Abstract
According to the great mathematician Henri Lebesgue, making direct comparisons of objects with regard to a property is a fundamental mathematical process for deriving measurements. Measuring objects by using a known scale first then comparing the measurements works well for properties for which scales of measurement exist. The theme of this paper is that direct comparisons are necessary to establish measurements for intangible properties that have no scales of measurement. In that case the value derived for each element depends on what other elements it is compared with. We show how relative scales can be derived by making pairwise comparisons using numerical judgments from an absolute scale of numbers. Such measurements, when used to represent comparisons can be related and combined to define a cardinal scale of absolute numbers that is stronger than a ratio scale. They are necessary to use when intangible factors need to be added and multiplied among themselves and with tangible factors. To derive and synthesize relative scales systematically, the factors are arranged in a hierarchic or a network structure and measured according to the criteria represented within these structures. The process of making comparisons to derive scales of measurement is illustrated in two types of practical real life decisions, the Iran nuclear show-down with the West in this decade and building a Disney park in Hong Kong in 2005. It is then generalized to the case of making a continuum of comparisons by using Fredholm’s equation of the second kind whose solution gives rise to a functional equation. The Fourier transform of the solution of this equation in the complex domain is a sum of Dirac distributions demonstrating that proportionate response to stimuli is a process of firing and synthesis of firings as neurons in the brain do. The Fourier transform of the solution of the equation in the real domain leads to nearly inverse square responses to natural influences. Various generalizations and critiques of the approach are included.
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References
Aczel, J. andAlsina, C., (1986). On synthesis of judgments,Socio-Economic Planning Sciences,20, 333–339.
Aczel, J. andSaaty, T. L., (1983). Procedures for synthesising ratio judgments,Journal of Mathematical Psychology,27, 93–102.
Arrow, K. J., (1963).Social Choice and Individual Values, 2nd ed., Wiley, New York.
Bauer, R. A., Collar, E. andTang, V., (1992).The Silverlake Project, Oxford University Press, New York.
Brillouet-Belluot, N., (1999). On a Simple Linear Functional Equation on Normed Linear Spaces, Ecole Centrale de Nantes, F-44 072 Nantes-cedex 03, France, October 1999.
Davis, P. J. andHersh, R., (1986).Descartes Dream, Harcourt Brace and Jovanovich, New York.
Fechner, G., (1966).Elements of Psychophysics, Adler, H. E. (Trans.), Vol.2, Holt, Rinehart and Winston, New York. See also Batschelet, S.,Introduction to Mathematics for Life Scientists, Springer, 1971.
Lebesgue, H., (1928).Leçons sur l’integration, 2nd ed., Gauthier-Villars, Paris.
Mackay, A. F., (1980).Arrow’s Theorem: The Paradox of Social Choice — A Case Study in the Philosophy of Economics, Yale University Press, New Haven.
Saaty, T. L., (2000).The Brain, Unraveling the Mystery of How it Works: The Neural Network Process, RWS Publications, 4922 Ellsworth Avenue, Pittsburgh, PA 15213.
Saaty, T. L., (2005).Theory and Applications of the Analytic Network Process, Pittsburgh, PA: RWS Publications, 4922 Ellsworth Avenue, Pittsburgh, PA 15213.
Saaty, T. L., (2006).Fundamentals of Decision Making ; the Analytic Hierarchy Process, Pittsburgh, PA: RWS 3. Publications, 4922 Ellsworth Avenue, Pittsburgh, PA 15213.
Saaty, T. L., andOzdemir, M., (2005).The Encyclicon, RWS Publications, 4922 Ellsworth Avenue, Pittsburgh, PA 15213
Saaty, T. L. andPeniwati, K., (2008).Group Decision Making, RWS Publications, 4922, Ellsworth Avenue, Pittsburgh, PA 15213.
Saaty, T. L. andTran, L. T., (2007). On the invalidity of fuzzifying numerical judgments in the Analytic Hierarchy Process,Mathematical and Computer Modelling,46, 7-8, 962–975.
Saaty, T. L. andVargas, L. G., (1993). Experiments on Rank Preservation and Reversal in Relative Measurement,Math. Comput. Modelling,17, 4/5, 13–18.
Saaty, T. L. andVargas, L. G., (1997). Implementing Neural Firing: Towards a New Technology,Mathl. Comput. Modelling,26, 4, 113–124.
Saaty, T. L. and Vargas, L. G., (2008). The Possibility of Group Choice : Pairwise Comparisons and Merging Functions, (in publication).
Saaty, T. L., Vargas, L. G. and Whitaker, R., (2008). Addressing Criticisms of the Analytic Hierarchy Process, (in publication).
Wilkinson, J. H., (1965).The Algebraic Eigenvalue Problem, Clarendon Press, Oxford.
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(To the Memory of my Beloved Friend Professor Sixto Rios Garcia)
The author has been awarded with the 2008 Informs Impact Prize by the Institute for Operations Research and the Management Sciences for his seminal work on the Analytic Hierarchy Process, and for its deployment and extraordinary impact.
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Saaty, T.L. Relative measurement and its generalization in decision making why pairwise comparisons are central in mathematics for the measurement of intangible factors the analytic hierarchy/network process. Rev. R. Acad. Cien. Serie A. Mat. 102, 251–318 (2008). https://doi.org/10.1007/BF03191825
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DOI: https://doi.org/10.1007/BF03191825
Keywords
- comparisons
- conflict resolution
- decision
- eigenvalue
- functional equation
- hierarchy
- intangibles
- judgment
- measurement
- network
- neural firing
- sensitivity
- synthesis