Scale Dependence and Ranking Intervals in Additive Value Models Under Incomplete Preference Information

A multiattribute additive value function that has been built from a complete specification of the decision maker's (DM's) preferences gives scale-independent decision recommendations, which do not depend on how the value function is normalized. In this paper, we show that if the preference specification is incomplete, many widely employed decision rules for comparing alternatives give scale-dependent decision recommendations in which the relative ranking of alternatives depends not only on the DM's preferences but also on the normalization of the value function. But because normalization does not involve preference statements, the recommendations should be scale independent so that they do not depend on the chosen normalization. To provide such recommendations, we propose ranking intervals , which show how a given alternative compares with all other alternatives for all value functions that are consistent with the stated preference information. These intervals can be computed from mixed integer linear optimization problems that are constrained by inequalities implied by the DM's preference statements. We illustrate the use of ranking intervals by analyzing university rankings and discuss their uses in project portfolio selection.