Creating an acceptable consensus ranking for group decision making

Abstract

This paper examines the problem of combining a set of ordinal rankings to form an acceptable consensus ranking. The objective of traditional group decision making problem is to determine the Minimum Violation Ranking. Motived by the applications of adjusted consensus in recent years, we study this problem from a new perspective, for obtaining an acceptable consensus ranking for group decision making. In this paper, every voter ranks a set of alternatives respectively, and we know the acceptability index, which represents the minimum adjustments that are allowed for each voter. The problem is to find the Minimum Acceptable Violation Ranking (MAVR) which minimizes the sum of voter’s unacceptable violations. Besides, we develop a branch and bound ranking algorithm to solve this problem. The suggested improvement include: (1) analysing the ranking preference by two ways: pairwise preference and ranking-based preference; (2) constructing the lower bound and upper bound, which exclude at most half of the feasible solutions in each iteration process. Furthermore, the effectiveness and efficiency of this algorithm are verified with an example and numerical experiments. Finally, we discuss two extensions of the basic MAVR problem: the Minimum Weighted Acceptable Violation problem, whose voters are accompanied with a set of weights or multiples, and the Minimum Hierarchy Acceptable Violation problem, which uses hierarchical acceptability indexes. In addition, our results can be applied to other ranking and subset selection problems in which provide consensus rankings over the alternatives.