Solution of the least squares method problem of pairwise comparison matrices

  • Sándor BozókiEmail author
Original Paper


The aim of the paper is to present a new global optimization method for determining all the optima of the Least Squares Method (LSM) problem of pairwise comparison matrices. Such matrices are used, e.g., in the Analytic Hierarchy Process (AHP). Unlike some other distance minimizing methods, LSM is usually hard to solve because of the corresponding nonlinear and non-convex objective function. It is found that the optimization problem can be reduced to solve a system of polynomial equations. Homotopy method is applied which is an efficient technique for solving nonlinear systems. The paper ends by two numerical example having multiple global and local minima.


Pairwise comparison matrix Least squares approximation Polynomial system Homotopy method Incomplete pairwise comparison matrix 


  1. Barzilai J, Cook WD, Golany B (1987) Consistent weights for judgements matrices of the relative importance of alternatives. Oper Res Lett 6: 131–134CrossRefGoogle Scholar
  2. Blankmeyer E (1987) Approaches to consistency adjustments. J Optim Theory Appl 54: 479–488CrossRefGoogle Scholar
  3. de Borda JC (1781) Mémoire sur les électiones au scrutin. Histoire de l’Académie Royale des Sciences, ParisGoogle Scholar
  4. Bozóki S (2003) A method for solving LSM problems of small size in the AHP. Central Eur J Oper Res 11: 17–33Google Scholar
  5. Bozóki S, Lewis R (2005) Solving the Least Squares Method problem in the AHP for 3 × 3 and 4 × 4 matrices. Central Eur J Oper Res 13: 255–270Google Scholar
  6. Bozóki S (2006) Weights from the least squares approximation of pairwise comparison matrices (in Hungarian, Súlyok meghatározása páros összehasonlítás mátrixok legkisebb négyzetes közelítése alapján). Alkal-mazott Matematikai Lapok 23: 121–137Google Scholar
  7. Buchberger B (1965) On Finding a Vector Space Basis of the Residue Class Ring Modulo a Zero Dimensional Polynomial Ideal (in German). Ph.D. Dissertation, University of Innsbruck, Department of Mathematics, AustriaGoogle Scholar
  8. Budescu DV, Zwick R, Rapoport A (1986) A comparison of the Eigenvector Method and the Geometric Mean procedure for ratio scaling. Appl Psychol Measure 10: 69–78CrossRefGoogle Scholar
  9. Chu ATW, Kalaba RE, Spingarn K (1979) A comparison of two methods for determining the weight belonging to fuzzy sets. J Optim Theory Appl 4: 531–538CrossRefGoogle Scholar
  10. Cook WD, Kress M (1988) Deriving weights from pairwise comparison ratio matrices: An axiomatic approach. Eur J Oper Res 37: 355–362CrossRefGoogle Scholar
  11. Condorcet M (1785) Essai sur l’Application de l’Analyse à la Probabilité des Décisions Rendues á la Pluralité des Voix, ParisGoogle Scholar
  12. Crawford G, Williams C (1985) A note on the analysis of subjective judgment matrices. J Math Psychol 29: 387–405CrossRefGoogle Scholar
  13. De Jong P (1984) A statistical approach to Saaty’s scaling methods for priorities. J Math Psychol 28: 467–478CrossRefGoogle Scholar
  14. DeGraan JG (1980) Extensions of the multiple criteria analysis method of T.L. Saaty (Technical Report m.f.a. 80-3) Leischendam, The Netherlands: National Institute for Water Supply. Presented at EURO IV, Cambridge, England, July 22-25Google Scholar
  15. Drexler FJ (1978) Eine Methode zur Berechnung sämtlicher Lösungen von Polynomgleichungssystemen. Numer Math 29: 45–58CrossRefGoogle Scholar
  16. Farkas A, Lancaster P, Rózsa P (2003) Consistency adjustment for pairwise comparison matrices. Numer Linear Algebra Appl 10: 689–700CrossRefGoogle Scholar
  17. Fechner GT (1860) Elemente der Psychophysik. Breitkopf und Härtel, LeipzigGoogle Scholar
  18. Fülöp J (2008) A method for approximating pairwise comparison matrices by consistent matrices. J Global Optim (in print)Google Scholar
  19. Gao T, Li TY, Wang X (1999) Finding isolated zeros of polynomial systems in \({\mathbb{C}^n}\) with stable mixed volumes. J Symbolic Comput 28: 187–211CrossRefGoogle Scholar
  20. Garcia CB, Zangwill WI (1979) Finding all solutions to polynomial systems and other systems of equations. Math Program 16: 159–176CrossRefGoogle Scholar
  21. Gass SI, Rapcsák T (2004) Singular value decomposition in AHP. Eur J Oper Res 154: 573–584CrossRefGoogle Scholar
  22. Golany B, Kress M (1993) A multicriteria evaluation of methods for obtaining weights from ratio-scale matrices. Eur J Oper Res 69: 210–220CrossRefGoogle Scholar
  23. Hashimoto A (1994) A note on deriving weights from pairwise comparison ratio matrices. Eur J Oper Res 73: 144–149CrossRefGoogle Scholar
  24. Jensen RE (1983) Comparison of Eigenvector, Least squares, Chi square and Logarithmic least square methods of scaling a reciprocal matrix. Working Paper 153.
  25. Jensen RE (1984) An Alternative Scaling Method for Priorities in Hierarchical Structures. J Math Psychol 28: 317–332CrossRefGoogle Scholar
  26. Kéri G (2005) Criteria for pairwise comparison matrices (in Hungarian. Kritériumok páros összehasonlítás mátrixokra). Szigma 36: 139–148Google Scholar
  27. Li TY (1997) Numerical solution of multivariate polynomial systems by homotopy continuation methods. Acta Numer 6: 399–436CrossRefGoogle Scholar
  28. Saaty TL (1980) The analytic hierarchy process. McGraw-Hill, New YorkGoogle Scholar
  29. Saaty TL, Vargas LG (1984) Comparison of eigenvalues, logarithmic least squares and least squares methods in estimating ratios. Math Model 5: 309–324CrossRefGoogle Scholar
  30. Thorndike EL (1920) A constant error in psychological ratings. J Appl Psychol 4: 25–29CrossRefGoogle Scholar
  31. Thurstone LL (1927) The Method of Paired Comparisons for Social Values. J Abnormal Social Psychol 21: 384–400CrossRefGoogle Scholar
  32. Zahedi F (1986) A simulation study of estimation methods in the analytic hierarchy process. Socio-Econ Plan Sci 20: 347–354CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Laboratory and Department of Operations Research and Decision SystemsComputer and Automation Research Institute, Hungarian Academy of SciencesBudapestHungary

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