Multiattribute Utility Functions: Decompositions Using Interpolation

Abstract

Methodologies developed in the last several years allow formal inclusion of two sources of complexity in the analysis of a decision problem—uncertainty and multiple conflicting objectives. Uncertainty can be handled by assessing the decision maker's attitude towards risk in the form of a von Neumann-Morgenstern utility function. Conflicting objectives may be handled by making the utility function multidimensional. A problem that then arises is in devising assessment protocols for these multidimensional (multiattribute) functions that make efficient use of the decision maker's time and effort. Mostly, such methods have concentrated on identifying structural properties of the function, such as separability, by asking questions of a general nature about the preferences and tradeoffs of the decision maker. These methods have been applied successfully on such problems as facilities siting, selection of appropriate medical treatment, optimal pest control and the evaluation of R&D proposals. The problem addressed in this paper is that of what to do about the assessment if no useful decomposition of the utility function can be identified. One possiblity is to make use of some of the properties that were most nearly satisfied and then attempt to support the resulting recommendations with a sensitivity analysis. This paper provides the analyst with an efficient, routine method of approximating the utility function to any degree of accuracy that the decision maker or the problem requires. The idea is to assess the function exactly only on a multidimensional grid and then to interpolate other values of the utility function from those on the grid. Evidently, a finer grid will, in general, provide a better approximation. The resulting utility function is continuous. It is not anticipated that many of the problems in which conflicting objectives are a factor will need this level of sophistication. But just as there are problems that require a substantial modelling effort to identify feasible consequences so, occasionally, an analysis would benefit from an accurate modelling of the objective function.