# Generalized Linguistic Hesitant Intuitionistic Fuzzy Hybrid Aggregation Operators

Authors: Wei Yang; Jiarong Shi; Yongfeng Pang
Journal: Mathematical Problems in Engineering
 (2015)

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## Abstract

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.


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## Body

## 1. 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 [21–23].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(s7)” which is (0.7,0.2) and the degree of “very good (s8)” is (0.6,0.3). The other expert think he belongs to the degree of “slight good (s6)” which is (0.8,0.1) and the degree “good (s7)” is (0.6,0.2). They cannot persuade each other and the evaluation values can be represented as hˇ={(s6,(0.8,0.1)),(s7,(0.7,0.2),(0.6,0.2)),(s8,(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 Section2, 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.


## 2. Linguistic Hesitant Intuitionistic Fuzzy Term Set

Definition 1 (see [7]).
LetX be a reference set. An HFS A on X is a function h that returns a subset of values in [0,1] when it is applied to X:(1)A=x,hAx∣x∈X,where hA(x) is a set of some different values in [0,1], representing the possible membership degrees of the element x∈X to A. hA(x) 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 thatS={si∣i=1,…,g} is a finite and totally ordered discrete term set, where si represents a possible value for a linguistic variable. A set of nine terms S [24] can be expressed as follows:(2)S=s1=extremely poor,s2=very poor,s3=poor,s4=slightly poor,s5=fair,s6=slightly good,s7=good,s8=very good,s9=extremely good.In order to preserve all the information, the discrete linguistic term setS can be extended to a continuous one S¯={sα∣s0≤sα≤sg, α∈[0,g]}. If sα∈S, it is called the original linguistic term and it is called virtual linguistic term otherwise.The distance measure betweensα and sβ can be defined as (3)dsα,sβ=1gα-β,where sα,sβ∈S¯ 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.
LetX be a reference set and let S¯={sα∣s0≤sα≤sg} be a linguistic term set. A hesitant fuzzy linguistic term set (HFLS) A¯ on X is an ordered finite subset of consecutive linguistic terms of S¯(4)A¯=xi,h¯A¯xi∣xi∈X,i=1,2,…,n,where h¯A¯(xi): X→S¯ denotes all the possible linguistic evaluation values of element xi∈X. For convenience, we call h¯A¯(xi) a hesitant fuzzy linguistic element (HFLE), which can be represented as(5)h¯A¯xi=si∣si∈h¯A¯xi,and here si 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.Definition 3.
LetX={x1,x2,…,xn} be a reference set and let S¯={sα∣s0≤sα≤sg} be a linguistic term set. A linguistic hesitant intuitionistic fuzzy term set (LHIFS) Aˇ on X is defined as(6)Aˇ=xi,hˇAˇxi∣xi∈X,i=1,2,…,n,where hˇAˇ(xi): X→H denotes all possible intuitionistic fuzzy linguistic evaluation values of element xi∈X. For convenience, we call hˇAˇ(xi) a linguistic hesitant intuitionistic fuzzy element (LHIFE), which can be represented as(7)hˇAˇxi=sθi,lhsθi∣sθi,lhsθi∈hˇAˇxi,where sθ(i) is a linguistic argument and lh(sθ(i)) is the set of intuitionistic fuzzy membership values sθ(i) satisfying xi. (sθ(i),lh(sθ(i))) is the linguistic intuitionistic fuzzy element (LIFE). H is the set of all LHIFEs.If all intuitionistic fuzzy membership(αi,βi) reduces to (μi,1-μi) in LIFEs, then LHIFE reduces to the linguistic hesitant fuzzy element (LHFE) as hAˇ(xi)={(sθ(i),m(sθ(i)))∣(sθ(i),m(sθ(i)))∈hAˇ(xi)}, where m(sθ(i)) is the set of fuzzy memberships with sθ(i) satisfying the attribute xi. If all intuitionistic fuzzy membership (αi,βi) reduces to (1,0), then the LHIFE reduces to the hesitant fuzzy linguistic element (HFLE).Definition 4.
