The Additive and the Multiplicative AHP

  • Freerk A. Lootsma
Part of the Applied Optimization book series (APOP, volume 8)


The Analytic Hierarchy Process (AHP) of Saaty (1980) is a widely used method for MCDA, presumably because it elicitates preference information from the decision makers in a manner which they find easy to understand. The basic step is the pairwise comparison of two so-called stimuli, two alternatives under a given criterion, for instance, or two criteria. The decision maker is requested to state whether he/she is indifferent between the two stimuli or whether he/she has a weak, strict, strong, or very strong preference for one of them. The original AHP has been criticized in the literature because the algorithmic steps do not properly take into account that the method is based upon ratio information. The shortcomings can easily be avoided in the Additive and the Multiplicative AHP to be discussed in the present chapter. The Additive AHP is the SMART procedure with pairwise comparisons on the basis of difference information. The Multiplicative AHP with pairwise comparisons on the basis of ratio information is a variant of the original AHP. There is a logarithmic relationship between the Additive AHP (SMART) and the Multiplicative AHP. Both versions can easily be fuzzified. The reasons why we deviate from the original AHP will be explained at the end of this chapter.


Fuzzy Logic Analytic Hierarchy Process Criterion Weight Final Grade Indifference Curve 
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References to Chapter 5

  1. 1.
    Barzilai, J., Cook, W.D., and Golany, B., “Consistent Weights for Judgement Matrices of the Relative Importance of Alternatives”. Operations Research Letters 6, 131–134, 1987.MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Barzilai, I, and Golany, B., “Deriving Weights from Pairwise Comparison Matrices: the Additive Case”. Operations Research Letters 9, 407–410, 1990.MATHCrossRefGoogle Scholar
  3. 3.
    Barzilai, J., and Golany, B., “AHP Rank Reversal, Normalization, and Aggregation Rules”. INFOR 32, 57–64, 1994.MATHGoogle Scholar
  4. 4.
    Barzilai, J., and Lootsma FA., “Power Relations and Group Aggregation in the Multiplicative AHP and SMART”. To appear in the Journal of Multi-Criteria Decision Analysis, 1997. In the same issue there will be critical comments by P. Korhonen, O. Larichev, and L.G. Vargas, and a response by F.A. Lootsma and J. Barzilai.Google Scholar
  5. 5.
    Belton, V., and Gear, A.E., “On a Shortcoming of Saaty’s Method of Analytical Hierarchies”. Omega 11, 227–230, 1983.CrossRefGoogle Scholar
  6. 6.
    Boender, C.G.E., Graan, J.G. de, and Lootsma, F.A., “Multi-Criteria Decision Analysis with Fuzzy Pairwise Comparisons”. Fuzzy Sets and Systems 29, 133–143, 1989.MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Budescu, D.V., Crouch, B.D., and Morera, O.F., “A Multi-Criteria Comparison of Response Scales and Scaling Methods in the AHP”. In W.C. Wedley (ed.), Proceedings of the Fourth International Symposium on the AHP. Simon Fraser University, Burnaby, B.C., Canada, 1996, pp. 280–291.Google Scholar
  8. 8.
    Cogger, K.O., and Yu, PL, “Eigenweight Vectors and Least-Distance Approximations for Revealed Preferences in Pairwise Weight Ratios”. Journal of Optimization Theory and Applications 46, 483–491, 1985.MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Crawford, G., and Williams, C, “A Note on the Analysis of Subjective Judgement Matrices”. Journal of Mathematical Psychology 29, 387–405, 1985.MATHCrossRefGoogle Scholar
  10. 10.
    Dijk, H.K. van, Kloek, T., and Boender, C.G.E., “Posterior Moments Computed by Mixed Integration”. Journal of Econometrics 29, 3–18, 1985.MATHCrossRefGoogle Scholar
  11. 11.
    Dyer, J.S., “Remarks on the Analytic Hierachy Process”. Management Science 36, 249–258, 1990. In the same issue there are apologies by T.L. Saaty (259–268), P.T. Harker and L. Vargas (269–273), and a further clarification by J.S. Dyer (274–275).MathSciNetCrossRefGoogle Scholar
