The consensus models with interval preference opinions and their economic interpretation☆
Introduction
Group decision making (GDM) mainly focuses on unstructured decision making problems that require experts׳ subjective judgment [1], [2], [3], [4]. Generally, experts consist of lots of individual decision makers (DMs) who represent different interests, values and preferences. And therefore, how to arrive at a consensus has become a hot topic in GDM research. The definition of consensus has been varied widely. The American Heritage Dictionary defines consensus as “an opinion or position reached by a group as a whole”. Bezdek et al. [5] and Spillman et al. [6] interpreted consensus as “a full and unanimous agreement”. In fact, these two definitions are concerned with the final outcome, while Ref. [7], [8], [9] stress the evolutionary process of reaching consensus. Besides, Ness et al. [10] take consensus to represent the case where most DMs “agree on a clear option”, the few DMs who oppose this option provide rational and essential suggestions, and eventually, all the DMs “agree to support the decision”. In addition, Steve [11] divided consensus into two categories: accidental consensus (i.e., the theory chosen by all individual DMs based on their own independent judgment) and essential consensus (i.e., the viewpoint determined by collective negotiations and discussions). Obviously, how to reach the consensus in the latter category actually involves a multistage process: each individual changes their opinions gradually and all the views tend to be unanimous after many rounds of discussion [12], [13]. More importantly, the above evolutionary process usually needs a worthy, effective and efficient moderator (i.e., super-individual defined by [14]) who dominates the whole process of consensus reaching and has strong team leadership and communication skills to convince individuals to change their opinions into an acceptable option [12], [14]. Therefore, this research is based on the following hypothesis: a moderator [15], [16] introduced in GDM is able to persuade DMs to change their opinions towards a consensus opinion by paying costs (i.e., consuming resources such as time or money), and DMs׳ opinions will gradually approach the consensus.
Actually, consensus decision making involves a multi-objective problem-solving process, in which each individual is the minimum decision making unit, and his/her opinion can be regarded as a sub-objective. In the multi-objective optimization theory, a non-inferior (but not globally optimized) solution that meets most goals is a satisfying choice and is good enough for many real-life situations. Generally, multi-objective problems are often transformed into single objective problems with some decision rules [17]. Such aggregation operators in GDM as the weighted averaging (WA) operator [18], the ordered weighted average (OWA) operator [19], [20], the power average (PA) operator [21], and the probabilistic weighted average operator [22], representing decision rules in mathematics, are determined by DMs or the moderator under specific GDM background. For example, Ben-Arieh and Chen [23] presented a method which aggregates experts׳ judgements into a collective opinion with the fuzzy linguistic OWA operators and calculates the consensus level. Zhang [24] adopted several new hesitant fuzzy aggregation operators to solve multiple attribute GDM problems in hesitant fuzzy environments. His research incorporates both the decision arguments and the relationships between them. Furthermore, Parreiras et al. [25] introduced a flexible consensus scheme for multi-criteria GDM under linguistic assessments based on fuzzy aggregation models. In fact, the optimal solution to a single objective problem is actually a Pareto optimal (non-inferior) solution to the corresponding multi-objective problem [26], meaning that an ideal consensus reached in GDM is merely an optimal solution in the single-objective decision making sense, or merely a Pareto optimal solution in the multi-objective decision making sense.
In 2007, Ben-Arieh and Easton [15] introduced the concept of minimum cost consensus, constructed a multi-criteria consensus model under linear cost opinion elasticity, and presented linear-time algorithms to find the minimum cost consensus. They then generalized their work to derive new algorithms for reaching it [16]. The minimum cost consensus model by Ben-Arieh and Easton [15], [16] is to aggregate the deviations between an individual׳s opinion and consensus opinion using a weighted arithmetical mean operator and to construct an optimization programming model based on it. In fact, the consensus model, proposed by Ben-Arieh et al., stems from Gonzalez-Pachon and Romeroc׳s distance-based goal programming (GP) models [4], [27], [28]. The widely used standard GP model, brought forward by Charnes and Cooper [29], aims to minimize the deviations attached to the goal and the aspiration levels determined by DMs. As a result, Gonzalez-Pachon and Romeroc׳s findings have laid a theoretical foundation for the research of Ben-Arieh et al. on the construction of the minimum cost consensus models.
