Abstract
A decision maker using the Analytic Hierarchy Process (AHP) could be consistent, and still provide no information in the resulting vector of priorities. An extreme example would be a pairwise comparison judgment matrix filled with 1s which is totally consistent under the various definitions of consistency, but has provided no information about the prioritization of alternatives resulting from the decision maker's judgments. In this paper, the quality of a consistent decision maker's judgments using the Analytic Hierarchy Process is placed in the context of the entropy of the resulting vector of priorities. Indeed, it is the purpose of this paper to provide a formal definition of this notion ofentropy of a priority vector, and to provide a framework for a quantitative measurement of the information content of consistent pairwise comparison judgment matrices of a decision maker who is using the Analytic Hierarchy Process. We will prove that the entropy of the vector of priorities for consistent matrices follows a normal distribution and discuss some general considerations of this result.
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References
Golden, Bruce L., and Wang, Qiwen: 1990, ‘An alternate measure of consistency’, in:The Analytic Hierarchy Process, Applications and Studies, Golden, B., Wasil, E., and Harker, P., (Eds.), Springer-Verlag, New York-Leipzig.
Golub, G. H., and Van Loan, C. F.: 1989,Matrix Computations, The Johns Hopkins University Press, Baltimore, Maryland.
Gray, R. M.: 1990,Entropy and Information Theory, Springer-Verlag, New York.
Harker, P. T.: 1990, ‘The art and science of decision making: The analytic hierarchy process’, in:The Analytic Hierarchy Process, Applications and Studies, Golden, B., Wasil, E., and Harker, P. (Eds.), Springer-Verlag, New York-Leipzig.
Hays, W. L.: 1988,Statistics, Holt, Rinehart and Winston Inc., New York.
Karni, R., Sánchez, P., and Tummala, V. M. R.: 1990, ‘A comparative study of multiattribute decision making methodologies’,Theory and Decision 29, 203–222.
Lane, E. F., and Verdini, W. A.: 1989, ‘A consistency test for AHP decision makers’,Decision Sciences 20, 575–590.
Lewis, P. A. W., Goodman, A. S., and Miller, J. M.: 1969, ‘A pseudo-random number generator for the System/360’,IBM Syst. J. 8(2), 136–146.
Massan, B. H.: 1988,Multi-Criteria Decision Making (MCDM) Techniques in Planning, Pergamon Press, Oxford-New York.
Ross, S.: 1984,A First Course in Probability, Macmillan, New York-London.
Saaty, T. L.: 1980,The Analytic Hierarchy Process, McGraw-Hill, New York.
Sánchez, P. and Noble, E.: 1990, ‘An improved measure of mean random inconsistencies for AHP decision makers’,Proceedings of the 1990 Annual Meeting 1, 212–215.
Schrage, L.: June 1979, ‘A more portable random number generator’,ACM Transactions on Mathematical Software 5(2), 132–138.
Theil, H.: 1979, ‘The measurement of inequality by components of income’,Econom. Lett. 2, 197–199.
Vargas, L. G.: 1982, ‘Reciprocal matrices with random coefficients’,Mathematical Modelling 3, 69–81.
Vargas, L. G.: 1983, ‘Analysis of sensitivity of reciprocal matrices’,Applied Mathematics and Computation 12, 201–220.
Zahedi, F.: 1986, ‘The analytic hierarchy process — a survey of the method and its applications’,Interfaces 16(4), 96–108.
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Noble, E.E., Sanchez, P.P. A note on the information content of a consistent pairwise comparison judgment matrix of an AHP decision maker. Theor Decis 34, 99–108 (1993). https://doi.org/10.1007/BF01074896
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DOI: https://doi.org/10.1007/BF01074896