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.

In the current account, we present an analysis of a non-discriminating criterion under simple additive weighting synthesis, considering a deep decision hierarchy. A non-discriminating criterion describes a criterion where all decision alternatives under consideration perform equally. We question eliminating such a criterion from the decision hierarchy in search of simpler problem representation and computational efficiency. Yet, we prove such an approach may result in order misrepresentations between decision alternatives. This analysis is performed in the form of four research questions that relate to the detection of certain conditions under which such distortions in the order integrity of decision alternatives will occur, calculating the change in their final performances, distinguishing the alternatives whose performances are consistent, and examining the role of the normalization procedure adopted in averting such distortions when the non-discriminating criterion is ignored. Along these lines, this study provides clear inferences which are of interest to researchers and decision makers, using simple additive weighting and similar methods that rely on additive synthesis.

In this manuscript, with grounding in Liu–Lin axioms of greyness degree and information content, we provide new results that relate to these concepts in consideration of a number of mathematical operations over a sequence of grey numbers. In particular, we derive greyness degree results of summation, conic combination, and convex combination of a sequence, as well as inverse of a number and normalization of a number over a sequence. Then, we turn our attention to prove information content results for the union and intersection of a sequence. We illustrate our results by using a simple Monte Carlo simulation in the multi-attribute decision-making context, and by using an interesting dice-rolling experiment. Through our analysis, we also provide some new definitions, such as for conic combination, convex combination, normalization, and union and intersection operations. The novelty of the derived results in this study is that they can help researchers and practitioners of grey systems in tracking probable intensifications and reductions in the greyness degree in successive application steps of their working methods. Moreover, researchers are provided with two results to calculate information content for the union and intersection of grey numbers in an uncomplicated manner.