Application of methods of tropical optimization for evaluating alternatives based on pairwise comparisons

N. Krivulin ; V. A. Ageev ; I. V. Gladkikh

Vestnik Sankt-Peterburgskogo universiteta. Prikladnaâ matematika, informatika, processy upravleniâ, Tome 13 (2017) no. 1, pp. 27-41

Résumé

Solution to the problem of evaluating alternatives based on pairwise comparisons is considered, using methods of tropical optimization. The problem of deriving the vector of scores for alternatives reduces to the approximation of pairwise comparison matrices by consistent matrices in the sense of the log-Chebyshev metric. Then the approximation problem is formulated and solved in terms of tropical mathematics. The results obtained are represented in compact vector form, ready for further analysis and practical computations. In the case when the solution is non-unique (up to a positive factor), it is suggested that the set of solutions be characterized via two solutions that are, in some sense, the best and worst solutions. As the best solution, the vector is taken which best differentiates between the alternatives with the highest and lowest scores, and as the worst, the vector which worth differentiates these alternatives. It is shown that these vectors can be obtained using methods of tropical optimization. To illustrate the results obtained, solution examples for problems of evaluating the scores of alternatives are given. Refs 23.

Détail

Keywords: tropical mathematics, idempotent semifield, tropical optimization, pairwise comparison matrix, log-Chebyshev metric
Mots-clés : consistent matrix, matrix approximation.

Bibliographie

[1] Gladkikh I. V., Svetlanova Zh. V., “The measurement of product characteristics importance in marketing research”, Vestnik of Saint Petersburg University. Series 8. Management, 2006, no. 2, 65–87 (In Russian)

[2] Thurstone L. L., “A law of comparative judgment”, Psychological Review, 34:4 (1927), 273–286 | DOI

[3] David H. A., The method of paired comparisons, Griffin's Statistical Monographs and Courses, 12, Griffin, London, 1963, 124 pp. | MR

[4] Saaty T. L., The analytic hierarchy process: planning, priority setting, resource allocation, McGraw-Hill, New York, 1980, 281 pp. | MR | Zbl

[5] Nogin V. D., “A simplified variant of a method for the analysis of hierarchies, based on a nonlinear convolution of criteria”, Journal of Computational Mathematics and Mathematical Physics, 44:7 (2004), 1261–1270 (In Russian) | Zbl

[6] Saaty T. L., Vargas L. G., “Comparison of eigenvalue, logarithmic least squares and least squares methods in estimating ratios”, Math. Modelling, 5:5 (1984), 309–324 | DOI | MR | Zbl

[7] Barzilai J., “Deriving weights from pairwise comparison matrices”, J. Oper. Res. Soc., 48:12 (1997), 1226–1232 | DOI | Zbl

[8] Chu M. T., “On the optimal consistent approximation to pairwise comparison matrices”, Linear Algebra Appl., 272:1–3 (1998), 155–168 | DOI | MR | Zbl

[9] Farkas A., Lancaster P., Rózsa P., “Consistency adjustments for pairwise comparison matrices”, Numer. Linear Algebra Appl., 10:8 (2003), 689–700 | DOI | MR | Zbl

[10] Elsner L., van den Driessche P., “Max-algebra and pairwise comparison matrices. II”, Linear Algebra Appl., 432:4 (2010), 927–935 | DOI | MR | Zbl

[11] Tran N. M., “Pairwise ranking: Choice of method can produce arbitrarily different rank order”, Linear Algebra Appl., 438:3 (2013), 1012–1024 | DOI | MR | Zbl

[12] Gursoy B. B., Mason O., Sergeev S., “The analytic hierarchy process, max algebra and multi-objective optimisation”, Linear Algebra Appl., 438:7 (2013), 2911–2928 | DOI | MR | Zbl

[13] Maslov V. P., Kolokol'tsov V. N., Idempotent analysis and its applications in optimal control, Fizmatlit Publ., M., 1994, 144 pp. (In Russian)

[14] Golan J. S., Semirings and affine equations over them, Mathematics and Its Applications, 556, Springer, New York, 2003, 256 pp. | MR

[15] Heidergott B., Olsder G. J., van der Woude J., Max plus at work, Princeton Series in Applied Mathematics, Princeton University Press, Princeton, 2006, 226 pp. | MR | Zbl

[16] Krivulin N. K., Methods of idempotent algebra for problems in modeling and analysis of complex systems, Saint Petersburg University Publ., Saint Petersburg, 2009, 256 pp. (In Russian)

[17] Krivulin N. K., Gladkikh I. V., “Tropical mathematics-based methods for constructing a consistent matrix of pairwise comparisons”, Models and methods of tropical mathematics with applications to problems in economics and management, VVM Publ., Saint Petersburg, 2013, 4–32 (In Russian)

[18] Krivulin N. K., Gladkikh I. V., “Computation of the consistent pairwise comparison matrix in marketing research by using methods of tropical mathematics”, Vestnik of Saint Petersburg University. Series 8. Management, 2015, no. 1, 3–43 (In Russian)

[19] Krivulin N., “Rating alternatives from pairwise comparisons by solving tropical optimization problems”, 12th Intern. Conference on Fuzzy Systems and Knowledge Discovery (FSKD), eds. Z. Tang, J. Du, S. Yin, L. He, R. Li, IEEE, 2015, 162–167 | DOI | MR

[20] Krivulin N., “Using tropical optimization techniques to evaluate alternatives via pairwise comparisons”, Proc. 7th SIAM Workshop on Combinatorial Scientific Computing, eds. A. H. Gebremedhin, E. G. Boman, B. Ucar, SIAM, Philadelphia, 2016, 62–72 | DOI

[21] Krivulin N., “Extremal properties of tropical eigenvalues and solutions to tropical optimization problems”, Linear Algebra Appl., 468 (2015), 211–232 | DOI | MR | Zbl

[22] Krivulin N., “Solving a tropical optimization problem via matrix sparsification”, Relational and Algebraic Methods in Computer Science, Lecture Notes in Comput. Sci., 9348, eds. W. Kahl, M. Winter, J. N. Oliveira, Springer, Cham, 2015, 326–343 | DOI | MR | Zbl

[23] Krivulin N., “A maximization problem in tropical mathematics: A complete solution and application examples”, Informatica, 27:3 (2016), 587–606 | DOI