Full Length ArticleDouble hierarchy hesitant fuzzy linguistic term set and MULTIMOORA method: A case of study to evaluate the implementation status of haze controlling measures
Introduction
Hesitant fuzzy linguistic term set (HFLTS), combined by hesitant fuzzy set (HFS) [1], [2], [3], [4] and fuzzy linguistic approach [5], was developed by Rodríguez et al. [6] in 2012. It is a useful tool to deal with qualitative information given that the HFLTS can represent the linguistic information that is much more in line with people's cognitions and expressions. In recent years, amounts of scholars have researched the HFLTS theory from different research directions including information aggregation [7], [8], [9], fuzzy measures [10], [11], [12], [13], [14], preference relations [13], [15], [16], [17], decision making [12], [13], [14], [18], [19], [20], [21], [22], [23], etc.
As the researches on HFLTSs have been studied in-depth and the HFLTS theory is becoming gradually mature, some shortcomings of HFLTSs, however, also emerged from two aspects:
- a)
In group decision making process, the aggregated hesitant fuzzy linguistic information cannot represent the important degree or the frequency of each linguistic term included in the HFLTS.
- b)
The HFLTS is not accurate enough to describe some more complex linguistic terms or linguistic term sets (LTSs).
For the first shortcoming, Pang et al. [24] defined a probability linguistic term set (PLTS) to generalize the HFLTSs by adding the probability information of each single linguistic term, which is a very reasonable method for saving all original linguistic information given by the experts in group decision making process. Furthermore, by utilizing the PLTSs, the experts can not only provide several linguistic evaluation values over an object (alternative or criterion), but also reflect the probability information of each element included in the LTS. Later, some scholars have studied the PLTSs from different aspects, among others: probabilistic linguistic preference relation and consistency measures [25], probabilistic linguistic vector-term sets to promote the application of multi-granular linguistic information [26], comparative procedure-based multiple criteria decision making (MCDM) problems [27], novel operational laws of PLTSs based on two equivalent transformation functions [28].
For the second shortcoming, it is obvious that sometimes the HFLTS cannot describe some complex linguistic terms or LTSs accurately. For example, let be a LTS, then we can utilize {s2, s3}, and {s2} to express the linguistic expressions “more than very high”, “between low and high” and “very high”. However, sometimes, we may need to use some more complex linguistic terms to represent our comprehensive opinions such that “entirely high”, “just right medium”, “a little high”, etc. Considering that we cannot use any method or theory to solve this problem, in this paper, we introduce a novel concept: double hierarchy linguistic term set (DHLTS). Generally, the DHLTS consists of two hierarchy LTSs (denoted by the first hierarchy LTS and the second hierarchy LTS). The second hierarchy LTS is a linguistic feature or detailed supplementary of each linguistic term included in the first hierarchy LTS. Let the above LTS S be the first hierarchy LTS, and be the second hierarchy LTS. Then we can describe “entirely high”, “just right medium”, “a little high” with DHLEs (the element included in the DHLTS), which are denoted as , and , respectively. Based on the DHLTS, we can develop a double hierarchy hesitant fuzzy linguistic term set (DHHFLTS). The DHHFLTS is a novel concept, which can be used to deal with some practical MCDM problems with linguistic information.
MCDM is one of the most important branches in decision analysis theory and many fruitful results and models have been achieved related to this area. Among the widely used MCDM methodologies, the multiple multi-objective optimization by ratio analysis (MULTIMOORA) method and its extensions have been investigated by many scholars [29], [30], [31], [32], [33], [34], [35], [36], [37], [38], [39], [40]. As an effective and comprehensive method, it combines three aspects including the ratio system, the reference point, and the full multiplicative form. The MULTIMOORA method and its extended forms have been applied to many fields such as transition economies [29], human resource management and performance management [30], EU Member States updating management [31], heating losses ranking in a building [32], supplier selection [34] and so on.
In this paper, we mainly develop a double hierarchy hesitant fuzzy linguistic MULTIMOORA (DHHFL-MULTIMOORA) method to deal with practical MCDM problems. We apply the DHHFL- MULTIMOORA method to a case of selecting the best city in China by evaluating the implementation status of haze controlling measures. Some comparisons between the DHHFL-MULTIMOORA method and the hesitant fuzzy linguistic TOPSIS method are provided to show the advantages of the proposed method.
