Elsevier

Information Fusion

Volume 51, November 2019, Pages 19-29
Information Fusion

Full Length Article
Managing incomplete preferences and consistency improvement in hesitant fuzzy linguistic preference relations with applications in group decision making

https://doi.org/10.1016/j.inffus.2018.10.011Get rights and content

Highlights

  • A method to compute missing elements of IHFLPRs is introduced.

  • An additive consistency improvement method of IHFLPRs is proposed.

  • A GDM model based on IHFLPRs is introduced.

Abstract

Incomplete hesitant fuzzy linguistic preference relations (IHFLPRs) are useful in decision making which combine advantages of hesitant fuzzy linguistic term sets and incomplete fuzzy preference relations. The existing researches on IHFLPRs pay little attention to missing elements, and the consistency improvement processes change original information greatly. Inspired by the worst and the best consistency indexes of hesitant fuzzy linguistic preference relations, this paper constructs several optimization models to calculate the missing elements of IHFLPRs. As a result, a complete hesitant fuzzy linguistic preference relation is obtained. Furthermore, an algorithm is introduced to improve the additive consistency of the hesitant fuzzy linguistic preference relation to an acceptable level. Finally a group decision making model based on IHFLPRs is introduced and an example is presented.

Introduction

Hesitant fuzzy set (HFS) [24], [25] is an important extension of fuzzy set [40], in which several values might be the membership degrees of an object belonging to a given set. Based on HFS, many extended forms are introduced, among which the hesitant fuzzy linguistic term set (HFLTS) [20] is a popular one, and many research results on it have been developed. It is noted that HFLTSs can also consist of unbalanced linguistic terms although in most cases they are composed of balanced linguistic terms [6]. Some comprehensive review on HFSs and HFLTSs is provided in [19], [22].

Fuzzy preference relations are commonly-used in group decision making (GDM), which permit decision makers to provide a pairwise comparison at one time, and they don’t need to consider other alternatives at the same time. The HFLTSs facilitate decision makers to express their opinions when they are hesitant under linguistic environment. Combining both advantages, the hesitant fuzzy linguistic preference relations (HFLPRs) are a convenient tool for decision makers to express their preferences in form of HFLTSs when they think that the preference of one alternative over another is suitable to be expressed in several linguistic terms rather than a single linguistic term. For the first time Rodríguez et al. [21] utilized HFLTSs in fuzzy preference relations and proposed a GDM model. Constructing a HFLPR requires decision makers to provide n(n1)/2 pairwise comparisons for n alternatives. But in some situations, some elements of the HFLPR are not provided due to lack of necessary knowledge, avoidance of elicitation of sensitive information, time pressure, etc. In these situations, an incomplete hesitant fuzzy linguistic preference relation (IHFLPR) is utilized, which is similar to a traditional incomplete fuzzy preference relation (IFPR). There are mainly two ways to deal with IFPRs. One way is deriving priority weights to rank alternatives by using some methods such as least-square method [7], chi-square method [31], logarithmic least squares method [34], eigenvector method [35]. The other way is estimating missing elements of the IFPR and further conducting calculations on the complete FPR [1], [2], [3], [11].

Based on IFPRs, researches on incomplete hesitant fuzzy preference relations (IHFPRs) and IHFLPRs mainly focus on the following issues:

  • 1)

    Deriving priority weights directly from known elements of an IHFPR or an IHFLPR. This method is utilized in [32], [45], [47].

  • 2)

    Constructing a consistent FPR or a fuzzy linguistic preference relation (FLPR) by using known elements. The obtained FPR or FLPR is generally totally different from the original IHFPR or IHFLPR. This method is utilized in [23], [48].

Consistency is an important issue in using HFLPRs or HFPRs. There are mainly three types of consistency discussed in literature: weak consistency, additive consistency and multiplicative consistency. They represent human’s different understanding of consistency. Considering that additive consistency is stricter than weak consistency, and is more simple than multiplicative consistency in computation, this paper considers additive consistency of HFLPRs. Regarding consistency improvement methods for HFLPRs or HFPRs, most of existing researches can be categorized as follows:

  • 1)

    Constructing a consistent HFLPR or a HFPR which is totally different from the original HFLPR or HFPR. Generally the adjusted HFLPR or HFPR is obtained by using an optimization model. Sometimes the obtained HFLPR or HFPR might not be accepted by decision makers since all of original opinions of decision makers are adjusted. Researches using this method are such as [23], [45], [50].