Lethˇ1, hˇ2, and hˇ be LHIFEs, λ>0. Consider that ak=(sθ(k),lh(sθ(k)))∈hˇ, ai=(sθ(i),lh(sθ(i)))∈hˇ1, and aj=(sθ(j),lh(sθ(j)))∈hˇ2. Some operation laws can be defined as follows:(1)
hˇ1⊕hˇ2=⋃ai∈hˇ1,aj∈hˇ2{(sθ(i)+θ(j), ⋃(μi,νi)∈lh(sθ(i)),(μj,νj)∈lh(sθ(j)){(μi+μj-μiμj,νiνj)})},(2)
hˇ1⊗hˇ2=⋃ai∈hˇ1,aj∈hˇ2{(sθ(i)θ(j), ⋃(μi,νi)∈lh(sθ(i)),(μj,νj)∈lh(sθ(j)){(μiμj,νi+νj-νiνj)})},(3)
λhˇ=⋃ak∈hˇ{(sλθ(k),⋃(μk,νk)∈lh(sθ(k)){(1-(1-μk)λ,νkλ)})},(4)
(hˇ)λ=⋃ak∈hˇ{(sθ(k)λ,⋃(μk,νk)∈lh(sθ(k)){(μkλ,1-(1-νk)λ)})}.Definition 5.
Letai=(sθ(i),lh(sθ(i))) be a LIFE, then the score function s(ai) of ai can be defined as(8)sai=θiglhsθi∑αi,βi∈lhsθiαi-βi,and the accuracy function h(ai) of ai can be defined as(9)hai=θiglhsθi∑αi,βi∈lhsθiαi+βi,where g is the number of linguistic arguments in linguistic term set S and lhsθi is the number of intuitionistic fuzzy memberships in lh(sθ(i)).Letai=(sθ(i),lh(sθ(i))) and aj=(sθ(j),lh(sθ(j))) be two LIFEs. Then, ai and aj can be ranked as follows:(1)
ifs(ai)<s(aj), then ai<aj,(2)
ifs(ai)=s(aj) and
(I)
ifh(ai)<h(aj), then ai<aj,(II)
ifh(ai)=h(aj), then ai=aj.Definition 6.
Lethˇ={(sθ(i),lh(sθ(i)))∣(sθ(i),lh(sθ(i)))∈hˇ} be LHIFE; the score function S(hˇ) can be defined as(10)Shˇ=1lhˇ∑i=1lhˇθiglhsθi∑αi,βi∈lhsθiαi-βi.The accuracy function A(hˇ) can be defined as(11)Ahˇ=1lhˇ∑i=1lhˇθiglhsθi∑αi,βi∈lhsθiαi+βi.Here, lhˇ is the number of LIFEs in hˇ and g is the number of linguistic arguments in linguistic term set S, and lhsθi is the count of intuitionistic fuzzy memberships in lh(sθ(i)).Based on the score function and accuracy function, we present the following method to compare LHIFEs. Lethˇ1 and hˇ2 be two LHIFEs as follows:(1)
ifS(hˇ1)<S(hˇ2), then hˇ1<hˇ2;(2)
ifS(hˇ1)=S(hˇ2) and
(I)
A(hˇ1)<A(hˇ2), then hˇ1<hˇ2,(II)
A(hˇ1)=A(hˇ2), then hˇ1=hˇ2.


## 3. 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.Definition 7.
LetX={x1,x2,…,xn} be a reference set and let hˇj(j=1,2,…,n) be a collection of LHIFEs on X. A generalized linguistic hesitant intuitionistic fuzzy hybrid weighted averaging (GLHIFHWA) operator is a mapping GLHIFHWA: Hn→H, which can be defined as follows:(12)GLHIFHWAhˇ1,hˇ2,…,hˇn=⨁j=1n⁡wjωhˇσjλhˇσjλ∑j=1nwjωhˇσjλ1/λ,where λ∈(-∞,+∞) and w=(w1,w2,…,wn) is the weight vector on the ordered set N={1,2,…,n} with wi≥0 and ∑j=1nwj=1. ω=(ωhˇ1,ωhˇ2,…,ωhˇn) is the weight vector of LHIFEs hˇj(j=1,2,…,n) such that ωhˇj≥0 and ∑j=1nωhˇj=1. (σ(1),σ(2),…,σ(n)) is a permutation of (1,2,…,n) such that ωhˇσ(j)hˇσ(j) is the jth largest ωhˇjhˇj.Definition 8.
