Axioms of the Analytic Hierarchy Process (AHP) and its Generalization to Dependence and Feedback: The Analytic Network Process (ANP)

The AHP/ANP are multicriteria decision-making theories that deal with both hierarchic structures when the criteria are independent of the alternatives and with networks when there is any dependence within and between elements of the decision. Both of them have been repeatedly used in practice by various researchers and practitioners. From the perspective of almost 40 years of practice in solving problems using both theories, some of their properties seem to be more important than others. The article indicates four of them as fundamental for understanding AHP/ANP. These are the axioms related to structure, computation, and expectation. The mathematical formulation of the axioms is preceded by an introduction explaining the motivation behind the introduced concepts. The article is expository and it is an improved and refined version of the work [1]. Introduction It is useful to seek an abstract understructure for all the systems we wish to axiomatize in order to determine what the axioms should be about and what aspects they relate to. Thus we begin by describing a system very briefly in order to organize our axioms accordingly. We find it convenient and aesthetically satisfying to think of systems in terms of four major attributes in which flow and function are part of the structure, as illustrated in Figure 1. They all have meaning individually and as they depend on each other. These are structure, flow, function and purpose1. To design a system such as the brain, one proceeds from the most general purpose to the particular functions of the parts, the flows necessary to perform the functions and the structure, which constrains and directs 1 Please note that structure, flow, function and purpose, underlies the methods of modeling and analysis of contemporary IT systems. For example, UML (Unified Modeling Language) structure diagrams describe the structure of the system, whilst UML behavior and interaction diagrams provide information on flows between different objects providing various functionalities [5]. 2 the flows. This framework also makes it possible to differentiate and categorize a variety of systems. We note that the whole universe may be regarded as one gigantic system. From a practical standpoint we do not deal with the entire universe every time we have a problem. Structure is used to refer to a wide variety of concrete objects and abstract ideas, yet it is generally possible to understand its meaning from the context in which it is used.