Min-Additive Utility Functions

This paper introduces the ―min-additive‖ (also called ―min-average‖) utility function. This function is a weighted combination of an additive utility function and a minimization over a set of single attribute utility functions. The weighting is accomplished by exploiting information already contained in the additive and minimization models. Four forms of the min-additive (MA) model are presented—basic, uniform, logistic, and relaxed. The basic MA model generalizes the additive and minimization models but does not require any additional parameters to be estimated. It can be employed in situations where the decisionmaker’s preferences violate the additive independence assumptions inherent in the additive model. The uniform MA model extends the basic MA model by adding ―location‖ and ―spread‖ parameters. The logistic MA model extends the uniform MA model by creating a continuously differentiable weighting function. This weighting function is shown to be a close approximation of a Gaussian cumulative distribution function. The relaxed MA model removes the non-negativity requirements on the weights. This version of the MA model is shown to be a generalization of the two-dimensional multi-linear utility function (and the two-dimensional multiplicative utility function). Numerical examples and graphical representations of the models are presented. The paper contains three appendices. Appendix A illustrates how the MA model can be nested in a decision preference hierarchy. Appendix B compares the MA model to the recently proposed ―limited average‖ and ―exponential-average‖ family of utility functions. Finally, Appendix C summarizes the complement to the MA model—the max-additive model. The maxadditive model is used in risk analysis and other situations involving disutilities.