LetX={x1,x2,…,xn} be a reference set and let hˇj(j=1,2,…,n) be a collection of LHIFEs on X. A generalized linguistic hesitant intuitionistic fuzzy hybrid geometric mean (GLHIFHGM) operator is a mapping GLHIFHGM: Hn→H, which can be defined as follows:(13)GLHIFHGMhˇ1,hˇ2,…,hˇn=1λ⨂j=1n⁡λhˇσjwjωhˇσjλ/∑j=1nwjωhˇσjλ,where λ∈(-∞,+∞) and w=(w1,w2,…,wn) is the weight vector on the ordered set N={1,2,…,n} with wi≥0 and ∑j=1nwj=1. ω=(ωhˇ1,ωhˇ2,…,ωhˇn) is the weight vector of LHIFEs hˇj(j=1,2,…,n) such that ωhˇj≥0 and ∑j=1nωhˇj=1. (σ(1),σ(2),…,σ(n)) is a permutation of (1,2,…,n) such that hˇσ(j)ωσ(j) is the jth largest hˇjωj.Lethˇj(j=1,2,…,n) be a collection of LHIFEs and ai=(sθ(i),lh(sθ(i)))∈hˇi. By using the operation laws of LHIFEs, we can get the values of the GLHIFHWA operator and the GLHIFHGM operator, which are all LHIFEs and(14)GLHIFHWAhˇ1,hˇ2,…,hˇn=⋃aσi∈hˇσis∑j=1nwjωhˇσjθσjλ/∑j=1nwjωhˇσjλ1/λ,⋃μσi,νσi∈lhsθσi1-∏j=1n1-μσjλwjωhˇσjλ/∑j=1nwjωhˇσjλ1/λ,1-1-∏j=1n1-1-νσjλwjωhˇσjλ/∑j=1nwjωhˇσjλ1/λ,GLHIFHGMhˇ1,hˇ2,…,hˇn=⋃aσi∈hˇσis∏j=1nθσjwjωhˇσj/∑j=1nwjωhˇσj,⋃μσi,νσi∈lhsθσi1-1-∏j=1n1-1-μσjλwjωhˇσj/∑j=1nwjωhˇσj1/λ,1-∏j=1n1-νσjλwjωhˇσj/∑j=1nwjωhˇσj1/λ.Now, we consider some special cases of the GLHIFHWA operator and the GLHIFHGM operator by considering some specialλ and the associated weight vectors w=(w1,w2,…,wn) and ω=(ω1,ω2,…,ωn).Remark 9.
If all(μi,νi)∈lh(sθ(i)) reduces to (μi,1-μi), then LHIFEs hˇj(j=1,2,…,n) become linguistic hesitant fuzzy elements (LHFEs) hj′(j=1,2,…,n) and GLHIFHWA operator becomes the following generalized linguistic hesitant fuzzy hybrid weighted averaging (GLHFHWA) operator:(15)GLHFHWAh1′,h2′,…,hn′=⨁j=1n⁡wjωhσj′hσj′λ∑j=1nwjωhσj′λ1/λ=⋃aσi′∈hσi′s∑j=1nwjωhσj′θσjλ/∑j=1nwjωhσj′λ1/λ,⋃μσi∈lhsθσi1-∏j=1n1-μσjλwjωhσj′λ/∑j=1nwjωhσj′λ1/λ.The GLHIFHGM operator becomes the generalized linguistic hesitant fuzzy hybrid geometric mean (GLHFHGM) operator as follows:(16)GLHFHGMh1′,h2′,…,hn′=1λ⨂nj=1⁡λhσj′wjωhσj′/∑j=1nwjωhσj′=⋃aσi′∈hσi′s∏j=1nθσjwjωhσj′/∑j=1nwjωhσj′,⋃μσi∈lhsθσi1-1-∏j=1n1-1-μσjwjωhσj′/∑j=1nwjωhσj′1/λ.
