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Algebraic Solution to Constrained Bi-Criteria Decision Problem of Rating Alternatives through Pairwise Comparisons

St. Petersburg State University, Universitetskaya Emb. 7/9, 199034 St. Petersburg, Russia
Academic Editor: Vassili Kolokoltsov
Mathematics 2021, 9(4), 303; https://doi.org/10.3390/math9040303
Received: 5 January 2021 / Revised: 26 January 2021 / Accepted: 29 January 2021 / Published: 4 February 2021
We consider a decision-making problem to evaluate absolute ratings of alternatives from the results of their pairwise comparisons according to two criteria, subject to constraints on the ratings. We formulate the problem as a bi-objective optimization problem of constrained matrix approximation in the Chebyshev sense in logarithmic scale. The problem is to approximate the pairwise comparison matrices for each criterion simultaneously by a common consistent matrix of unit rank, which determines the vector of ratings. We represent and solve the optimization problem in the framework of tropical (idempotent) algebra, which deals with the theory and applications of idempotent semirings and semifields. The solution involves the introduction of two parameters that represent the minimum values of approximation error for each matrix and thereby describe the Pareto frontier for the bi-objective problem. The optimization problem then reduces to a parametrized vector inequality. The necessary and sufficient conditions for solutions of the inequality serve to derive the Pareto frontier for the problem. All solutions of the inequality, which correspond to the Pareto frontier, are taken as a complete Pareto-optimal solution to the problem. We apply these results to the decision problem of interest and present illustrative examples. View Full-Text
Keywords: idempotent semifield; tropical bi-objective optimization; Pareto-optimal solution; constrained bi-criteria decision problem; pairwise comparisons idempotent semifield; tropical bi-objective optimization; Pareto-optimal solution; constrained bi-criteria decision problem; pairwise comparisons
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MDPI and ACS Style

Krivulin, N. Algebraic Solution to Constrained Bi-Criteria Decision Problem of Rating Alternatives through Pairwise Comparisons. Mathematics 2021, 9, 303. https://doi.org/10.3390/math9040303

AMA Style

Krivulin N. Algebraic Solution to Constrained Bi-Criteria Decision Problem of Rating Alternatives through Pairwise Comparisons. Mathematics. 2021; 9(4):303. https://doi.org/10.3390/math9040303

Chicago/Turabian Style

Krivulin, Nikolai. 2021. "Algebraic Solution to Constrained Bi-Criteria Decision Problem of Rating Alternatives through Pairwise Comparisons" Mathematics 9, no. 4: 303. https://doi.org/10.3390/math9040303

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