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AN ALGEBRAIC REPRESENTATION VIA DIFFERENTIAL EQUATIONS FOR PAIRWISE COMPARISONS OF AHP

Takafumi Mizuno

Abstract


We propose a simple algebraic representation for pairwise comparisons of AHP. The representation is an associative relation between the importances of elements and consists of basic arithmetic operations. First, we define a ratio, which is estimated by decision makers by comparing the importances of elements, as a partial differentiation of importances (Section 2). Then, we construct systems of differential equations. Algebraic representations of the importances are derived as formal solutions of the equations. We analyze pairwise comparisons and the construction of the importances from them with the representations (Section 3). The validity of using eigenvectors and C.I. in AHP is illustrated by deriving a particular solution of the equations.

https://doi.org/10.13033/ijahp.v9i1.278


Keywords


pairwise comparison method; ternary comparison method; ternary diagram

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References


Saaty, T. (1980). The Analytic Hierarchy Process, New York: McGraw-Hill. Doi: http://dx.doi.org/10.1080/00137918308956077

Saaty, T. (1977). A scaling method for priorities in hierarchical structure. Journal of Mathematical Psychology, 15, 234-281. Doi: http://dx.doi.org/10.1016/0022-2496(77)90033-5

Cogger, K. O. & Yu., P. L. (1985). Eigenweight vectors and least-distance approximation for revealed preference in pairwise weight ratios. Journal of Optimization Theory and Applications, 46(4), 483-491. Doi: 10.1007/BF00939153

Harker, P. and Vargas, L. (1987). The theory of ratio scale estimation: Saaty's analytic hierarchy process. Management Science, 33(11), 1383–1403. Doi: http://dx.doi.org/10.1287/mnsc.33.11.1383




DOI: http://dx.doi.org/10.13033/ijahp.v9i1.278