Measurement and preference function modelling

The application of the principles of mathematical modelling to measurement theory is reviewed and the implications of a classification of measurement models by the mathematical operations that are applicable to the resultant scales are summarized. The conditions for the applicability of the operations of addition and multiplication imply that the models of classical measurement theory do not enable the application of these operations. For example, measurement without units does not enable the operations of addition and multiplication. Ratios of preference scales are shown to be undefined thus invalidating the analytic hierarchy process. The models of classical measurement theory, including utility theory, are weak and do not enable the operations of addition and multiplication as well.