An algebraic framework for uniformly analyzing interval pairwise comparison matrices

This paper reports an algebraic framework for uniformly analyzing multiple types of interval pairwise comparison matrices (IPCMs). First, a possibility degree formula of ranking interval numbers is developed by using an isomorphic mapping. The partial order of intervals is established such that the equivalence classes can be constructed. Based on the binary operation and partial order of intervals, an Abelian linearly ordered (Alo) group is formed. Second, the reciprocal property and consistency of IPCMs are redefined within the framework of Alo group. It is found that the shortcoming of some existing consistency definitions of IPCMs is overcome. Third, a novel index vector is proposed to measure the inconsistency level of IPCMs. The partial order of the index vector is established for defining acceptable consistency of IPCMs. The results show that the isomorphisms between Alo groups offer a theoretical basis to uniformly investigate the properties of various IPCMs.