Mathematical Methods of Operations Research

, Volume 64, Issue 2, pp 271–284 | Cite as

Inferring Efficient Weights from Pairwise Comparison Matrices

  • R. Blanquero
  • E. Carrizosa
  • E. CondeEmail author
Original Article


Several multi-criteria-decision-making methodologies assume the existence of weights associated with the different criteria, reflecting their relative importance.One of the most popular ways to infer such weights is the analytic hierarchy process, which constructs first a matrix of pairwise comparisons, from which weights are derived following one out of many existing procedures, such as the eigenvector method or the least (logarithmic) squares. Since different procedures yield different results (weights) we pose the problem of describing the set of weights obtained by “sensible” methods: those which are efficient for the (vector-) optimization problem of simultaneous minimization of discrepancies. A characterization of the set of efficient solutions is given, which enables us to assert that the least-logarithmic-squares solution is always efficient, whereas the (widely used) eigenvector solution is not, in some cases, efficient, thus its use in practice may be questionable.


Decision analysis Multi-objective Optimization Fractional programming 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Bryson N (1995) A goal programming method for generating priority vectors. J Oper Res Soc 46:641–648zbMATHCrossRefGoogle Scholar
  2. Carrizosa E, Conde E, Fernández FR, Muñoz FR, Pareto J (1995) Pareto optimality in linear regression. J Math Anal Appl 190:129–141zbMATHCrossRefMathSciNetGoogle Scholar
  3. Chankong V, Haimes Y (1983) Multiobjective decision making. North-Holland, AmsterdamzbMATHGoogle Scholar
  4. Choo EU, Wedley WC (2004) A common framework for deriving preference values from pairwise comparison matrices. Comput Oper Res 31:893–908zbMATHCrossRefGoogle Scholar
  5. Connell A, Corless RM (1993) An experimental interval arithmetic package in Maple. Interval Comput 2:120–134Google Scholar
  6. Cook WD, Kress M (1988) Deriving weights from pairwise comparison ratio matrices: an axiomatic approach. Eur J Oper Res 37(3):355–362zbMATHCrossRefMathSciNetGoogle Scholar
  7. Hoffman AJ (1960) Some recent applications of the theory of linear inequalities to extremal combinatorial analysis. In: Proceedings of symposia in applied mathematics, Vol 10. Bellman R, Hall M, Jr (eds), American Mathematical Society, Providence, pp113–127Google Scholar
  8. Lootsma FA (1996) A model for the relative importance of the criteria in the multiplicative ahp and smart. Euro J Oper Res 94:467–476zbMATHCrossRefGoogle Scholar
  9. Martos B (1975) Nonlinear Programming. Theory and methods. North-Holland, AmsterdamzbMATHGoogle Scholar
  10. McCormick ST (1997) How to compute least infeasible flows. Math Program Ser B 78(2):179–194CrossRefMathSciNetGoogle Scholar
  11. Monagan M, Geddes K, Heal K, Labahn G, Vorkoetter S (1997) Maple V programming guide for release 5. Springer, Berlin Heidelberg New YorkGoogle Scholar
  12. Ramanathan R (1997) A note on the use of goal programming for the multiplicative ahp. J Multi-criteria Decis Anal 6:296–307zbMATHCrossRefGoogle Scholar
  13. Saaty TL (1977) A scaling method for priorities in hierarchical structures. J Math Psychol 15:234–281zbMATHCrossRefMathSciNetGoogle Scholar
  14. Saaty TL (1980) Multicriteria decision making: the Analytic hierarchy Process. McGraw-Hill, New YorkGoogle Scholar
  15. Saaty TL (1990) Eigenvector and logaritmic least squares. Euro J Oper Rese, 48:156–160zbMATHCrossRefGoogle Scholar
  16. Saaty TL (1994) Fundamentals of decision making. RSW Publications, PittsburgGoogle Scholar
  17. Saaty TL (1994) How to make a decision: The analytic hierarchy process. Interfaces 24:19–43CrossRefGoogle Scholar
  18. Schaible S (1995) Fractional programming. Horst R, Pardalos PM, (eds), In: Handbook of global optimization, pp 495–608Google Scholar
  19. Steuer R (1986) Multiple criteria optimization: theory, computation, application. Wiley, New YorkGoogle Scholar
  20. Zeleny M (1982) Multiple criteria decision making. McGraw-Hill, New YorkzbMATHGoogle Scholar

Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  1. 1.Facultad de MatemáticasUniversidad de SevillaSevilleSpain

Personalised recommendations