The general case of dependence in hierarchic decision theory

Thomas Saaty
Joseph M. Katz Graduate School of Business
University of Pittsburgh
United States

Publication date: Jan, 1987

Journal: Toward Interactive and Intelligent Decision Support Systems
- Pages: 239-248

Abstract: A hierarchy is a simple structure used to represent the simplest type of functional (contextual or semantic) dependence of one level or component of a system on another in a sequential manner. It is also a convenient way to decompose a complex problem in search of causeeffect explanations in steps which form a linear chain. One result of this approach is to assume the functional independence of an upper part, component, or cluster from its lower parts. This often does not imply its structural independence from the lower parts which involves information on the number of elements, their measurements, etc. But there is a more general way to structure a problem involving functional dependence. It allows for feedback between components. It is a network system of which a hierarchy is a special case. In both hierarchies and networks the elements within each component may be also dependent on each other (see Saaty and Takizawa, 1986). Figure 1 below shows two drawings which depict the structural difference between the two frameworks. In this figure, a loop means that there is inner dependence of elements within a component.

Keywords: Principal eigenvalue, Priority vector, Absolute priority, Nonlinear network structural independence