Extension of the TOPSIS Method for Decision-making Problem Under Dual Hesitant Fuzzy Language Environment

: In order to comprehensively and actually describe the evaluation process, the dual hesitant fuzzy linguistic (DHFL) set is introduced in this paper, which includes more decision-making information, such as fuzzy state, hesitant process and language information. Specifically, some basic concepts of DHFL set are illustrated and a new distance measure for DHFL information is proposed, which is suitable for overcoming the irrational traditional methodology upon the general distance measure and basic probability concepts. Then, technique for order preference by similarity to ideal solution (TOPSIS) method is extended in dual hesitant fuzzy language environment, a novel TOPSIS method using the DHFL set is presented. Finally, the sensitivity analysis is performed to verify the feasibility and stability of the developed method, then the advantages of the proposed method are also confirmed by detailed comparative analysis.

the comprehensiveness and rationality of group MCDM process, Hatami and Kangi (2017) introduced three types of fuzzy TOPSIS methods for handling the imprecise information. On the basis of hesitant fuzzy correlation coefficient, Sun et al., (2018) presented a new TOPSIS method to deal with the negative-value information. Sajjad et al., (2018) extended the TOPSIS approach through the combination of Choquet integral-based distance and IVPFCIG operator for MAGDM problems, while Hajek and Froelich (2019) presented an IVIFCM-TOPSIS method, which can model the interactions among imprecise criteria for MCGDM problems. Furthermore, some novel TOPSIS methods are extended to hesitant Pythagorean fuzzy set (Liang and Xu 2017), spherical fuzzy sets (Kutlu Gündoğdu and Kahraman 2019), interval-valued hesitant fuzzy N-soft set (Akram and Adeel 2019), interval-valued spherical fuzzy sets , ordered fuzzy numbers (Kacprzak 2019), and etc.
Recently, decision making under linguistic environment has become a hotspot, and there are a series of papers focus on this research (Wei et al. 2016;Yang and Ju 2014;Zhang et al. 2019a;Zhang et al. 2019b;Zhang et al. 2018). But to make a right decision, there are many other factors that need to be considered. Obviously, because of the complex socioeconomic environment and vague human thinking, hesitant and uncertain information is usually appeared when making decisions, so that the DMs may could not precisely express their decisions by using linguistic expressions, and the criteria weights and the preference values of DMs are frequently ambiguous, which cannot be represented by crisp numerical value of the classical TOPSIS methods. With the purpose of expressing the preferences of the DMs more comprehensively, the dual hesitant fuzzy linguistic set (DHFLS) is introduced to evaluate linguistic terms (Yang and Ju 2015). Compared with other sets, the DHFLS has both consider the influence of the membership hesitancy degree as well as the non-membership hesitancy degree, which is an efficient tool to comprehensively describe various types of uncertainty. Therefore, DHFLS could better indicate the evaluation information in MCDM problems. Aiming at eliminate the uncertainty caused by ambiguous information and describe the preferences of DMs better, a MCDM method with mutually supportive arguments under DHFL environments is constructed in this paper.
The remainder of this article is arranged as follows: In Sect. 2, some basic definitions of DHFLS, distance measures and dual hesitant language fuzzy MCDM problem are briefly introduced. In Sect. 3, the procedure to solve the MCDM problem in DHFL environment using the proposed method is described in details. In Sect. 4, the sensitivity analysis, comparative study and illustrative example are used to prove the rationality of the presented method. Finally, concluding remarks and suggestions of further research are made in Sect. 5.

Concept and definition description
Some basic definitions and notations concerning DHFLS, distance measures and dual hesitant language fuzzy MCDM problem are introduced in this section.

Definition 1 (Yang and Ju 2014)
. Suppose X is a reference set, then a DHFLS on X can be expressed as follows: ,, is a DHFLE, the score function and accuracy function of  are given as: Zadeh 1975).

Definition 3 (Yang and Ju 2014). Let
DHFLEs, the comparison order of DHFLEs has the following situations:

Distance measures
As we know, the main idea of the existing methods (Chen et al. 2011;Torra 2010;Xia and Xu 2011) is to compare the number of elements in the hesitant fuzzy sets firstly. If the result of the comparison is not equal, then the maximum/minimum degree of the membership/non-membership is added to the set with fewer number of elements several times until the number in the two sets is same. However, there are two problems with this approach: (1) Adding the maximum/minimum evaluation value, which greatly highlights the subjectivity of the DMs; (2) Judgement and determination of the DMs' risk attitude is a hard work. For the purpose of computing the distance between two DHFL variables, a new distance measure for DHFL information is introduced and is suitable for overcoming the irrational traditional methodology upon the well-known distance measure and basic probability concepts, which are computed directly from DHFL variables and any maximum/minimum value does not need to be added to the evaluation set. , then the normalized Euclidean distance for two DHFLEs are described by: Where # k h , # l h , # k g and # l g are the numbers of values in k h , l h , k g and l g respectively, such that , , ,

Dual hesitant language fuzzy MCDM problem
To describe the MCDM process under DHFL information in detail, suppose  

Extension of the TOPSIS under DHFL environment
The TOPSIS method will be extended to the DHFL environment for handling the hesitant and uncertain information in this section. Subsequently, according to the comprehensive indexes, all alternatives are ranked, the specific steps are as follows: Step 1. Determine the PIS and NIS based on DHFL information.
Step 2. Use equations (6) and (7) to calculate the weighted Hamming distance of each alternative to PIS and NIS, respectively.
Step 3. According to the ideal solution determined in step 1, the relative closeness of each alternative is calculated. Take alternative i A as an example, the formula is as follows: Step 4. Rank all alternatives i A and choose the optimal one(s) according to   ii A  .
Step 5. Give a sensitivity analysis of the solution.

Illustration of the presented approach
As we all know, maintenance services, as a backup for repairable products, can effectively improve the customer satisfaction of particular manufacturing companies (Davies and Isaac 1999). In fact, maintenance services have been regarded as an important part of the product. In order to successfully achieve the operational goals, choosing a service agent with good performance is an extremely significant decision making problem , the result is shown in Table 1.
In what follows, the proposed method is utilized to choose the most desirable one.
Step 1. For each criterion, the PIS and NIS are determined, which is presented in Table 2. Step 2. Using the formulas (6) and (7) to calculate the separation measures, which are distances of each individual decision from PIS and NIS, respectively. A and 5 A concerning w Figure 1. Sensitivity analysis results by TOPSIS method

Conclusion
Considering that DHFLS is more suitable than other fuzzy sets for dealing with fuzzy and hesitant information in complex environments, to solve the MCDM problem with DHFL information, an DHFLS based innovative TOPSIS method was proposed. The new method is straight forward and easy to implement on computer, where the form of attribute data is the DHFL numbers. The basic concepts of DHFL set and extended definition of hamming distance are introduced. In order to prove the efficiency of the presented approach, an example is used in this paper. Additionally, a sensitivity analysis of the final ranking results to the weights on the TOPSIS method is studied to test the stability of proposed method. In future, MCDM problems involving other types of fuzzy hesitant information can be further studied, and to provide more effective ways for DMs, the decision-making methods such as TODIM and VIKOR can also be applied in the DHFL environment.