The chapter focuses on the analysis of scaling constants when constructing a utility function over multi-dimensional prizes. Due to fuzzy rationality, those constants are elicited in an interval form. It is assumed that the decision maker has provided additional information describing the uncertainty of the scaling constants’ values within their uncertainty interval. The non-uniform method is presented to find point estimates of the interval scaling constants and to test their unit sum. An analytical solution of the procedure to construct the distribution of the interval scaling constants is provided, along with its numerical realization. A numerical procedure to estimate pvalue of the statistical test is also presented. The method allows making an uncertainty description of constants through different types of probability distributions and fuzzy sets.
TopIntroduction
Quantitative decision analysis is based on subjective statistics (Jeffrey, 2004) and utility theory (Von Neumann & Morgenstern, 1947). It has its main objective to provide guidance to a rational choice of an alternative from the side of the decision maker (DM). The concept of rational decision making lies on sets of axioms, which prescribe a way to model decision problems, measure preferences (by utilities) and uncertainty (by probabilities), and at a final step – balance this quantitative data according to expected utility (French & Insua, 2001; von Neumann & Morgenstern, 1947).
A major step in the decision analysis process is to construct an adequate model of consequences, which follows from the objectives of the DM in the particular problem. The adopted approach is to represent consequences as vectors, whose coordinates (attributes) measure the degree to which the consequence complies with the DM’s goals. Generally, there are many objectives, hence many vector coordinates to measure these. Therefore the default structure of consequences is a d-dimensional (multidimensional) vector 
=(x1, x2, …, xd), whose coordinates Xi (the attributes) measure different aspects in the problem that are important for the DM. Let’s denote the set of all prizes as X. Then 
in the set of prizes, is a subset of the d-dimensional Euclidean space.
There are three types of attribute independence which facilitate the process of constructing a utility function over 
. The most important one from a practical stand point is utility independence (Keeney & Raiffa, 1993). Let’s divide the set of attributes {X1, X2, …, Xd} into the greatest possible number of mutually utility independent fundamental vector attributes Y1, Y2, …, Yn that are n
{2,3,…,d} non-empty non-overlapping subsets. Each Yi is a system of random variables with an arbitrary realization 
and then prizes in X may be presented as 
=(
,
,…,
). Mutual utility independence allows to decompose the multi-dimensional utility function to fundamental utility functions over groups of fundamental vector attributes Y1, Y2, …, Yn. Mutual utility independence allows constructing a normalized fundamental utility function ui(.) over each fundamental vector attribute Yi: u(
)=f [u1(
), u2(
),…, un(
)]. The importance of each vector attribute for the overall preferences of the DM over the multi-dimensional prizes is given by their scaling constants.