Elsevier

Information Sciences

Volume 489, July 2019, Pages 93-112
Information Sciences

Group decision making with double hierarchy hesitant fuzzy linguistic preference relations: Consistency based measures, index and repairing algorithms and decision model

https://doi.org/10.1016/j.ins.2019.03.037Get rights and content

Highlights

  • We define double hierarchy hesitant fuzzy linguistic preference relation (DHHFLPR).

  • A novel method is developed to calculate the consistency thresholds.

  • Two consistency repairing algorithms is proposed to improve the DHHFLPR.

  • A method is set up to deal with GDM problems with DHHFLPRs.

  • The proposed method is validated by a practical group decision making problem.

Abstract

Group decision making, refers to inviting a group of decision makers to evaluate, prioritize or select the optimal one among some available alternatives in the actual decision making process. Considering that the double hierarchy hesitant fuzzy linguistic term set can describe natural languages clearly, in this paper, we define the concept of double hierarchy hesitant fuzzy linguistic preference relation (DHHFLPR) and propose some additive consistency measures. To judge whether a DHHFLPR is of acceptable consistency or not, we introduce a consistency index, and develop some novel threshold values for judging whether a DHHFLPR is of acceptable consistency or not. Furthermore, we develop two consistency repairing algorithms based on the automatic improving method and the feedback improving method respectively, to improve the DHHFLPR with unacceptable consistency. Additionally, a method is set up to deal with group decision making problems with double hierarchy hesitant fuzzy linguistic preference information. Finally, the proposed method is validated by a case study that is used to evaluate the water resource situations of some important cities in Sichuan Province, and some comparative analyses are given to show the efficiency of the proposed method.

Introduction

Group decision making refers to inviting a group of decision makers to evaluate, prioritize or select the optimal one among some available alternatives in the actual decision making process. During group decision making, linguistic information is more in line with the real thoughts of decision makers and Zadeh [33], [34], [35] proposed a fuzzy linguistic approach to deal with it. As well as he proposed a concept of Computing with words (CW). Zadeh [32] also explained CW by “Computing with words is a system of computation in which the objects of computation are words, phrases and propositions drawn from a natural language. The carriers of information are propositions. It is important to note that Computing with words is the only system of computation which offers a capability to compute with information described in a natural language.” And he divided CW into two levels. In Level 1 CW (CW1), the objects of computation are some simple linguistic terms such as words, phrases and simple propositions. In Level 2 CW (CW2), the objects of computation include possibly complex propositions, and semantics of natural languages play an important role. Motivated by the CW2, in recent years, lots of linguistic models based on fuzzy set theory were developed to represent complex linguistic information such as hesitant fuzzy linguistic term set (HFLTS) [7], [13], [16], [18], 2-tuple linguistic model [11], [14], virtual linguistic term model [30], [31], and type-2 fuzzy sets [2], [15], [28].

Complex linguistic information can be found around us in our daily lives. For example, a teacher is hesitant when he/she gives the mark of a student, and he/she may utilize a HFLTS {good, very good, perfect} to express his/her opinion. However, all the linguistic terms included in this HFLTS have the same important degrees, which is not always adequate when representing the real thoughts of people. Therefore, one question is raised: How should we represent natural languages more accurately? With this in mind, four novel proposals have been developed to solve this problem: Firstly, Pang et al. [17] proposed a probabilistic linguistic term set (PLTS), which mainly consists of two parts: one is to utilize weights to represent the important degrees of natural languages given by people directly; the other one is to show the frequencies of linguistic terms. However, considering that sometimes the weights of linguistic terms included in complex linguistic information cannot be expressed clearly by PLTS such as “more than fast”, Durand and Truck [5] developed a mapping function to compute weights and assigned them to corresponding linguistic terms. Additionally, Zhang et al. [36] introduced a probabilistic distribution of several linguistic terms, and developed the concept of distribution linguistic preference relations. Obviously, all of the above linguistic models consist of linguistic terms and numerical values simultaneously. To only utilize linguistic labels to represent complex linguistic information, Gou et al. [9] proposed a double hierarchy linguistic term set, and its hesitant extension named double hierarchy hesitant fuzzy linguistic term set. Double hierarchy linguistic term set adds a second hierarchy linguistic term set and uses linguistic labels to represent the important degrees of complex linguistic terms rather than numerical values, but the second hierarchy linguistic terms of different first hierarchy linguistic terms have no inevitable relation. In fact, all these four linguistic models belong to the CW2. Considering that the double hierarchy hesitant fuzzy linguistic term set can be used to reflect complex linguistic information intuitively, it will serve as the basis for this study and its basic element is called a double hierarchy hesitant fuzzy linguistic element (DHHFLE).

In group decision making, preference relations are popular and powerful techniques for decision maker preference modeling [24]. A large number of preference relations have been proposed in the literature such as the fuzzy preference relations [23], the linguistic preference relations [36], the multiplicative preference relations [19], and the hesitant fuzzy linguistic preference relation (HFLPR) [8], [25], [29], [37], [38]. Consistency measures of preference relations are the vital basis of group decision making and have been studied extensively, which show that the supplied preferences satisfy some transitive properties [29]. Consistency measures include two parts: (1) judging whether each preference relation is of acceptable consistency; (2) improving the preference relation with unacceptable consistency.

Up to now, two critical defects of existing consistency measures are being more and more apparent:

  • 1)

    It is common that the normalization procedure is very necessary for making calculations expediently. But almost all methods complete it by adding or deleting some linguistic terms [38]. Obviously, these methods may cause the original information loss and make calculations complex.

