Generalized Linguistic Hesitant Intuitionistic Fuzzy Hybrid Aggregation Operators

Some hybrid aggregation operators have been developed based on linguistic hesitant intuitionistic fuzzy information. The generalized linguistic hesitant intuitionistic fuzzy hybrid weighted averaging (GLHIFHWA) operator and the generalized linguistic hesitant intuitionistic fuzzy hybrid geometric mean (GLHIFHGM) operator are defined. Some special cases of the new aggregation operators are studied and many existing aggregation operators are special cases of the new operators. A new multiple attribute decision making method based on the new aggregation operators is proposed and a practical numerical example is presented to illustrate the feasibility and practical advantages of the new method.


Introduction
Fuzziness and uncertainty exist extensively in decision making process.Many useful tools have been developed to model fuzzy and uncertain information such as fuzzy set [1], intuitionistic fuzzy set [2], interval-valued intuitionistic fuzzy set [3], linguistic arguments [4,5], and hesitant fuzzy set [6,7].Hesitant fuzzy set was developed by Torra and Narukawa, which is the generalization of fuzzy set, and permits the membership having a set of possible values.The envelope of the hesitant fuzzy set is the intuitionistic fuzzy set.Hesitant fuzzy set has been studied and applied extensively since it can model fuzzy and uncertain information more accurately and flexibly.Some aggregation operators have been developed.Xia and Xu [8] discuss the relationship between the hesitant fuzzy set and intuitionistic fuzzy set.Zhu et al. [9] presented the hesitant fuzzy geometric Bonferroni mean and the hesitant fuzzy Choquet geometric Bonferroni mean.Wei [10] defined some prioritized aggregation operators for aggregating hesitant fuzzy information.Zhang et al. [11] presented induced generalized hesitant fuzzy ordered weighted averaging (IGHFOWA) operator and induced generalized hesitant fuzzy ordered weighted geometric (IGHFOWG) operator.A family of hesitant fuzzy Hamacher operators is proposed for aggregating hesitant fuzzy information based on the Hamacher t-norm and t-conorm [12].Zhang [13] defined some hesitant fuzzy power aggregation operators.Some distance measures, similarity measures, and correlation measures have been defined.Xu and Xia [14] defined the distance and correlation measures for hesitant fuzzy information.The relationship between the entropy and the similarity measure and the distance measure for hesitant fuzzy sets (HFSs) and interval-valued hesitant fuzzy sets (IVHFSs) has been studied by Farhadinia [15].Ye [16] proposes a correlation coefficient between dual hesitant fuzzy set as a new extension of existing correlation coefficients for hesitant fuzzy sets and intuitionistic fuzzy sets.Some multiple attribute decision making methods have been generalized to accommodate hesitant fuzzy information.Chen and Xu [17] developed a new approach, named HF-ELECTRE II approach, that combines the idea of HFSs with the ELECTRE II method.Zhang and Wei [18] extended the VIKOR to hesitant fuzzy set setting.Hesitant fuzzy set has been generalized to accommodate interval values [19], triangular fuzzy values [20], and linguistic arguments [21][22][23].
Though many useful methods have been developed for hesitant fuzzy information, there are still many decision making problems cannot be solved properly by using the existing methods since decision problems have become more and more complex with the developing of science and technology.Linguistic models have been developed and used extensively in decision problems since fuzzy nature of human thinking can be reflected.In evaluating process, decision makers express some hesitation, which can be modeled effectively by intuitionistic fuzzy values.For example, in evaluating performance of a candidate, one expert thinks he belongs to the degree of "good ( 7 )" which is (0.7, 0.2) and the degree of "very good ( 8 )" is (0.6, 0.3).The other expert think he belongs to the degree of "slight good ( 6 )" which is (0.8, 0.1) and the degree "good ( 7 )" is (0.6, 0.2).They cannot persuade each other and the evaluation values can be represented as ȟ = {( 6 , (0.8, 0.1)), ( 7 , (0.7, 0.2), (0.6, 0.2)), ( 8 , (0.6, 0.3))}.
The information is defined as linguistic hesitant intuitionistic fuzzy information.In this paper, we develop the linguistic hesitant intuitionistic fuzzy set based on the linguistic term set and intuitionistic fuzzy set.We define the generalized linguistic hesitant intuitionistic fuzzy hybrid weighted averaging (GLHIFHWA) operator.A wide range of aggregation operators are special cases of the GLHIFHWA operator, including generalized linguistic hesitant intuitionistic fuzzy ordered weighted averaging (GLHIFOWA) operator, the generalized linguistic hesitant intuitionistic fuzzy weighted averaging (GLHIFWA) operator, the linguistic hesitant intuitionistic fuzzy hybrid weighted averaging (LHIFHWA) operator, and the linguistic hesitant intuitionistic fuzzy hybrid weighted quadratic averaging (LHIFHWQA) operator.We define the generalized linguistic hesitant intuitionistic fuzzy hybrid geometric mean (GLHIFHGM) operator.Many exiting aggregation operators are special cases of the GLHIFHGM operator including the generalized linguistic hesitant hybrid geometric mean (GLHHGM) operator, the linguistic hesitant intuitionistic fuzzy hybrid geometric mean (LHIFHGM) operator, the linguistic hesitant intuitionistic fuzzy hybrid quadratic geometric mean (LHIFHQGM) operator, and the generalized linguistic hesitant intuitionistic fuzzy ordered geometric mean (GLHIFOGM) operator.Based on the two new aggregation operators, we develop a new multiple attribute decision making method.Finally, we apply the new method to a numerical example to illustrate its feasibility and practical advantages.This paper is organized as follows.In Section 2, we first briefly introduce the concepts of hesitant fuzzy sets.Then, we present linguistic hesitant intuitionistic fuzzy set and some operational laws of the linguistic hesitant intuitionistic fuzzy elements.In Section 3, we present the GLHIFHWA operator and the GLHIFHGM operator.Some special cases of the new aggregation operators are studied.In Section 4, a new multiple attribute decision making method based on the new aggregation operators is developed.In Section 5, a numerical example is presented to illustrate the new method.The conclusions are presented in the last section.

