Group Assessment of Comparable Items from the Incomplete Judgments

Abstract

In this study, we propose an approach to derive a group assessment of an item as its weight vector on multiple viewpoints. When there are a group of decision makers who give the judgments of the item as comparison matrices on the viewpoints, it is reasonable that the weight vector of the item is the core of those by all the decision makers. Each decision maker’s weight vector basically includes his/her given comparison matrix, which represents only a part of his/her thinking. Namely, there is an inclusion relation between a comparison and a ratio of the corresponding weights. In addition, there are items other than the target one. A decision maker gives the comparison matrices of some of the other items if s/he knows them, as well as the target one. It is natural that there is a correlation between the judgment of the target item to those of the others. The correlation is taken into consideration from the aspect of consistency of his/her judgments. We define a fuzzy degree of the consistency with all the comparison matrices s/he gives. As the consistency degree for a comparison matrix increases, it may become unable to satisfy the inclusion relation between the comparison matrix and the weight vector. Hence, we introduce a fuzzy degree of inclusion relation in order to relax it. There is a trade-off between them. Therefore, by maximizing both degrees we obtain the weight vector of the target item from the comparison matrices of the target item considering the consistency of each decision maker’s judgments. The proposed approach is applicable even in the case that a group of given comparison matrices is incomplete such that some comparisons in a comparison matrix are missing and/or the comparison matrices of some items are missing.