  • 2)

    Considering that there are some unreasonable places in the calculations of consistency thresholds under linguistic preference information environment, it is necessary to improve the existing consistency thresholds as the novel references for consistency improving processes.

To solve these two defects successfully, whilst considering that the double hierarchy hesitant fuzzy linguistic term set can describe linguistic evaluation information comprehensively and correctly, as well as there exists no any research available regarding its preference information. In this paper, the decision makers’ linguistic evaluation information can establish some preference matrices with double hierarchy hesitant fuzzy linguistic information, denoted as double hierarchy hesitant fuzzy linguistic preference relation (DHHFLPR). In addition, to avoid the occurrence of some self-contradictory situations, it is very important to carry out the consistency checking and improving process for each DHHFLPR in a group decision making process. In this paper, we discuss some additive consistency measures for DHHFLPRs and the main contributions of this paper are summarized as follows:

  • a)

    For the first defect above, we develop a new normalization method by utilizing the linguistic expected-value of each DHHFLE to transform the DHHFLPR into the normalized DHHFLPR equivalently. The linguistic expected-value of the DHHFLE can be obtained by aggregating all elements of a DHHFLE into a double hierarchy linguistic term. With this method, we will not lose any linguistic terms and can make the calculations simpler.

  • b)

    For the purpose of judging whether a DHHFLPR is of acceptable consistency or not, we define a consistency index of the DHHFLPR and develop a novel method to improve the existing methods for calculating the consistency thresholds. Then we present two convergent consistency repairing algorithms based on automatic improving method and feedback improving method respectively to improve the consistency index of a given DHHFLPR with unacceptable consistency.

  • c)

    We propose a weight-determining method for obtaining the weight information of each decision maker, and then develop an algorithm to deal with the group decision making problem with double hierarchy hesitant fuzzy linguistic preference information.

Nowadays, the Sichuan water resource is an important water system in China. The protection of water quality of Sichuan water resources has become a crucial issue for the economic and social stability and rapid development of China. Therefore, the evaluation of water resource situations is a very important study carried out every year. In this paper, a case study is set up to apply our method to deal with a practical group decision making problem which is to evaluate the water resource situations of some important cities in Sichuan province.

To do so, the rest of this paper is organized as follows: Section 2 mainly discusses some basic concepts. Section 3 defines DHHFLPR, the additive consistent DHHFLPR, and the consistency index of DHHFLPR. Section 4 develops two convergent consistency repairing algorithms. Section 5 develops an algorithm to deal with the group decision making problem with DHHFLPRs. Section 6 sets up a case study to handle the Sichuan water resource management problem, and makes some comparative analyses with existing methods. Section 7 gives some discussions for highlighting the advantages of the proposed methods. Finally, we make some conclusions and propose some future research directions in Section 8.

Section snippets

Introducing the double hierarchy linguistic term set and the hesitant extension

In this section, we discuss three essential issues regarding double hierarchy linguistic term set and its hesitant extension with the aim of understanding them better.

DHHFLPR: Additive consistency and index

As we mentioned in the Introduction, there are some shortcomings such as the normalization methods, consistency index and consistency thresholds in existing consistency measures. In this section, a novel concept of DHHFLPR is defined first, then we develop an additive consistency measure method and a consistency index of DHHFLPR on the basis of the distance measure of DHHFLEs to judge whether a DHHFLPR is of acceptable consistency or not.

Consistency repairing algorithms

In some practical decision making processes with DHHFLPRs, it is common for there to be a DHHFLPR H˜SO=(hSOij)m×m of unacceptable consistency, namely, CI(H˜SO)>CI¯(H˜SO). In this case, we need to repair the DHHFLPR H˜SO until it reaches the consistency threshold. To improve the consistency, two existing methods have been developed: the automatic method [38] and the feedback-based method [1], [6], [12], [38]. Similarly, we establish two consistency repairing algorithms based on the automatic

Group decision making with DHHFLPRs

In this section, we first describe the group decision making problem with DHHFLPRs. Then a decision maker weight-determining method is developed on the basis of information entropy theory. Finally, an algorithm is proposed to deal with the group decision making problem with DHHFLPRs.

Case study: Sichuan water resource management

In this section, the proposed method is validated by a case study of evaluating the water resource situations of some cities in Sichuan Province, and some comparative studies with others methods are made.

Discussion

Based on the decision making processes above and the basic characteristics of different consistency checking and repairing models [25], [29], [38], the details of these models are summarized as follows:

  • (1)

    Firstly, we have discussed two different consistency repairing algorithms for the DHHFLPR of unacceptable consistency. The automatic optimization method mainly improves the DHHFLPR of unacceptable consistency by utilizing the adjusted parameter θ (0 ≤ θ ≤ 1). We can obtain different results if we

Concluding remark

In this paper, we have defined the concept of DHHFLPR and developed some additive consistency measures. Then, utilizing the linguistic expected-value of DHHFLE, we have proposed a new normalization method to transform the DHHFLPR into the normalized DHHFLPR equivalently. Additionally, for the purpose of judging whether a DHHFLPR is of acceptable consistency or not, we have defined a consistency index of the DHHFLPR and develop a novel method to improve the existing method for calculating the

Acknowledgments

This study was funded by the National Natural Science Foundation of China (Nos. 71571123, 71771155, 71501135, 71771156 and 71801174), the Major Program of the National Social Science Fund of China (No. 17ZDA092), the Scholarship from China Scholarship Council (No. 201706240012), and the Fundamental Research Funds for the Central Universities (No. 2012017yjsy121).

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