Distance-Based Decision Making, Consensus Building, and Preference Aggregation Systems: A Note on the Scale Constraints

Distance metrics and their extensions are widely accepted tools in supporting distance-based decision making, consensus building, and preference aggregation systems. For several models of this nature, it may be necessary to elucidate the problem output in the original input domain. When a particular parameter of interest is desired to be produced in this original domain, i.e., the scale, the decision makers simply resort to constraints that function in parallel with this goal. However, there exist some cases where such a membership is guaranteed by the mathematical properties of the distance metric utilized. In this paper, we argue that the scale constraints utilized in this manner under the distance-metric optimization framework are, in some cases, completely redundant. We provide necessary mathematical proofs and illustrate our arguments through an abstract physical system, examples, a case study, and a brief computational experiment.