A note on group decision-making procedure based on incomplete reciprocal relations

In a very recent paper by Xu and Chen (Soft Comput 12:515-521, 2008), a novel procedure for group decision making with incomplete reciprocal relations was developed. In this note, we examine the function between the fuzzy preference relation and its corresponding priority vector developed by Xu and Chen with a numerical example and show that the function does not hold in general cases. Then, we deduce an exact function between the additive transitivity fuzzy preference relation and its corresponding priority vector. Based on this, we develop a procedure for the decision making with an incomplete reciprocal relation and also develop a procedure for the group decision making with incomplete reciprocal relations. In order to compare the performances of our method with Xu and Chen's method in fitting the reciprocal relation, we introduce some criteria. Theoretical analysis and numerical examples have shown that the function deduced by us is more reasonable and effective than Xu and Chen's.