  • 2)

    Adjusting the HFLPR or HFPR iteratively until an acceptable consistency is reached, and only partial elements are modified in the original HFLPR or HFPR. In such a method, the consistency is calculated in an average sense. Researches using this method are such as [15], [27], [36], [49].

Recently, Li et al. [13] introduced the interval consistency index (ICI), average consistency index (ACI), worst consistency index (WCI), and best consistency index (BCI) of HFLPRs. The WCI and BCI are obtained by computing the worst and the best additive consistency of FLPRs associated with a HFLPR, respectively. The ACI is the arithmetic mean of consistency indexes of all associated FLPRs, and ICI is an interval with lower bound and upper bound being WCI and BCI, respectively. By considering these consistency indexes, one can understand the overall consistency state of the HFLPR. Similar method is also utilized in [12] to compute the optimistic consistency index and pessimistic consistency index of FLPRs, and in [5] to calculate the classical consistency index, the boundary consistency index and the average consistency index of interval fuzzy preference relations. In [14] it is introduced an optimization model maximizing the average consistency index to obtain the personalized numerical values for linguistic terms.

Considering that IHFLPRs are a convenient way for decision makers, in this paper we focus on methods to manage incomplete elements in an IFHLPR and afterwards improve additive consistency of the obtained HFLPR. The existing researches dealing with IHFLPR or IHFPR do not attach much importance on missing elements. Thus we introduce a method to calculate each missing element based on WCI and BCI of a HFLPR. We then propose a direct additive consistency improvement approach improving the worst additive consistency to an acceptable level rather than improving the average additive consistency in most of the existing researches. This approach follows the second consistency improvement method, which improves consistency in an iterative way. It seems that this approach is more acceptable by decision maker since some of their original information is preserved.

The novelty of this paper is as follows:

  • 1)

    An algorithm to calculate missing elements in an IHFLPR is proposed. Motivated by WCI and BCI in [13], this algorithm considers the most hesitant situation and obtains the lower and upper bounds of for each missing element.

  • 2)

    A direct way to improve additive consistency of the obtained HFLPR is introduced. This method improves the WCI and the BCI of the HFLPR in an iterative way until an acceptable consistency is reached.

  • 3)

    A GDM model based on IHFLPRs is introduced. In this model, closeness of each expert to other experts is reflected in decision makers’ weights in the aggregation process.

The remainder of this paper is structured as follows. Section 2 reviews the linguistic 2-tuple model, HFLTSs, HFLPRs and IHFLPRs. Section 3 presents a method to compute missing elements of an IHFLPR. Section 4 introduces an additive consistency improvement process, as well as some numerical examples. Section 5 presents a GDM model and an example. Finally Section 6 concludes the whole paper.

Section snippets

Preliminaries

In this section we briefly review the linguistic 2-tuple model, HFLTSs, HFLPRs and IHFLPRs.

Computing missing elements of IHFLPRs

In this section, we develop a method to calculate missing elements of an IHFLPR.

Assume that H=(Hij)n×n is an IHFLPR. If Hij=x is missing, then we know Hji is also missing. For simplicity, we only give known elements Hij for i < j, and Hij, i > j can be obtained by using reciprocity. The models to compute WCI and BCI in [13] can be applied here.

First we denote the set of known elements as Ω={(i,j)|Hijisknown}, and the set of missing elements as Ω¯={(i,j)|Hijismissing}, We then introduce 0–1

Additive consistency improvement process for HFLPRs

In this section, we introduce a method to improve the additive consistency of the obtained HFLPRs, and present several numerical examples.

A GDM model based on IHFLPRs

In GDM with uncertainty and hesitation, sometimes decision makers provide their preferences in form of IHFLPRs due to some reasons. In [23] it is introduced a GDM method to select the best solution. They consider the situation that weights of decision makers are known. In this paper we consider a different situation that the weights of decision makers are unknown, but they are to be determined by each decision maker’s proximity to other decision makers.

Conclusions

The HFLTSs facilitate decision makers for their elicitation of hesitant and fuzzy assessments. Combining HFLTSs with FLPRs provides a useful tool for decision makers to compare alternatives with hesitation. Due to knowledge insufficiency or some other reasons, IHFLPRs are sometimes utilized. Based on some research results on IHFLPRs, we propose a new approach to estimate missing elements and improve its additive consistency, and further utilize this method in GDM.

The main results in this paper

Acknowledgements

This work is supported by the Key Scientific Research Funds of Henan Provincial Department of Education (16A630038); the National Natural Science Foundation of China (11872175); the Doctoral Research Start-up Funding Project of Zhengzhou University of Light Industry and Henan University of Economics and Law (BSJJ2013053, 800234).

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