A review of soft consensus models in a fuzzy environment
Introduction
The term “consensus” has been used for years, even centuries, in a variety of context and areas. It generally concerns various situations in which there is a set of experts who present their testimonies which basically concern their opinions on some alternatives, topics, courses of action, etc., in question. The experts can be both individuals as such, smaller groups, larger groups, even organizations if they can be considered uniform with respect to the issues considered and testimonies [1], [2].
Consensus can be meant in various ways and group decision making (GDM) contexts. First, consensus is related with the state of agreement in a group in the sense that the individuals exhibit a state of common feeling as to the values in question. Strictly speaking, consensus has been meant from this perspective as a full and unanimous agreement [3], [4], though it has been deemed questionable if such a state is possible in virtually all real world situations [5]. Second, which is related to the first sense given above, consensus is meant as a way to reach consensus. This involves an evolution of the testimonies of the group towards consensus with respect to their testimonies; this evolution can be free or facilitated (moderated) by a special individual [1], [6]. Third, consensus can be meant as a way in which decisions should be meant in multi-person settings [7]. Basically, consensus decision making aims at attaining the consent, not necessarily the agreement, of the participants by accommodating views of all parties involved to attain a decision that will yield what will be beneficial to the entire group, not necessarily to the particular individuals who may give consent to what will not necessarily be their first choice but because, for instance, they wish to cooperate with the group. The full consent, however, does not mean that each individual is in full agreement [1]. Therefore, consensus boils down to cooperation in contrast to most GDM setting, notably voting, gaming, etc., which boil down to a competition.
It is clear that, ideally, consensus should refer to unanimity of individuals because the option or course of action attained will be best representative for the entire group. Obviously, unanimity may be difficult to attain, in particular in large and diversified groups of individuals as is the case in real world settings, and that is why milder benchmarks (definitions) of consensus have been employed exemplified by [1], [8]: unanimity minus one (U − 1), i.e., that all individuals but one support the decision, unanimity minus two (U − 2), i.e., all but two support the decision, unanimity minus three, (U − 3), etc. Moreover, some measures like 80%, 2/3, etc., can be employed, and even the so called rough consensus may be assumed which does not assume any specific rule which is to be determined later. Notice that all these milder definitions of consensus are still crisp and do not involve any imprecise (fuzzy) specification. We can well imagine here a fuzzy majority exemplified by “most”, “almost all”, “much more than a half” and so on [9], [10]. In such a way, we could introduce the concept of soft consensus as more adequate to model the GDM problems. We will discuss such fuzzy specifications, notably the concepts of a fuzzy majority and soft consensus [11], later in detail because they constitute a main topic here.
Virtually, all consensus reaching processes proceed in a multistage setting, i.e., the individuals change their opinions step by step until, possibly, some consensus is reached [1], [2]. Of course, this presupposes that the individuals are committed to those changes. At this junction, one can clearly see two situations. First, the process of changes of the testimonies proceeds in one way or another and can be modeled. Second, that process is moderated (facilitated) by a special person (“super-individual”) called a facilitator, moderator, etc., who is responsible for running the consensus reaching session in question by persuading the individuals to change their testimonies by rational argument, persuasion, etc., and keeping the process within a period of time considered [1]. The second option of a moderator running a consensus reaching process is usually more effective and efficient. However, this paradigm has been predominant in recent times only [5], [11], [12], [13].
Given the importance of obtaining an accepted solution by the whole group, the consensus has attained a great attention and it is virtually a major goal of GDM problems. The objective of this paper is to present a comprehensive presentation of the state of the art of all known consensus approaches, with an in-depth analysis of the respective problems and solutions as well as more relevant challenges. In particular, we focus on the consensus approaches in which there is a moderator, as they are more promising in practice and the most used in the literature, and based on the concept of fuzzy majority, which is more human-consistent and suitable for reflecting human perceptions of the meaning of consensus, i.e., the soft consensus.
This contribution is set out as follows. In Section 2, we describe the consensus processes based on moderator and the usual fuzzy GDM framework. In Section 3, we highlight the pioneer and most relevant contributions existing on consensus. In Section 4, the main fuzzy consensus approaches are described. The current trends and prospects in the development of consensus models are shown in Section 5. Finally, in Section 6, we present some concluding remarks.
Section snippets
Preliminaries
This section is devoted to describe the usual fuzzy GDM framework to develop consensus processes. In particular, we define the GDM problem, the usual consensus process, the formats of preferences used to express experts’ opinions, and the concepts of fuzzy majority and linguistic quantifiers.
Pioneer and prominent contributions
In this section we provide an historical perspective on the consensus approaches in decision making. To do so, we revise the pioneer and prominent contributions in the field.
The first mathematical approaches of consensus reaching processes started with the pioneering works by French and his collaborators in the late 1940s and early 1950s.
L. Coch, J.R.P. French. Overcoming resistance to change. Human Relations 1(4) (1948) 512–532.
J.R.P. French. A formal theory of social power. Psychological
Consensus approaches in GDM
In the literature, we can find different consensus approaches according to different criteria as reference domain, concept of coincidence, generation method of recommendations, and guiding measures. According to the reference domain used to compute the soft consensus measures, we find some approaches based on the expert set and others on the alternative set. According to the coincidence concept used to compute the soft consensus measures, we find some consensus approaches based on strict
Current trends in the development of consensus models
In this section, we present some current trends in the field of consensus models together with some open questions and prospects about them. We identify four current trends:
- 1.
Adaptive consensus models.
- 2.
Trust based consensus models.
- 3.
Dynamic and changeable consensus models.
- 4.
Consensus models based on agent theory.
Concluding remarks
In this paper, we have comprehensively analyzed consensus approaches based on soft consensus measures in which the consensus reaching process is guided by a moderator. To do so, we have introduced some basic concepts to understand the topic, we have highlighted both the pioneer and most relevant contributions on consensus models, we have described several approaches of consensus in GDM according to different criteria, and we have shown the current trends in the development of fuzzy consensus
Acknowledgments
This work has been developed with the financing of FEDER funds in FUZZYLING-II Project TIN2010-17876, the Andalusian Excellence Projects TIC-05299 and TIC-5991, and “Programa José Castillejo 2011 (JC2011-0002)”.
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