Linguistic Hesitant Intuitionistic Fuzzy Term Set
Definition 1 (see [7]).Let  be a reference set.An HFS  on  is a function ℎ that returns a subset of values in [0, 1] when it is applied to : where ℎ  () is a set of some different values in [0, 1], representing the possible membership degrees of the element  ∈  to . ℎ  () is called a hesitant fuzzy element (HFE).
Linguistic arguments are used extensively since uncertainty of information and fuzzy nature of human thinking can be effectively modeled.Suppose that  = {  |  = 1, . . ., } is a finite and totally ordered discrete term set, where   represents a possible value for a linguistic variable.A set of nine terms  [24] can be expressed as follows: (2) In order to preserve all the information, the discrete linguistic term set  can be extended to a continuous one  = {  |  0 ≤   ≤   ,  ∈ [0, ]}.If   ∈ , it is called the original linguistic term and it is called virtual linguistic term otherwise.
The distance measure between   and   can be defined as where   ,   ∈  are two linguistic terms.As generalization of hesitant fuzzy sets (HFSs), Zhang and Wu [25] develop hesitant fuzzy linguistic term sets (HFLSs), in which a linguistic variable has several linguistic terms.Definition 2. Let  be a reference set and let  = {  |  0 ≤   ≤   } be a linguistic term set.A hesitant fuzzy linguistic term set (HFLS)  on  is an ordered finite subset of consecutive linguistic terms of where ℎ  (  ):  →  denotes all the possible linguistic evaluation values of element   ∈ .For convenience, we call ℎ  (  ) a hesitant fuzzy linguistic element (HFLE), which can be represented as and here   is a linguistic argument.
In decision making process, the expert may think some linguistic term can be used when evaluating, but he/she may have hesitation.Intuitionistic fuzzy value can be used to model hesitation existing in decision making process.The hesitant intuitionistic fuzzy linguistic term set can be defined by using the linguistic arguments and intuitionistic fuzzy values, which can be defined as follows.

