We establish that none of the classical theories of choice can serve as a proper foundation for decision theory (and hence for decision-based design) and we construct a new theory of measurement that provides such a foundation. A proper theory of selection in engineering design cannot be founded on scales and variables to which the mathematical operations of addition and multiplication are not applicable. Yet it has not been proven that addition and multiplication are applicable to von Neumann and Morgenstern's utility scales or to any scales based on classical decision theory (which, in turn, is based on the classical theory of measurement). In fact, addition and multiplication are not applicable to utility scales, value scales, ordinal voting scales or any scales based on the classical theory of measurement whether the underlying variables are physical or subjective (i.e., psychological).

Selection is an important problem in engineering design (see [1], Chapter 3). By definition, selection means making choices and choice is synonymous to preference since we choose those objects that are preferred. Therefore, the scientific foundation of selection in engineering design (and elsewhere) is the measurement of preference. Consequently, our goal is the construction of preference scales that serve similar purposes as scales for measurement of physical variables such as time, energy and position. In Part 1 of this chapter, we consider the issues of the mathematical foundations for scale construction.