operations Research Society, Hamburg, .