  12. 12.
    French, S., “Decision Theory, an Introduction to the Mathematics of Rationality”. Ellis Horwood, Chichester, 1988.MATHGoogle Scholar
  13. 13.
    Gennip, C.G.E. van, Hulshof, J.A.M., and Lootsma, F.A., “A Multi-Criteria Evaluation of Diseases in a Study for Public-Health Planning”. To appear in the European Journal of Operational Research, 1997.Google Scholar
  14. 14.
    H. Johnson, C.R. Beine, W.B., and Wang, T.J., “Right-Left Asymmetry in an Eigenvector Ranking Procedure”. Journal of Mathematical Psychology 19, 61–64, 1979.MathSciNetCrossRefGoogle Scholar
  15. 15.
    Keeney, R., and Raiffa, H., “Decisions with Multiple Objectives: Preferences and Value Trade-offs”. Wiley, New York, 1976.Google Scholar
  16. 16.Laarhoven, P.J.M. van, and Pedrycz, W., “A Fuzzy Extension of Saaty’s Priority Theory”. Fuzzy Sets and Systems 11, 229–241, 1983.MathSciNetCrossRefGoogle Scholar
  17. 17.
    Lootsma, F.A., “Saaty’s Priority Theory and the Nomination of a Senior Professor in Operations Research”. European Journal of Operational Research 4, 380–388, 1980.CrossRefGoogle Scholar
  18. 18.
    Lootsma, F.A., “Modélisation du Jugement Humain dans l’Analyse Multicritère au Moyen de Comparaisons par Paires”. RAIRO/Recherche Opérationnelle 21, 241–257, 1987.MathSciNetMATHGoogle Scholar
  19. 19.
    Lootsma, F.A., “Numerical Scaling of Human Judgement in Pairwise-Comparison Methods for Fuzzy Multi-Criteria Decision Analysis”. In G. Mitra (ed.), “Mathematical Models for Decision Support”. Springer, Berlin, 1988, pp. 57–88.CrossRefGoogle Scholar
  20. 20.
    Lootsma, F.A., “Fuzzy Performance Evaluation of Nonlinear Optimization Methods, with Sensitivity Analysis of the Final Scores”. Journal of Information and Optimization Sciences 10, 15–44, 1989.MathSciNetMATHGoogle Scholar
  21. 21.
    Lootsma, F.A., “Scale Sensitivity in the Multiplicative AHP and SMART”. Journal of Multi-Criteria Decision Analysis 2, 87–110, 1993.MATHCrossRefGoogle Scholar
  22. 22.
    Lootsma, F.A., Boonekamp, P.G.M., Cooke, R.M., and Oostvoorn, F. van, “Choice of a Long-Term Strategy for the National Electricity Supply via Scenario Analysis and Multi-Criteria Analysis”. European Journal of Operational Research 48, 189–203, 1990.CrossRefGoogle Scholar
  23. 23.
    Mintzberg, H., “Power in and around Organizations”. Prentice-Hall, Englewood Cliffs, N.J., 1983.Google Scholar
  24. 24.
    Saaty, T.L., “A Scaling Method for Priorities in Hierarchical Structures”. Journal of Mathematical Psychology 15, 234–281, 1977.MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    Saaty, T.L., “The Analytic Hierarchy Process, Planning, Priority Setting, and Resource Allocation”. McGraw-Hill, New York, 1980.Google Scholar
  26. 26.
    Saaty, T.L., and Vargas, LG., “Inconsistency and Rank Preservation”. Journal of Mathematical Psychology 28, 205–214, 1984.MathSciNetMATHCrossRefGoogle Scholar
  27. 27.
    Stewart, T.J., “A Critical Survey on the Status of Multi-Criteria Decision Making Theory and Practice”. Omega 20, 569–586, 1992.CrossRefGoogle Scholar
  28. 28.
    Takeda, E., Cogger, K.O., and Yu, P.L., “Estimating Criterion Weights using Eigenvectors: a Comparative Study”. Omega 20, 569–586, 1987.Google Scholar
  29. 29.
    Torgerson, W.S., “Distances and Ratios in Psycho-Physical Scaling”. Acta Psychologica XIX, 201–205, 1961.CrossRefGoogle Scholar
  30. 30.
    Triantaphyllou, E., Lootsma, F.A., Pardalos, P.M., and Mann, S.H., “On the Evaluation and Application of Different Scales for Quantifying Pairwise Comparisons in Fuzzy Sets”. Journal of Multi-Criteria Decision Analysis 3, 133–155, 1994.MATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 1997

Authors and Affiliations

  • Freerk A. Lootsma
    • 1
  1. 1.Delft University of TechnologyThe Netherlands

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