An AHP-ANP Comparative Study on the Structural Impact of Adding or Deleting Indifferent Criteria

Abstract

This study deals with addition of indifferent or wash criteria on which the alternatives have equal performance in the Analytic Hierarchy Process (AHP) against the misguided argument as shortcoming of the AHP. In such case, it is noted that careful attention and a correct approach are required in order not to lead towards the wrong decision of rank reversal when composing priorities of criteria and performances of alternatives according to the criteria. It is proposed that prioritizing effect of added or deleted indifferent criteria requires consistency with the change of conceptual definition in regards to the related elements as well as that of the hierarchical structure if needed. Corresponding to such importance in ranking alternatives, two principles of the AHP need to be considered simultaneously: the completeness for expectation and the outer dependence of a hierarchy on which the argument of rank reversal focuses only. The completeness implies all criteria are represented in the hierarchy or the network. Then the decision makers make sure that their ideas are adequately represented for the outcome to match their expectations. The outer dependence is concerned with hierarchical decomposition, which implies containment of the small elements by large clusters or levels. If one is to omit this axiom, the principle of hierarchic composition would no longer be applied. Through simply adding a new criterion or deleting a criterion in a given hierarchy with no other dependency except outer-dependency, we can show how to adjust the current global priorities by the addition or deletion corresponding to the hierarchic level. This is based on the fact that the conceptual/substantial definition of corresponding higher criterion needs to change as well. As a result, this study shows the wrong example in the previous study overlooked the correct meaning of expectation. This approach coincides with the outcome by supermatrix approach as a general matrix form dealing with a hierarchy. Logically, the adding or deleting of subcriterion in a level of criterion requires the change of the related criteria definition in upper levels and impacts on their priorities according to the related equation which is accompanying with its local priority matrix. When the meaning of completeness is considered to change global vector, not only changing local vector but also changing the else is required. Therefore, it is required to consider the outer-dependency and completeness principles for expectation simultaneously when using hierarchical approach. This study also show that the Analytic Network Process (ANP) as a general approach can cover the case of adding feedback from alternatives to the given hierarchic criteria or subcriteria, where the input and output can be represented in the supermatrix. At first, we show the necessary calculation of feedback from the assumption that the new criterion is indifferent between the two alternatives. Finally, we also show that there is no rank reversal between the two given alternatives, and total priorities with regards to the alternative coincide with those before adding a new but indifferent criterion with the same values to alternatives. Additionally, we find out a feedback characteristic that the priorities of criteria are transferred to the lower subcriteria, which is from the assumption of bidirectional dependency between the criteria and two alternatives in the limiting supermatrix. The other is from the assumption of cycle, the direction of feedback to criteria and subcriteria. We can find the same result as that of the hierarchic approach in this example, and there is no rank reversal. Finally, this paper discusses the way in which there is no irrational/unreasonable rank reversal with simple addition of indifferent criteria in addition to the deletion of the indifferent criteria, which helps AHP/ANP application.