Article collection "Mathematical Problems of Cybernetics" №5, Moscow, 1994

Authors:Sholomov L.A.

Research of relations in criterion spaces and design of group choice operators

Abstract:

The ordinal relation x ̃ρy ̃, x ̃=(x_{1},…,x_{n}), y ̃=(y_{1}…,y_{n})ϵ R^{n} is completely determined by the relations (more, less, equal) of components x_{i} and y_{i}, i= 1,…,n. In terms of the representing functions g_{ρ}, a complete description of the ordinal relations ρ for the main classes used in multi-criteria choice problems is found. These are classes of L linear orders, W weak orders, S semi-orders, I interval orders, P partial orders, and A acyclic relations, L⊂W⊂S⊂I⊂P⊂A. We also studied the problem of design of (R_{1}^{n}→R_{2})-operators of group choice. It consists in describing the operators that aggregate any n-tuple of individual relations from the class R_{1} into a group relation of the class R_{2}. The article solves this problem for all pairs of classes (R_{1},R_{2}), R_{1},R_{2}∈ {L,W,S,I,P,A} satisfying a naturally occurring condition R_{1}⊆R_{2}. Recognition problems of whether operator in question is a (R_{1}^{n}→R_{2})-operator of a given type, are investigated for NP-hardness and polynomiality.

Keywords:

multi-criteria choice, group choice, ordinal relation, description of ordinal relations, group choice operator, design of operators