Distance and similarity measures for hesitant fuzzy linguistic term sets and their application in multi-criteria decision making Huchang Liao a,c , Zeshui Xu b,⇑ , Xiao-Jun Zeng c a Antai College of Economics and Management, Shanghai Jiao Tong University, Shanghai 200052, China b Business School, Sichuan University, Chengdu, Sichuan 610065, China c School of Computer Science, University of Manchester, Manchester M13 9PL, United Kingdom article info Article history: Received 4 December 2012 Received in revised form 20 February 2014 Accepted 21 February 2014 Available online 1 March 2014 Keywords: Hesitant fuzzy linguistic term set Distance measure Similarity measure Multi-criteria decision making abstract The hesitant fuzzy linguistic term sets (HFLTSs), which can be used to represent an expert’s hesitant preferences when assessing a linguistic variable, increase the flexibility of eliciting and representing linguistic information. The HFLTSs have attracted a lot of attention recently due to their distinguished power and efficiency in representing uncertainty and vagueness within the process of decision making. To enhance and extend the applicability of HFLTSs, this paper investigates and develops different types of distance and similarity measures for HFLTSs. The paper first proposes a family of distance and similarity measures between two HFLTSs. Then a variety of weighted or ordered weighted distance and similar- ity measures between two collections of HFLTSs are proposed and analyzed for discrete and continuous cases respectively. After that, the application of these measures to multi-crite- ria decision making problems is given. Based on the proposed distance and similarity mea- sures, the satisfaction degrees for different alternatives are established and are then used to rank alternatives in multi-criteria decision making. Finally a practical example concerning the evaluation of the quality of movies is given to illustrate the applicability and advantage of the proposed approach and the differences between the proposed distance and similar- ity measures. Ó 2014 Elsevier Inc. All rights reserved. 1. Introduction Hesitant fuzzy sets (HFSs), which were first introduced by Torra [30] as an extended form of fuzzy sets, have attracted a lot of attention recently due to their effectiveness and efficiency in representing uncertainty and vagueness [13–16,18,30,46,52]. The motivation for introducing HFSs was that it is sometimes difficult to determine the membership degree of an element to a set, and in some circumstances this difficulty is caused by a doubt between a few different values [30]. Since the HFS permits the membership degree of an element to a given set represented by several possible values between 0 and 1, it can express a decision maker’s hesitancy efficiently, especially when two or more sources of vagueness appear simultaneously. It should be noted that the HFS was introduced to handle the problems that are represented in quantitative situations. In many cases, however, uncertainty is produced by the vagueness of meanings whose nature is qualitative rather than quan- titative [4,8,9,24]. For example, when evaluating the ‘‘speed’’ of a car, the linguistic terms such as ‘‘fast’’, ‘‘very fast’’, ‘‘slow’’ may be used; when evaluating the ‘‘performance’’ of a company, the terms such as ‘‘good’’, ‘‘medium’’, and ‘‘bad’’ can be used. http://dx.doi.org/10.1016/j.ins.2014.02.125 0020-0255/Ó 2014 Elsevier Inc. All rights reserved. ⇑ Corresponding author. Tel.: +86 25 84483382. E-mail addresses: liaohuchang@163.com (H. Liao), xuzeshui@263.net (Z. Xu), x.zeng@manchester.ac.uk (X.-J. Zeng). Information Sciences 271 (2014) 125–142 Contents lists available at ScienceDirect Information Sciences journal homepage: www.elsevier.com/locate/ins