Mathematical Problems in Engineering 3
A linguistic hesitant intuitionistic fuzzy term set (LHIFS) Ǎ on  is defined as where ȟ Ǎ (  ):  →  denotes all possible intuitionistic fuzzy linguistic evaluation values of element   ∈ .For convenience, we call ȟ Ǎ (  ) a linguistic hesitant intuitionistic fuzzy element (LHIFE), which can be represented as where  () is a linguistic argument and ℎ( () ) is the set of intuitionistic fuzzy membership values  () satisfying   .

Aggregation Operators for Linguistic Hesitant Intuitionistic Fuzzy Information
In this section, we define some hybrid aggregation operators based on the linguistic hesitant intuitionistic fuzzy information.

Numerical Examples
A numerical example is presented to illustrate the efficiency and practical advantages of the proposed procedure.
Consider the decision making problem adapted from [26].An investor wants to invest some money in an enterprise to get the highest possible profits.He considers five possible alternatives: ( 1 ) computer company, ( 2 ) real estate company, ( 3 ) food company, ( 4 ) car company, and ( 5 ) communication company.Multiple experts are invited to evaluate alternatives with respect to the following possible situations for the economic environment: ( 1 ) negative growth rate, ( 2 ) low growth rate, ( 3 ) medium growth rate, and ( 4 ) high growth rate.The new method is used to rank alternatives.
Step 1.The experts evaluate the alternatives with respect to the attributes using the linguistic arguments and intuitionistic fuzzy memberships and the results are shown in Table 1.
Step 3. The GLHIFHSWA operator is used to aggregate the alternatives' evaluation values into collective ones, ȟ  ( = 1, 2, . . ., 5).For example, in order to calculate ȟ 1 for For other , we can get the results similarly and the results are omitted here for space limit.
From the above results, we can see that the ranking of alternatives may be different due to different aggregation operators. 4 is the optimal alternative if  = 0.1 or  = 0.2, since the smallest evaluation values are relatively large in  4 .The best alternative is  2 if the GLHIFHSWA 2 operator and the GLHIFHSWA 5 operator are used, since these operators focus on the values with the largest and second largest   . 5 become the best alternative if the following linguistic hesitant intuitionistic fuzzy aggregation operators are used: the GLHIFHSWA 0.5 operator, the LHIFHSWA operator, the GLHIFHSWA 10 operator, and the GLHIFHSWA 20 operator.The GLHIFHSWA 0.5 operator and the LHIFHSWA operator focus on the average values of the evaluation values.The GLHIFHSWA 10 operator and the GLHIFHSWA 20 operator focus on the value with the largest weight   .With the increasing , the largest weights   play more and more important role in aggregation. can be seen as the risk attitude of the decision maker.In real decision making problem, decision makers can select the corresponding operator according to real need and his/her interests.

Conclusions
In this paper, we develop new aggregation operators based on the new fuzzy sets named LHIFSs.Linguistic arguments and intuitionistic fuzzy memberships have been used for evaluation and linguistic hesitant fuzzy information has been gotten, which can reflect the hesitancy, uncertainty, and fuzziness existing in the decision making process.We develop the generalized linguistic hesitant intuitionistic fuzzy hybrid weighted averaging (GLHIFHWA) operator and the generalized linguistic hesitant intuitionistic fuzzy hybrid geometric mean (GLHIFHGM) operator.Some special cases of the new aggregation operators have been studied and many existing aggregation operators are the special cases of the new operators.A new multiple attribute decision making method based on the new operators has been developed and financial product selection problem is presented to illustrate the feasibility and practical advantages of the new methods.

Table 2 :
Ranking results based on the GLHIFHSWA operator with different .