MATHEMATICAL ASPECTS OF IMPLEMENTING THE ANALYTIC HIERARCHY PROCESS
Alexander K. Cherkashin
V. B. Sochava Institute of Geography
The article discusses current problems of forming metatheoretic foundations of modeling in the implementation of analytic hierarchy process (AHP), namely, issues of mathematical justification of the scale of judgments based on interrelated observations, pairwise comparisons of factors, criteria and alternatives, calculation of priorities and their synthesis in the final estimates. It is shown that the basic AHP rules (axioms) are directly related to the properties of partial derivatives that characterize the marginal rate of factors substitution, and to the Euler differential equation for homogeneous functions of many variables. At the metatheoretic level, these relationships are due to the procedures of fibering spaces and sets over elements of bundle bases with different contents. The AHP methodology is based on bundles of numerical sets, tangent bundles over manifolds, and knowledge bundles on system theories, which provides the universality of its application. Mathematical analysis of AHP procedures highlights the requirement of linearity of the research space and the need to linearize indicators at the beginning of data processing with a further transition from the initial absolute indicators to relative ones in order to form a local coordinate system of the tangent fiber (cluster) as local, linear, metric, and bounded space. The Euler equation relates the values of the coordinates of the local space of the tangent fiber in the form of metric dependencies as linear and nonlinear estimation functions known in AHP. To determine the preference scale at different levels of formalization, natural and integer numbers, rational numbers, real and hyperreal numbers of standard and nonstandard analysis are used. It is determined the restriction in AHP procedures, when having a matrix of paired evaluation and not knowing the type of ranking function, it is possible to calculate priorities and choose the desired solution. The calculated priorities are the relative factors sensitivities of the evaluation functions, and their ratios are the rates of substitution of the considered factors with unit values. Equations are derived for the synthesis of evaluation of the global priority based on local priorities across all criteria. The final evaluation function can be represented as a product of vectors and comparison matrices of different hierarchical levels (polylinear form).
Analytic hierarchy process (AHP), mathematical framework, procedure of bundle, tangent bundle, Euler equation, evaluation functions, pairwise comparison matrix