Design of an Algorithm Regarding Analytic Hierarchy Process

Abstract

People make three general types of judgments to express importance, preference, or likelihood and use them to choose the best among alternatives in the presence of environmental, social, political, and other influences. They base these judgments on knowledge in memory or from analyzing benefits, costs, and risks. From past knowledge, we sometimes can develop standards of excellence and poorness and use them to rate the alternatives one at a time. This is useful in such repetitive situations as student admissions and salary raises that must conform with established norms. Without norms one compares alternatives instead of rating them. Comparisons must fall in an admissible range of consistency. The analytic hierarchy process (AHP) includes both the rating and comparison methods. Rationality requires developing a reliable hierarchic structure or feedback network that includes criteria of various types of influence, stakeholders, and decision alternatives to determine the best choice [8].

for well-known statement that the maximal eigenvalue \(\lambda_{\max }\) is equal to \(n\) for the eigenvector problem \(\mathrm{A} w=\lambda w\) where \(A\) is, so called, the consistent matrix of pairwise comparisons of type \(\mathrm{n} \times \mathrm{n}(\mathrm{n} \geq 2)\) with the solution vector \(w\) that represents the probability components of disjoint events. Moreover, we suggest an algorithm for the determination of the eigenvalue problem solution \(A w=\mathrm{n} w\) as well as the corresponding flowchart. The algorithm for arbitrary consistent matrix \(A\) can be simply programmed and used.