The earlier consensus research based on total cost only takes into account the moderator׳s point of view, while in fact, individuals have to continuously modify their opinions to obtain a compromise consensus [28], and therefore it is necessary to take their interests into consideration as they deserve to be compensated. A case in point is the realization of IPCC (Intergovernmental Panel on Climate Change) consensus, which is obtained through multiple rounds of negotiations and constant compromises among political groups as well as experts in various fields. In the process, the consensus will be unilateral, or may be swayed by sectional groups if only the viewpoints of some center groups (i.e., moderator) are considered. And hence, to obtain a more scientific IPCC consensus, we should consider the voice of the developing countries or relatively weak groups (i.e., individual DM) who have abandoned their own interests to arrive at the generally accepted result. As an exchange, they will expect some compensation (e.g. lower taxes or more carbon quotas) and the more the better. Therefore, this paper constructs two linear optimization models based on the minimum cost and maximum return to explore from both the moderator׳s and the individual DMs׳ perspectives the cost consensus problems in GDM. With the help of the primal-dual linear programming theory, we analyze the relationships between all the variables and present the economic interpretation of the models proposed. This paper is an extension of the minimum cost consensus model presented by Ben-Arieh and Easton [15], [16].
The rest of this paper is arranged as follows. Section 2 introduces the distance measure on consensus decision making and then proposes two multi-objective programming models based on minimum cost and maximum return. Section 3 establishes the minimum cost and maximum return models for reaching consensus, which are based on primal-dual theory; and also explores the economic significance of these two models. Section 4 investigates the properties of these two primal-dual linear programming models and further discusses the theoretical meaning and economic significance of the primal-dual models. To further explain the proposed models, we provide an illustrative example in Section 5. Lastly, the conclusion and directions for future research are provided in Section 6.
Section snippets
Multi-objective programming models based on minimum cost and maximum return
Suppose that there are m decision makers (DMs) taking part in a GDM. Let , , represent the opinion of DM , , in the GDM. Each opinion of an individual DM represents an individual interest, so we define it to be an individual opinion. In consensus decision making, the ideal state is where there exists an ideal opinion such that . That is, when all opinions are equal to the same ideal opinion , the group has arrived at unanimity. Such an ideal opinion in
Construction and interpretation of minimum cost and maximum return consensus models
According to multi-objective programming theory, it is easy to find a set of Pareto efficient solutions by transforming the multi-objective programming model into a single-objective one with some aggregation operators. If we consider the total cost of all individuals as a whole, then the above model (1) can be transformed into a single-objective programming model. Generally speaking, we only need to find the optimal solution — also the Pareto optimal solution (non-inferior
Further interpreting the relation between the primal problem LP(w) and its dual problem DLP(w)
This section further interprets the economic significance of the consensus model by discussing the other relations between the primal problem LP(w) and its dual problem DLP(w). In other words, we will explore the condition which can lead to the consensus of all individuals and the moderator; we discuss the practical significance of the consensus opinion and the unit return; in addition, we will try to figure out the relations between the individual unit return, interval opinion, the moderator׳s
Numerical example
This example further interprets the relation between the primal problem LP(w) and its dual problem DLP(w). Suppose that there are four DMs , , , and in a GDM and their corresponding interval opinions are , , , and , respectively. The unit cost paid to the four DMs by the moderator are , , and , respectively, and the consensus opinion is supposed to be . The optimal consensus decision making model based on minimum cost is
Conclusions and future research
When dealing with inevitably imprecise or not fully reliable opinions, it is difficult for a DM to develop a precise, clear opinion. An interval opinion indicates a range of opinion, in which the left and right points are known but the true value cannot be fully determined. The optimization method is one of the most effective means for obtaining such a true value from an interval. In this paper, we first regarded an individual opinion as an interval, then, by introducing a distance-based
Acknowledgments
The work in this paper was supported by the Excellence Andalusian Project TIC-5991, KAU-project 27-135-35/HiCi, the National Project TIN2013-40658-P, the Innovation Foundation of Postgraduate Science and Technology in Jiangsu Province (KYLX_0858), a Marie Curie International Incoming Fellowship within the 7th European Community Framework Programme (Grant no. FP7-PIIF-GA-2013-629051), the National Natural Science Foundation of China (71171115, 71173116, 70901043), the Natural Science Foundation
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