On the Consensus Modeling with the Grey Interval Preferences

摘要

In group decision making (GDM), grey (interval) opinion accurately reflects the range of the decision maker's subjective judgement. To achieve the optimal consensus, a moderator has to persuade each decision maker (DM) to change his/her grey opinion by paying the cost, while each DM expects return from the moderator for his/her changing opinion. A consensus model based on minimum cost from the moderator's viewpoint and a consensus model based on maximum return from the DM's viewpoint is developed. This paper first transforms a grey interval opinion into two discrete numbers, the opinion of membership degree and non-membership degree, then constructs a primal-dual linear programming models with grey interval preference to obtain the (optimal) consensus opinion. The primal consensus model is based on the minimum cost, and its objective function and restrictions consider both the consensus opinion of membership degree and non-membership degree; and the dual consensus model whose objective function and restrictions consider both the shadow price of membership degree and non-membership degree. Based on the principle of complementary slackness of primal-dual linear programming theory, the objective function and the restriction of dual consensus model are simplified. The relationship between individual DM's opinion, consensus opinions of membership degree and non-membership degree, shadow price of membership degree and non-membership degree, and the unit compensation of the moderator is analyzed. Meanwhile, the economic significance based on the fundamental properties of primal-dual linear programming problem is also explored.