Some interesting properties of the fuzzy linguistic model based on discrete fuzzy numbers to manage hesitant fuzzy linguistic information

Highlights

• Properties of the fuzzy linguistic model based on discrete fuzzy numbers are analysed.

• This model is used to handle hesitant fuzzy linguistic information.

• This model includes the hesitant fuzzy linguistic term sets model.

• Some advantages of the model based on discrete fuzzy numbers are pointed out.

• A fuzzy decision making model based on discrete fuzzy numbers is proposed.

Abstract

The management of hesitant fuzzy information is a topic of special interest in fuzzy decision making. In this paper, we focus on the use and properties of the fuzzy linguistic modelling based on discrete fuzzy numbers to manage hesitant fuzzy linguistic information. Among these properties, we can highlight the existence of aggregation functions with no need of transformations or the possibility of a greater flexibilization of the opinions of the experts, even using different linguistic chains (multigranularity). Furthermore, based on these properties we perform a comparison between this model and the one based on hesitant fuzzy linguistic term sets, showing the advantages of the former with respect to the latter. Finally, a fuzzy decision making model based on discrete fuzzy numbers is proposed.

Introduction

The presence of uncertainty is a common feature that characterizes a wide range of real problems related to decision making. Thus, for instance, a group of experts should decide the feasibility of a financial transaction from information which is often vague or incomplete, or similarly, when a medical team must make a diagnosis based on preliminary tests. Frequently, the uncertainty appears when we use assessments not necessarily quantitative but we deal with terms or qualitative information when making the decision. For this reason, the fuzzy linguistic approximations have emerged as a tool which allow to properly handle the qualitative information. In this sense, we highlight the symbolic linguistic model based on ordinal scales (wherein an order is considered among the different linguistic labels) [11], [14]; the linguistic 2-tuples model [15], which introduces the symbolic translation to the linguistic representation; the linguistic model based on type-2 fuzzy sets representation [32], which represents the semantics of the linguistic terms using type-2 fuzzy membership functions; the proportional 2-tuple model [34], which extends the 2-tuple model by using two linguistic terms with their proportion to model the information, or the linguistic model based on PSO and granular computing of linguistic information [2], which proposes to model the linguistic information like expressed in terms of information granules defined as sets, among many others. A common feature of many of these models is that experts should express their valuations choosing a single linguistic level associated with the linguistic variable. This kind of information is usually interpreted using a linguistic scale like this,

$$\mathcal{L}=\{\mathrm{EB},\mathrm{VB},\mathrm{B},\mathrm{F},\mathrm{G},\mathrm{VG},\mathrm{EG}\}$$

where the linguistic terms correspond to the expressions “Extremely Bad”, “Very Bad”, “Bad”, “Fair”, “Good”, “Very Good” and “Extremely Good” respectively. However in many cases the experts’ opinions do not correspond exactly to a particular linguistic term. On the contrary, expressions like “better than good” or “between fair and very good” or, even more complex ones, are commonly used by experts in order to make their opinions. Recently, V. Torra [31] introduced the hesitant fuzzy sets as a possible generalization of the classical fuzzy sets. Based on this idea, different authors [29], [30], [36], [37] have proposed a computational linguistic model based on hesitant fuzzy linguistic term sets, allowing that the expert could consider several possible linguistic values or richer expressions than a single term for an indicator, alternative, variable, etc; increasing the richness of linguistic elicitation based on fuzzy linguistic approach. Several methods based on hesitant fuzzy linguistic term sets for group decision-making have been recently proposed (see [6], [17], [18], [35], [40]). On the other hand, the authors [4], [5], [21], [26], [27], [28] have considered another linguistic computational model based on discrete fuzzy numbers [33], which allows to interpret the qualitative information in a more flexible way and also enables to aggregate the information they expressed using this type of fuzzy subsets directly. Although at a first glance the two models seems to be different, they are in fact quite connected. As we will prove, the hesitant fuzzy linguistic term sets used by the first model can be interpreted as particular cases of discrete fuzzy numbers and although the aggregation functions used in the aggregation phase presented in the original articles (see [29] for the hesitant fuzzy sets based one and [21] for the discrete fuzzy numbers based one) are different (as it is the exploitation phase), if both models used the adequate aggregation functions and the same exploitation method, the results would coincide.

In this article we want to make a step further and the main goal will be to show the advantages of using the model based on discrete fuzzy numbers with respect to the hesitant fuzzy sets based one. These advantages are related to the existence of aggregation functions on the set of discrete fuzzy numbers which allow us to aggregate the opinions without any transformation or loss of information, to the possibility of a greater flexibilization of the opinions given by the experts and finally, to the possibility of the experts to evaluate using different linguistic chains (multigranularity [23]). We will also present an example of a multi-criteria decision making problem using the linguistic model based on discrete fuzzy numbers, showing the commented advantages of this method.

The paper is organized as follows. In Section 2, we make a brief review of discrete fuzzy numbers and hesitant fuzzy linguistic term sets. In Section 3, we explain the main characteristics of the fuzzy linguistic model based on discrete fuzzy numbers, comparing this method with the model based on hesitant fuzzy linguist term sets. Throughout the process, we prove that the discrete fuzzy numbers allow a greater flexibilization and an easier management of the opinions given by the experts. In Section 4, we propose a multi-criteria decision making approach based on these particular fuzzy subsets using extensions of two well known discrete parametric compensatory aggregation functions [8], [20]. The last section is devoted to give some conclusions and the future work that we want to develop.