If all(μi,νi)∈lh(sθ(i)) reduces to (1,0), then LHIFEs hˇi(i=1,2,…,n) reduce to linguistic hesitant elements hi=sθi(i=1,2,…,n). The GLHIFHWA operator becomes the generalized linguistic hesitant hybrid weighted averaging (GLHHWA) operator as follows:(17)GLHHWAh1,h2,…,hn=⨁j=1n⁡wjωhσjhσjλ∑j=1nwjωhσjλ1/λ=⋃sθσi∈hσis∑j=1nwjωhσjθσjλ/∑j=1nwjωhσjλ1/λ.The GLHIFHGM operator becomes the generalized linguistic hesitant hybrid geometric mean (GLHHGM) operator as follows:(18)GLHHGMh1,h2,…,hn=1λ⨂nj=1⁡λhσjwjωhσj/∑j=1nwjωhσj=⋃sθσi∈hσis∏j=1nθσjwjωhσj/∑j=1nwjωhσj.Remark 10.
Ifλ=1, then the GLHIFHWA operator becomes the linguistic hesitant intuitionistic fuzzy hybrid weighted averaging (LHIFHWA) operator(19)LHIFHWAhˇ1,hˇ2,…,hˇn=⨁j=1n⁡wjωhˇσjhˇσj∑j=1nwjωhˇσj=⋃aσi∈hˇσis∑j=1nwjωhˇσjθσj/∑j=1nwjωhˇσj,⋃μσi,νσi∈lhsθσi1-∏j=1n1-μσjwjωhˇσj/∑j=1nwjωhˇσj,∏j=1nνσjwjωhˇσj/∑j=1nwjωhˇσj,and the GLHIFHGM operator becomes the linguistic hesitant intuitionistic fuzzy hybrid geometric mean (LHIFHGM) operator(20)LHIFHGMhˇ1,hˇ2,…,hˇn=⨂nj=1⁡hˇσjwjωhˇσj/∑j=1nwjωhˇσj=⋃aσi∈hˇσis∏j=1nθσjwjωhˇσj/∑j=1nwjωhˇσj,⋃μσi,νσi∈lhsθσi∏j=1nνσjwjωhˇσj/∑j=1nwjωhˇσj,1-∏j=1n1-μσjwjωhˇσj/∑j=1nwjωhˇσj.Remark 11.
Ifλ=2, then the GLHIFHWA operator reduces to be the linguistic hesitant intuitionistic fuzzy hybrid weighted quadratic averaging (LHIFHWQA) operator as follows:(21)LHIFHWQAhˇ1,hˇ2,…,hˇn=⋃aσi∈hˇσis∑j=1nwjωhˇσjθσj2/∑j=1nwjωhˇσj21/2,⋃μσi,νσi∈lhsθσi1-∏j=1n1-μσj2wjωhˇσj2/∑j=1nwjωhˇσj21/2,1-1-∏j=1n1-1-νσj2wjωhˇσj2/∑j=1nwjωhˇσj21/2.The GLHIFHWA operator becomes the linguistic hesitant intuitionistic fuzzy hybrid quadratic geometric mean (LHIFHQGM) operator(22)LHIFHQGMhˇ1,hˇ2,…,hˇn=⋃aσi∈hˇσis∏j=1nθσjwjωhˇσj/∑j=1nwjωhˇσj,⋃μσi,νσi∈lhsθσi1-1-∏j=1n1-1-μσj2wjωhˇσj/∑j=1nwjωhˇσj1/2,1-∏j=1n1-νσj2wjωhˇσj/∑j=1nwjωhˇσj1/2.
If all(μi,νi)∈lh(sθ(i)) reduces to (μi,1-μi), then LHIFHWQA operator becomes the linguistic hesitant fuzzy hybrid weighted quadratic averaging (LHFHWQA) operator as follows: (23)LHFHWQAh1′,h2′,…,hn′=⋃aσi′∈hσi′s∑j=1nwjωσj2/∑j=1nwjωhˇσj2θσj21/2,⋃μσi∈lhsθσi1-∏j=1n1-μσj2wjωhˇσj2/∑j=1nwjωhˇσj21/2.The LHIFHQGM operator becomes the linguistic hesitant fuzzy hybrid quadratic geometric mean (LHIFHQGM) operator(24)LHIFHQGMh1′,h2′,…,hn′=⋃aσi∈hσi′s∏j=1nθσjwjωhσj′/∑j=1nwjωhσj′,⋃μσi,νσi∈lhsθσi1-1-∏j=1n1-1-μσj2wjωhσj′/∑j=1nwjωhσj′1/2.
