Solving multicriteria problems of rating alternatives based on pairwise comparisons. Part II

  • Nikolai Krivulin Saint Petersburg State University, 7–9 Universitetskaya emb., 199034, Saint Petersburg, Russia http://orcid.org/0000-0003-3070-9355
  • Denis Yakovlev Saint Petersburg State University, 7–9 Universitetskaya emb., 199034, Saint Petersburg, Russia
Keywords: multicriteria decision making problems, pairwise comparisons, analytical hi erarchy process, tropical mathematics

Abstract

A number of well-known multicriteria problems of evaluating alternatives based on pairwise comparisons are considered. In these problems, given matrices containing results of paired comparisons of criteria and alternatives, one needs to find an absolute rating (priority, weight) of each alternative for decision making. Solutions to the problems are presented obtained using the method of analytical hierarchy process, the method of weighted geometric means, and the method of log-Chebyshev approximation of pairwise comparison matrices. The results obtained show that for some problems, solutions found by different methods may significantly differ from each other. In such cases, the decision to choose the best alternative may be based on additional analysis and comparison of the results of the problem solution obtained by all the methods used.

Author Biographies

Nikolai Krivulin, Saint Petersburg State University, 7–9 Universitetskaya emb., 199034, Saint Petersburg, Russia

Nikolai Krivulin, Doctor of Sciences (Phys.-Math.), Professor, Department of Statistical Modeling, St. Petersburg State University, nkk@math.spbu.ru

Denis Yakovlev, Saint Petersburg State University, 7–9 Universitetskaya emb., 199034, Saint Petersburg, Russia

Student, Saint Petersburg State University, denis.yakovlev03@bk.ru

References

V. V. Podinovskii and V. D. Nogin, Pareto-optimal’nye resheniya mnogokriterial’nykh zadach [Paretooptimal solutions to multicriteria problems], Moscow: Nauka, 1982 (in Russian).

T. Saati, Prinyatie reshenij. Metod analiza ierarkhij [Decision making. Hierarchy analysis method], Moscow: Radio i svyaz’, 1993 (in Russian).

V. D. Nogin, Prinyatie reshenii v mnogokriterial’noi srede: kolichestvennyi podkhod [Decision making in a multicriteria environment: a quantitative approach], Moscow: Fizmatlit, 2002 (in Russian).

T. L. Saaty, “A scaling method for priorities in hierarchical structures,” Journal of Mathematical Psychology, vol. 15, no. 3, pp. 234–281, 1977; doi:10.1016/0022-2496(77)90033-5

T. L. Saaty, “On the measurement of intangibles: A principal eigenvector approach to relative measurement derived from paired comparisons,” Notices of the American Mathematical Society, vol. 60, no. 2, pp. 192–208, 2013; doi:10.1090/noti944

R. Narasimhan, “A geometric averaging procedure for constructing supertransitive approximation to binary comparison matrices,” Fuzzy Sets and Systems, vol. 8, no. 1, pp. 53–61, 1982; doi:10.1016/0165-0114(82)90029-X

G. Crawford and C. Williams, “A note on the analysis of subjective judgment matrices,” Journal of Mathematical Psychology, vol. 29, no. 4, pp. 387–405, 1985; doi:10.1016/0022-2496(85)90002-1

J. Barzilai, W. D. Cook, and B. Golany, “Consistent weights for judgements matrices of the relative importance of alternatives,” Operations Research Letters, vol. 6, no. 3, pp. 131–134, 1987; doi:10.1016/0167-6377(87)90026-5

N. Krivulin and S. Sergeev, “Tropical implementation of the Analytical Hierarchy Process decision method,” Fuzzy Sets and Systems, vol. 377, pp. 31–51, 2019; doi:10.1016/j.fss.2018.10.013

N. K. Krivulin and V. A. Ageev, “Methods of tropical optimization in multicriteria problems of rating alternatives from pairwise comparisons,” Vestnik of St Petersburg University. Applied

Mathematics. Computer Science. Control Processes, vol. 15, no. 4, pp. 472–488, 2019 (in Russian); doi:10.21638/11702/spbu10.2019.405

N. Krivulin, “Application of tropical optimization for solving multicriteria problems of pairwise comparisons using log-Chebyshev approximation,” International Journal of Approximate Reasoning, vol. 169. p. 109168, 2024; doi:10.1016/j.ijar.2024.109168

F. L. Baccelli, G. Cohen, G. J. Olsder, and J. P. Quadrat, Synchronization and linearity, Wiley Series in Probability and Statistics, Chichester, UK: Wiley, 1993.

V. P. Maslov and V. N. Kolokol’tsov, “Idempotentnyi analiz i ego primenenie v optimal’nom upravlenii” [Idempotent analysis and its application in optimal control], Moscow: Fizmatlit, 1994 (in Russian).

J. S. Golan, Semirings and affine equations over them, Mathematics and Its Applications, vol. 556, New York: Springer, 2003; doi:10.1007/978-94-017-0383-3

B. Heidergott, G. J. Olsder, and J. van der Woude, Max Plus at Work, Princeton Series in Applied Mathematics, Princeton, NJ: Princeton Univ. Press, 2006.

N. K. Krivulin, Metody idempotentnoi algebry v zadachakh modelirovaniya i analiza slozhnykh sistem [Idempotent algebra methods in modeling and analysis of complex systems], St. Petersburg: Izdatel’stvo Sankt-Peterburgskogo universiteta, 2009.

T. L. Saaty, “The modern science of multicriteria decision making and its practical applications: The AHP/ANP approach,” Operations Research, vol. 61, no. 5, pp. 1101–1118, 2019 (in Russian); doi:10.2307/24540487

T. L. Saaty, “How to make a decision: The analytic hierarchy process,” European Journal of Operational Research, vol. 48, no. 1, pp. 9–26, 1990; 10.1016/0377-2217(90)90057-I

P. Bagchi and R. P. Rao, “Decision making in mergers: An application of the analytic hierarchy process,” Managerial and Decision Economics, vol. 13, no. 2, pp. 91–99, 1992; doi:10.1002/mde.4090130202

D. S. K. Goshal, S. K. Naskar, and D. D. Bose, “AHP in assessing performance of diploma institutes—A case study,” Journal of Technical Education and Training, vol. 3, pp. 67–81, 2012.

N. K. Krivulin, T. Abildaev, V. D. Gorshechnikova, D. Kapatsa, E. A. Magdich, and A. A. Mandrikova, “On solving multicriteria decision making problems based on pairwise comparisons,” Computer Tools in Education, no. 2, pp. 27–58, 2020 (in Russian); doi:10.32603/2071-2340-2020-2-27-58

N. K. Krivulin, D. S. Bulgakova, D. A. Grigoriev, K. I. Nagumanova, A. S. Prinkov, Y. A. Salova, and A. A. Filatova, “Solving multicriteria problems of rating alternatives based on pairwise comparisons,” Computer Tools in Education, no. 2, pp. 5–29, 2024 (in Russian); doi:10.32603/2071-2340-2024-2-5-29

Published
2025-12-13
How to Cite
Krivulin, N., & Yakovlev, D. (2025). Solving multicriteria problems of rating alternatives based on pairwise comparisons. Part II. Computer Tools in Education, (2), 5-25. https://doi.org/10.32603/2071-2340-2025-2-5-23
Section
Algorithmic mathematics and mathematical modelling