Two consensus models based on the minimum cost and maximum return regarding either all individuals or one individual

In some important group decision making, a moderator representing the collective interest, who has predetermined, and possesses an effective leadership and strong interpersonal communication and negotiation skills, is crucial to the consensus reaching. In the process of consensus reaching, the moderator needs to persuade each individual to change his/her opinion towards a consensus opinion by paying a minimum cost, while the individuals have to modify and to gradually approach this consensus opinion by expecting to obtain a maximum compensation. This paper, which proposes two kinds of minimum cost models with regard to all the individuals and one particular individual respectively, shows the economic significance of these two models by exploring their dual models grounded in the primal–dual linear programming theory, and builds the conditions under which these two models have the same optimal consensus opinion. The validity of the theoretical analysis is confirmed by numerical examples.