Consensus Models At Minimum Cost And Its Economic Interpretation

Reaching a consensus in group decision-making (GDM) usually needs lots of time and resources, so how to minimize the total cost and to construct the minimum cost consensus model become the hot topics of the decision-making field. Based on Ben-Arieh and Easton’s work, this paper further explores the minimum cost consensus problem by introducing the primal-dual programming theory, which expands and improves the theory of the minimum cost consensus: firstly, obtaining a consensus is actually a result of two-way communication and negotiation conducted between the moderator and individuals, therefore, the consensus decision-making should also take individual’s interest into consideration. Using the primal-dual programming theory, the maximum return model expected by all the individuals is proposed in this paper, on the basis of the minimum cost consensus model, deeply improving the consensus decision-making theory. Secondly, the economic meanings of the minimum cost and maximum return models are given in this paper, as well as their relations and properties, which is of great importance to explore the practical significance of the above two consensus models. Finally, the consensus models of the minimum cost and maximum return are expanded into several common decision scenarios:on the one hand, the expression of individual’s opinion in the minimum cost consensus model is changed from crisp number into interval number; on the other hand, the method to measure the opinion deviation is changed from the distance form into the least square form. The properties and the relations, as well as the economic interpretation, of the consensus models under the four decision scenarios are deeply discussed in this paper. As a result, this paper provides a new way to apply the minimum cost consensus model into the real world, and it improves the applicability and feasibility in reality. The main conclusions of this paper are listed as follows:(1) Under the situation of normal GDM/opinion defender, the distance form is adopted to measure the consensus deviation. Once a consensus is achieved, the unit return expected by the individual/opinion defender has economic significance, and the amount of all the maximum return expected by individual DMs is equal to that of the total cost paid by the moderator; besides, the relations between the unit return expected by each individual/opinion defender and the unit cost paid by the moderator are:if the consensus opinion is greater than an individual’s or the opinion defender’s opinion, then their unit return is negative, and vice versa; if individual DM’s or the opinion defender’s unit return is less than the unit cost paid by the moderator, then the value of the their opinion must equal to that of the consensus opinion. In addition, under some certain conditions, the optimal solutions for the minimum cost consensus model Pk(ε) and its basic model p(w) can both exist, and the ideal consensus opinions solved from them will be same. Furthermore, an intuitive method to solve the optimal solution of the basic minimum cost consensus model is given in this paper.(2) In normal GDM, the distance form is used to measure the consensus deviation and interval number is used to denote the individual’s preference. Once the consensus is reached, the unit return expected by each individual has its economic significance, and the amount of the total return acquired by all the individuals is equivalent to that of the total cost consumed by the moderator. Results show that compact correlations exist between the unit return expected by the individual for changing their opinions and the location of the consensus opinion within the individual’s interval. Similarly, there also exist compact correlations between the group opinion and the location of the individual DM’s unit return within the moderator’s positive/negative unit cost interval.(3) The least square form is applied to measure the consensus deviation. In GDM, the opinions of the moderator and the representative, who is on behalf of all the DMs, both have their economic significance. Once the consensus is reached, the two variables are equal, and the maximum total return of all DMs and the minimum total cost of the moderator are equal. Under the situation of opinion defender, similar properties can also be obtained. In addition, the shadow prices of the stakeholders have important economic significance and actually have the same definition domain as the consensus opinion. The relations between the stakeholders’shadow prices and the consensus opinion are:if the group opinion completely falls in the deviation range accepted by the stakeholder, then the return expected by the stakeholder is zero; if the expected return of some stakeholder is not zero, then it indicates that the optimal opinion is determined by this individual DM, further revealing the indirect influence of other individual’s acceptable consensus deviation on the final ideal group opinion.