Scale dependence in weight and rate multicriteria decision methods

This paper investigates the effects of constructed scales used to evaluate criteria, and monotonic perturbations of those scales, on the ranking of decision alternatives. The analysis focuses on the widely used ‘weight and rate’ method of multicriteria decision-making, but the findings also provide perspectives on scale dependence in numerous settings. We introduce a simulation method to generate ‘weight and rate’ decisions uniformly and use simulation to characterize the sensitivity of the obtained rankings to monotonic transformations. We define a weight-rate function and cumulative weight-rate function to characterize decision alternatives and draw on the results of stochastic dominance to address the sensitivity of the rankings to the choice of the constructed scale analytically for certain classes of decisions. We define absolute, first-order, and second-order weight-rate dominance conditions. We also define a special case where the domain of the weight-rate function of one alternative is contained within the domain of another. We show that this special case is especially sensitive to the constructed scale. We also show via simulations that increasing the number of attributes or the number of alternatives increases sensitivity to the constructed ratings scale. The numerical and theoretical results show the ranking of decision alternatives is sensitive to the constructed scale, except in limited cases such as absolute and first-order weight-rate dominance. The results provide insights for practitioners and government officials into this widely used method of decision-making and cautions them to the sensitivity of their results to the rating and weighting scales that are used.