The analytic hierarchy and analytic network measurement processes: Applications to decisions under risk

Mathematics applications largely depend on scientific practice. In science measurement depends on the use of scales, most frequently ratio scales. A ratio scale there is applied to measure various physical attributes and assumes a zero and an arbitrary unit used uniformly throughout an application. Different ratio scales are combined by means of formulas. The formulas apply within structures involving variables and their relations under natural law. The meaning and use of the outcome is then interpreted according to the judgment of an expert as to how well it meets understanding and experience or satisfies laws of nature that are always there. Science derives results objectively, but interprets their significance subjectively. In decision making, there are no set laws to characterize structures in which relations are predetermined for every decision. Understanding is needed to structure a problem and then also to use judgments to represent importance and preference quantitatively so that a best outcome can be derived by combining and trading off different factors or attributes. From numerical representations of judgments, priority scales are derived and synthesized according to given rules of composition. In decision making the priority scales can only be derived objectively after subjective judgments are made. The process is the opposite of what we do in science. This paper summarizes a mathematical theory of measurement in decision making and applies it to real-life examples of complex decisions.