If all(μi,νi)∈lh(sθ(i)) reduces to (1,0), then LHIFHWQA operator becomes the linguistic hesitant hybrid weighted quadratic averaging (LHHWQA) operator as follows: (25)LHHWQAh1,h2,…,hn=⋃sθσi∈hσis∑j=1nwjωhσjθσj2/∑j=1nwjωhσj21/2.The LHIFHQGM operator becomes the linguistic hesitant hybrid weighted geometric mean (LHHWGM) operator (26)LHHWGMh1,h2,…,hn=⋃sθσi∈hσis∏j=1nθσjwjωhσj/∑j=1nwjωhσj.Remark 12.
Ifwj=1/n(j=1,2,…,n), the GLHIFHWA operator degenerates to the following generalized linguistic hesitant intuitionistic fuzzy weighted averaging (GLHIFWA) operator:(27)GLHIFWAhˇ1,hˇ2,…,hˇn=⋃ai∈hˇis∑j=1nωhˇjλ/∑j=1nωhˇjλθjλ1/λ,⋃μi,νi∈lhsθj1-∏j=1n1-μjλωhˇjλ/∑j=1nωhˇjλ1/λ,1-1-∏j=1n1-1-νjλωhˇjλ/∑j=1nωhˇjλ1/λ.The GLHIFHGM operator becomes the generalized linguistic hesitant intuitionistic fuzzy weighted geometric mean (GLHIFWGM) operator:(28)GLHIFWGMhˇ1,hˇ2,…,hˇn=⋃ai∈hˇis∏j=1nθjωhˇj,⋃μi,νi∈lhsθi1-1-∏j=1n1-1-μjλωhˇj1/λ,1-∏j=1n1-νjλωhˇj1/λ.
Ifλ=1 and wj=1/n for all j=1,2,…,n, the GLHIFHWA operator becomes the linguistic hesitant intuitionistic fuzzy weighted averaging (LHIFWA) operator:(29)LHIFWAhˇ1,hˇ2,…,hˇn=⨁nj=1⁡ωhˇjhˇj=⋃ai∈hˇis∑j=1nωhˇjθj,⋃μi,νi∈lhsθi1-∏j=1n1-μjωhˇj,∏j=1nνjωhˇj.
The GLHIFHGM operator becomes the linguistic hesitant intuitionistic fuzzy weighted geometric mean (LHIFWGM) operator(30)LHIFWGMhˇ1,hˇ2,…,hˇn=⋃ai∈hˇis∏j=1nθjωhˇj,⋃μi,νi∈lhsθi∏j=1nμjωhˇj,1-∏j=1n1-νjωhˇj.Remark 13.
Ifωhˇj=1/n(j=1,2,…,n), the GLHIFHWA operator degenerates to the generalized linguistic hesitant intuitionistic fuzzy ordered weighted averaging (GLHIFOWA) operator:(31)GLHIFOWAhˇ1,hˇ2,…,hˇn=⨁j=1nwjhˇσjλ1/λ=⋃aσi∈hˇσis∑j=1nwjθσjλ1/λ,⋃μσi,νσi∈lhsθσi1-∏j=1n1-μσjλwj1/λ,1-1-∏j=1n1-1-νσjλwj1/λ.The GLHIFHGM operator becomes the generalized linguistic hesitant intuitionistic fuzzy ordered geometric mean (GLHIFOGM) operator:(32)GLHIFOGMhˇ1,hˇ2,…,hˇn=⋃aσi∈hˇσis∏j=1nθσjwj,⋃μσi,νσi∈lhsθσi1-1-∏j=1n1-1-μσjλwj1/λ,1-∏j=1n1-νσjλwj1/λ.
Ifλ=1 and ωhˇj=1/n(j=1,2,…,n), the GLHIFHWA operator becomes the linguistic hesitant intuitionistic fuzzy ordered weighted averaging (LHIFOWA) operator as follows: (33)LHIFOWAhