Generalized desirability functions: a structural and topological analysis of desirability functions

We have Derringer and Suich's desirability functions in mind, especially, the two-sided ones in our analysis throughout this study. We propose and develop a finite partitioning procedure of the individual desirability functions over their compact and connected interval which leads to the definition of generalized desirability functions. We call the negative logarithm of an individual desirability function having a max-type structure and including a finite number of nondifferentiable points as a generalized individual desirability function. By introducing continuous selection functions into desirability functions and, especially, employing piecewise max-type functions, it is possible to describe some structural and topological properties of these generalized functions. Our aim with this generalization is to show the mechanism that gives rise to a variation and extension in the structure of functions used in classical desirability approaches.