Optimization by the Analytic Hierarchy Process

TLDR: In this article, the authors studied the transitivity of preference through a new approach to consistency, which need not always strictly hold for the results to be acceptable, and not all alternatives need to be directly comparable.

Abstract: : The Analytic Hierarchy Process serves as a framework for people to structure their own problems and provide their own judgements based on knowledge, reason or feelings, to derive a set of priorities for activities to which they, for example, wish to allocate effort or resources. In this process transitivity of preference is studied through a new approach to consistency - which need not always strictly hold for the results to be acceptable. Also since hierarchic structures may not be complete, not all alternatives need to be directly comparable. It is necessary to construct a pairwise comparison matrix of the relative contribution or impact of each element on each governing objective or criterion in the adjacent upper level. In such a matrix of the elements by the elements, the elements are compared in a pairwise manner with respect to a criterion in the next level. In comparing the i,j elements, people prefer to give a judgement which indicates the dominance as an integer. Thus, if the dominance does not occur in the i,j position while comparing the ith element with the jth element then it is given in the j,i position as a ji and its reciprocal is automatically assigned to aij.