Generalized polynomial decomposable multiple attribute utility functions for ranking and rating multiple criteria discrete alternatives1

摘要

This paper describes a class of nonlinear polynomial functions which are far more complex than linear and additive functions and other types of well-known utility functions. Although the assessment of such nonlinear polynomial functions may require more time and possibly more cognitive burden, the advantage of using more complex and nonlinear functions is that more complex preferential behaviors can be represented, and hence a larger set of decision problems can be solved. In this paper, we develop a generalized decomposable multiattribute utility function (GDMAUF) for representing the Decision Maker's (DM's) preferential behavior. We show that any polynomial function can be represented by the GDMAUF, and that such a function can be assessed to represent the DM's behavioral preferences (or so-called the utility function). Our approach is based on the use of holistic responses (numerical rankings and rating of alternatives) from which we assess the gradients of the DM's unknown utility function. From assessed gradients, we then construct a generalized decomposable multiattribute utility function that mimics and represent the DM's preferential behavior. This assessed generalized decomposable multiattribute utility function can be used directly to rank and rate all alternatives without any further interaction from the DM. We also demonstrate that the generalized utility function (GDMAUF) is more general and flexible than some other well-known utility structures such as additive, multiplicative, multilinear, quasi-concave and quasi-convex. The underlying assumptions for GDMAUF are that it is differentiable and that it can be expressed in a polynomial form. Some classes, extensions and properties of the generalized utility function are explored to simplify its assessment. Several examples and experimental results are provided.