The highlights of this paper are summarized as follows:
- (1)
We define the DHLTS and the DHHFLTS, both of them can be used to describe the linguistic information more accurately.
- (2)
The DHHFL-MULTIMOORA method with double hierarchy hesitant fuzzy linguistic information, developed in this paper, can comprehensively consider three aspects’ information, which ensures the decision making result much more convincing.
- (3)
This paper mainly solves a practical MCDM problem, which is to select the optimal city in China by evaluating the implementation status of haze controlling measures.
The rest of this paper are organized as follows: We review some concepts and operational laws of HFLTSs in Section 2. In Section 3, we propose the concepts of DHLTS and DHHFLTS, the basic components of which can be denoted as double hierarchy linguistic terms (DHLTs) and double hierarchy hesitant fuzzy linguistic elements (DHHFLEs), respectively. Then two equivalent transformation functions between the DHLTs (DHHFLEs) and the evaluations in [0,1] (HFE) are established. Furthermore, some basic operational laws and properties of DHHFLEs are developed in this section. In Section 4, we first propose a MCDM model with double hierarchy hesitant fuzzy linguistic information, and then develop a novel DHHFL-MULTIMOORA method. In Section 5, we apply the DHHFL-MULTIMOORA method to deal with a practical case about selecting the best city in China by evaluating the implementation status of haze controlling measures. Moreover, we make some comparisons between the DHHFL-MULTIMOORA method and the hesitant fuzzy linguistic TOPSIS method. Finally, we finish this paper with some concluding remarks and future research directions in Section 6.
Section snippets
Hesitant fuzzy linguistic terms set: concept and operational laws
In 2010, Torra [1] proposed the concept of HFS on X as a function that when applied to X returns a subset of [0,1]. To be easily understood, Xia and Xu [34] expressed the HFS by a mathematical symbol where hA(x) is a set of some values in [0,1], denoting the possible membership degrees of the element x ∈ X to the set A. Additionally, can be called a hesitant fuzzy element (HFE) and Θ being the set of all HFEs.
In 2012, Rodríguez et al. [6] defined the concept of HFLTS
Double hierarchy linguistic term set and double hierarchy hesitant fuzzy linguistic term set
In this section, we mainly define the concept of DHLTS, and then apply it to express hesitant fuzzy information and develop the DHHFLTS. Additionally, some operational laws and properties are proposed.
Double hierarchy hesitant fuzzy linguistic MULTIMOORA method
This section main introduces the double hierarchy hesitant fuzzy linguistic MULTIMOORA method. We first give a brief description about the MCDM problems with the complicated linguistic information. Then, we summarize the general process of the traditional MULTIMOORA method. After that, the procedure of double hierarchy hesitant fuzzy linguistic MULTIMOORA method is developed and the algorithm is given for the convenience of application.
Case Study: the evaluation of air pollution control measures for treating haze
In this section, we mainly apply the DHHFL-MULTIMOORA method to deal with a practical MCDM problem concerning the evaluation over the air pollution control measures for treating haze. Moreover, some comparisons with the hesitant fuzzy linguistic TOPSIS method are provided to show the advantages of the proposed method.
Conclusions and future research directions
In this paper, we have introduced the concept of DHLTS, and then we have utilized it to develop the DHHFLTS which is more accurate and comprehensive than the HFLTS in information representation. Some operational laws and properties of the DHHFLEs have been developed based on the equivalent transformation functions. We have drawn some figures to facilitate the understandings of the DHLTS and the DHHFLTS. Furthermore, we have proposed a DHHFL-MULTIMOORA method for dealing with the MCDM problems
Acknowledgments
The authors would like to thank the editors and the anonymous referees for their insightful and constructive comments and suggestions that have led to this improved version of the paper. The work was supported in part by the National Natural Science Foundation of China (Nos. 71571123, 71501135, 61273209 and 71532007), the China Postdoctoral Science Foundation (No. 2016T90863), and the Central University Basic Scientific Research Business Expenses Project (Nos. skgt201501 and skqy201649).
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