# The Additive and the Multiplicative AHP

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

## Abstract

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.

## Keywords

Fuzzy Logic Analytic Hierarchy Process Criterion Weight Final Grade Indifference Curve
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## 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.
2. 2.
Barzilai, I, and Golany, B., “Deriving Weights from Pairwise Comparison Matrices: the Additive Case”. Operations Research Letters 9, 407–410, 1990.
3. 3.
Barzilai, J., and Golany, B., “AHP Rank Reversal, Normalization, and Aggregation Rules”. INFOR 32, 57–64, 1994.
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.
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.
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.
9. 9.
Crawford, G., and Williams, C, “A Note on the Analysis of Subjective Judgement Matrices”. Journal of Mathematical Psychology 29, 387–405, 1985.
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.
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).
12. 12.
French, S., “Decision Theory, an Introduction to the Mathematics of Rationality”. Ellis Horwood, Chichester, 1988.
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.
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.
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.
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.
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.
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.
21. 21.
Lootsma, F.A., “Scale Sensitivity in the Multiplicative AHP and SMART”. Journal of Multi-Criteria Decision Analysis 2, 87–110, 1993.
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.
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.
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.
27. 27.
Stewart, T.J., “A Critical Survey on the Status of Multi-Criteria Decision Making Theory and Practice”. Omega 20, 569–586, 1992.
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.
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.