A Semiring-based study of judgment matrices: properties and models

In decision making and group decision making, multiplicative reciprocal judgment matrices and additive reciprocal judgment matrices are used as two kinds of important preference information. In this paper, semirings are applied to the discussion of judgment matrix properties. First, two special semirings are constructed. Second, the properties of the consistent judgment matrices are given as a set of equations (all in the semiring sense), which include idempotency equations and fixed point equations. We find that there exists one and only one specially constrained fixed point as the priority vector of a consistent judgment matrix. Third, optimization models for inconsistent judgment matrices are presented. Finally, we offer some simple illustrative examples.