Identifying and correcting the defects of the Saaty analytic hierarchy/network process: A comparative study of the Saaty analytic hierarchy/network process and the Markov chain-based analytic network process

The Saaty analytic network process (Saaty-ANP) is a generalization of the analytic hierarchy process. The Markov chain-based ANP (MC-ANP) is another decision-making approach suitable for general structures. Both ANPs use a relative measurement (paired comparisons with ratio scales) to estimate tangible and intangible factors, use a stochastic matrix (SM) to solve feedback problems and obtain the same results under some conditions. The SaatyANP does not define the basic concepts, nor does it check the rationality of the structure, which may lead to meaningless solutions and ignore a subclass of feedback decision problems. MC-ANP separates the alternatives from the criteria and defines the attributes, criteria, criterion dominated relations (CDRs) and reasonable constraints of the CDRs by means of digraphs; it also represents CDRs as Markov chain transition diagrams and corresponding (stochastic) adjacency matrices and obtains solutions from a system of linear equations. With the MC-ANP, for the real alternative problems (Class I), the solutions are priorities of the alternatives obtained by the parametric positive left eigenvectors of the SM, and for the nominal alternative problems (Class II), the solutions are priorities of the criteria obtained by the nonnegative right eigenvector of the SM. We analyze the conditions and causes of rank reversal; note that rank reversal does not appear in Class II problems; the study offers a rank reversal ANP example (with feedback) and presents a rank-preserving method for Class I problems. We discuss the contribution of MC-ANP, how to compensate for the defects of Saaty-AHP/ANP, and present issues that need further consideration.