(b) The dissent of e is defined as σm(e) =
m
∑
l=1 σΛlGl(e)
m .
(c) The precision-dissension profile of e is defined asΦm(e) = (m, πm(e), σm(e)).
With Φm(e) one can know the characteristics of the assessments of expert e
regarding precision and dissension after evaluating m different sets of alternatives within their respective groups.
4.6
Conclusions and future work
Based on the weak points of existing consensus measures for GDM by means of HFLTSs, two consensus measures are defined in this chapter in order to capture dif- ferences among situations in which the previous measures are not able to make a difference.
On the one hand, a consensus level is defined for the whole group on a specific alternative as a normalization of the addition of distances from a central opinion to the opinion of each expert of the group, and an analogous definition is given for a set of several alternatives instead of just one of them. On the other hand, the consensus level is defined for each expert with respect to the rest of the group based on the distance between his/her opinion and the central opinion for both one specific alternative and a set of alternatives.
Additionally, a study is carried out to compare the presented measures with the similar existing ones and concludes that the measures presented in this chapter are more accurate in situations in which existing measures consider the level of agree- ment to be the same but where common sense suggests they should be different. Moreover, the comparison study also shows that the collective degree of consensus presented in this chapter has a lower time complexity than the existing measures.
Lastly, a profile of an expert is presented to keep track of the precision and dissen- sion in his/her assessments with a view to using this information for future experts selection processes.
Future work will focus on two main directions. From a theoretical point of view, a dynamical study will be carried out on both the consensus-reaching process and the precision-dissension profile of DMs in several GDM processes. In particular, the proposed consensus measures will be used to measure polarization in this kind of scenarios. From a practical point of view, all the introduced concepts are already being implemented in a real case example framed in the city tourism management field.
Chapter 4. Consensus, Dissension and Precision in Group Decision Making by means of an Algebraic Extension of Hesitant Fuzzy Linguistic Term Sets
Acknowledgements
This research has been partially supported by the INVITE Research Project (TIN2016- 80049-C2-1-R and TIN2016-80049-C2-2-R (AEI/FEDER, UE)), funded by the Span- ish Ministry of Science and Information Technology.
Chapter 5
Free Double Hierarchy Hesitant
Fuzzy Linguistic Term Sets: An
Application on the TOPSIS
Methodology
5.1
Introduction
Fuzzy sets were introduced by Zadeh to deal with uncertain decision-making pro- cesses [77]. Several extensions have been presented since then such as the Intuition- istic Fuzzy Sets [5, 6] or the Hesitant Fuzzy Sets (HFSs) [28, 62, 72]. However, in some areas, people prefer to use a qualitative reasoning better than a quantitative reasoning. To this end, Zadeh also introduced the concept of linguistic variable [78]. From then on, several studies have been developed on that field [24, 39, 40,42, 43,
65].
With the aim of combining HFSs and qualitative reasoning, Rodríguez et al. intro- duced the concept of Hesitant Fuzzy Linguistic Term Set (HFLTS) [53] that was later redefined in a mathematical form by Liao et al. [35]. So far, several contributions pre- sented in the literature have studied several aspects of the HFLTSs such as hesitant fuzzy linguistic information aggregation techniques [19,68], hesitant fuzzy linguistic measure methods [20, 33, 35], hesitant fuzzy linguistic operational laws [18], hes- itant fuzzy linguistic preference relations [35, 36, 81] and hesitant fuzzy linguistic decision-making methods [20,45,46].
Nonetheless, in some situations, HFLTSs are not able to depict with enough de- tails the complexity inherent in human reasoning when evaluating with linguistic assessments. Some authors have studied how to define linguistic expressions more complex than single linguistic terms as reviewed by Rodríguez et al. [54]. In order to provide a more precise tool, Gou et al. presented the concept of Double Hierarchy Hesitant Fuzzy Linguistic Term Sets (DHHFLTSs) [21]. This structure allows each decision maker to choose one term from a first hierarchy Linguistic Term Set (LTS) and later choose another term from a second hierarchy LTS gaining more accuracy on the linguistic assessment.
Chapter 5. Free Double Hierarchy Hesitant Fuzzy Linguistic Term Sets: An Application on the TOPSIS Methodology
DHHFLTSs are a very useful tool to deal with qualitative assessments, yet they present some shortcomings given that the second hierarchy LTS has to be the same for every single term of the first hierarchy LTS. This leads us to three main issues:
Firstly, misleading or meaningless linguistic expressions may appear as a result of using a fixed second hierarchy LTS. For instance, while “extremely” has a strong positive meaning on “good”, it does not have the same positive meaning when it is applied to “regular”. In addition, while a term like “close to” makes sense when applied to “perfect”, it should not be applied to a linguistic term like “normal”. This is due to the fact that “close to normal” can be understood in two different meanings (worse than average or better than average).
Secondly, not all linguistic terms need the same range of precision for their cor- responding second hierarchy LTSs. As an example, linguistic terms such as “bad” or “good”, in general, accept a much wider variety of precision than terms such as “null” or “perfect”.
Lastly, all decision makers are forced to use the same second hierarchy LTS. It is known that the decision makers have their own preferences about which linguis- tic expressions to use. For instance, for the linguistic term “perfect”, one decision maker could prefer to use the second hierarchy LTS {“not far from”, “almost”, “com- pletely”}, and another one could feel more comfortable by using {“close to”, “to- tally”}.
In this chapter, we present a new structure that overcomes these three issues called Free Double Hierarchy Hesitant Fuzzy Linguistic Term Sets (FDHHFLTSs), whose elements are called Free Double Hierarchy Hesitant Fuzzy Linguistic Elements (FD- HHFLEs). Based on the introduced structure, each decision maker involved in Group Decision-Making (GDM) situation is allowed to choose the second hierarchy LTS that he or she thinks that suits it better, with as many terms as desired.
Furthermore, an order and a distance between FDHHFLEs are also presented in this chapter in order to compare and quantify distances between linguistic assess- ments provided by the decision makers by means of the aforementioned structure. These order and distance are used to introduce a free double hierarchy approach based on the well-known multi-criteria decision-making TOPSIS ranking method, enabling us to rank alternatives that have been assessed by means of free double hierarchy hesitant fuzzy linguistic information.
The rest of this chapter is structured as follows: First, Section 5.2 summarizes basic concepts already introduced in the literature that will be used throughout the work. The new free double hierarchy structure is introduced in Section 5.3. Sec- tion5.4introduces an order and a distance for FDHHFLEs. A free double hierarchy approach based on the TOPSIS method is presented in Section5.5as well as a sim- ulated example to illustrate the presented approach. Finally, Section5.6summarizes the main conclusions and points out the directions of future research.