Algorithms for improving consistency or consensus of reciprocal [0,1]-valued preference relations

We investigate the consistency and consensus of reciprocal [0,1]-valued preference relations (also called fuzzy preference relations by many authors) based on the multiplicative consistency property, which is an important issue in fuzzy set theory. An algorithm is first developed to improve the consistency level of a reciprocal [0,1]-valued preference relation, and the corresponding algorithm for the incomplete reciprocal [0,1]-valued preference relation is also developed. We further propose the consensus improving algorithms for individual reciprocal [0,1]-valued preference relations or incomplete ones. The convergence and robustness of the algorithms are proven and some important conclusions are obtained. In addition, the proposed algorithms can improve the consistency or consensus of reciprocal [0,1]-valued preference relations with less interactions with the decision makers, which can save a lot of time and obtain the results quickly.