Soft Computing

, Volume 23, Issue 2, pp 393–405 | Cite as

Hesitant fuzzy Lukasiewicz implication operation and its application to alternatives’ sorting and clustering analysis

  • Miaomiao Wen
  • Hua Zhao
  • Zeshui XuEmail author


Hesitant fuzzy set (HFS) takes several possible values as the membership degree of an element to a set to express the decision makers’ hesitance when making decisions. Since its appearance, the HFS has been widely used in many fields, such as decision making, clustering analysis. Lukasiewicz implication operator, an indispensable part of implication operators, can grasp more nuances compared with the others. In this paper, we shall combine the Lukasiewicz implication operator with HFSs to realize a direct clustering analysis algorithm and a novel alternative sorting method in decision making under hesitant fuzzy environment. To do that, we first apply the Lukasiewicz implication operator to deal with HFEs by getting a hesitant fuzzy Lukasiewicz implication operator, and then construct a hesitant fuzzy triangle product and a hesitant fuzzy square product based on the new implication operator. After that, the hesitant fuzzy square product is applied to define the similarity degree between HFSs, and based on which, we develop a direct clustering algorithm for hesitant fuzzy information. Meanwhile, the hesitant fuzzy triangle product is used to induce a new alternative sorting method. Finally, two numerical examples are given to illustrate the effectiveness and practicability of our method and algorithm, one of which involves the evaluation analysis of the Arctic development risk.


Hesitant fuzzy set Lukasiewicz implication operator Clustering analysis Hesitant fuzzy square product Alternative sorting 


Compliance with ethical standards

Conflict of interest

We declare that we do not have any commercial or associative interest that represents a conflict of interest in connection with the work submitted.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Basic EducationPLA Army Engineering UniversityNanjingChina

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