Extremum in the problem of paired comparisons

Objectives. An analysis of the problem of evaluating alternatives based on the results of expert paired comparisons is presented. The importance and relevance of this task is due to its numerous applications in a variety of fields, whether in the technical and natural sciences or in the humanities, ranging from construction to politics. In such contexts, the problem frequently arises concerning how to calculate an objective ratings vector based on expert evaluations. In terms of a mathematical formulation, the problem of finding the vector of objective ratings can be reduced to approximating the matrices of paired comparisons by consistent matrices.

Methods. Analytical analysis and higher algebra methods are used. For some special cases, the results of numerical calculations are given.

Results. The theorem stating that there is always a unique and consistent matrix that optimally approximates a given inversely symmetric matrix in a log-Euclidean metric is proven. In addition, derived formulas for calculating such a consistent matrix are presented. For small dimensions, examples are considered that allow the results obtained according to the derived formula to be compared with results for other known methods of finding a consistent matrix, i.e., for calculating the eigenvector and minimizing the discrepancy in the log-Chebyshev metric. It is proven that all these methods lead to the same result in dimension 3, while in dimension 4 all results are already different.

Conclusions. The results obtained in the paper allow us to calculate the vector of objective ratings based on expert evaluation data. This method can be used in strategic planning in cases where conclusions and recommendations are possible only on the basis of expert evaluations.

1. Korobov V.B. Teoriya i praktika ekspertnykh metodov (Theory and Practice of Expert Methods). Moscow: INFRA-M; 2019. 279 p. (in Russ.). ISBN 978-5-16015053-6. https://doi.org/10.12737/monography_5caee0067f1835.43206494

2. Andreichikov A.V., Andreichikova O.N. Analiz, sintez, planirovanie reshenii v ekonomike (Analysis, Synthesis, Planning of Decisions in the Economy). Moscow: Finansy i statistika; 2004. 467 p. (in Russ.). ISBN 5-279-02901-7

3. Saaty T. Prinyatie reshenii. Metod analiza ierarkhii (Decision Making. Hierarchy Analysis Method). Moscow: Radio i svyaz’; 1993. 341 р. (in Russ.). ISBN 5-25600443-3

4. Saaty T. Prinyatie reshenii pri zavisimostyakh i obratnykh svyazyakh: analiticheskie seti (Decision Making with Dependencies and Feedbacks: Analytical Networks). Moscow: URSS; 2010. 357 p. (in Russ.). ISBN 978-5397-01622-3

5. Breiman L. Random forests. Machine Learning. 2001;45(1): 5–32. https://doi.org/10.1023/A:1010933404324

6. Belov V., Tatarintsev A., Nikulchev E. Comparative characteristics of big data storage formats. J. Phys.: Conf. Ser. 2021;1727(1):012005. http://doi.org/10.1088/1742-6596/1727/1/012005

7. Belov V., Tatarintsev A., Nikulchev E. Choosing a data storage format in the Apache Hadoop system based on experimental evaluation using Apache Spark. Symmetry. 2021;13(2):195. https://doi.org/10.3390/sym13020195

8. Moro Visconti R., Morea D. Big data for the sustainability of healthcare project financing. Sustainability. 2019;11(13):3748. https://doi.org/10.3390/su11133748

9. Gusev A., Ilin D., Nikulchev E. The dataset of the experimental evaluation of software components for application design selection directed by the artificial bee colony algorithm. Data. 2020;5(3):59. https://doi.org/10.3390/data5030059

10. Munir R.F., Abelló A., Romero O., Thiele M., Lehner W. A cost-based storage format selector for materialized results in bigdata frameworks. Distrib. Parallel Databases. 2020;38(3):335–364. https://doi.org/10.1007/s10619-019-07271-0

11. Gusev A., Ilin D., Kolyasnikov P., Nikulchev E. Effective selection of software components based on experimental evaluations of quality of operation. Eng. Lett. 2020;28(2):420–427.

12. Krivulin N.K., Ageev V.A., Gladkikh I.V. Application of methods of tropical optimization for evaluating alternatives based on pairwise comparisons. Vestnik Sankt Peterburgskogo universiteta. Prikladnaya matematika. Informatika. Protsessy upravleniya = Vestnik of Saint Petersburg University. Applied Mathematics. Computer Science. Control Processes. 2017;13(1):27–41. https://doi.org/10.21638/11701/spbu10.2017.103

13. Litvinov G.L. The Maslov dequantization, idempotent and tropical mathematics: a briff introduction. Zapiski nauchnykh seminarov Sankt-Peterburgskogo otdeleniya matematicheskogo instituta im. V.A. Steklova RAN (Zapiski Nauchnykh Seminarov POMI). 2005;326(13):145–182 (in Russ.).

14. Gantmakher F R. Teoriya matrits (Matrix Theory). Moscow: Fizmatlit; 2004. 560 p. (in Russ.). ISBN 5-92210524-8

15. Evseeva O.A., Pulkin I.S., Tatarintsev A.V. On the problem of processing expert judgments. In: Innovatsionnye tekhnologii v elektronike i priborostroenii: sbornik trudov konferentsii (Innovative technologies in electronics and instrumentation: collection of conference proceedings). Moscow: MIREA; 2021. V. 1. P. 355–359 (in Russ.).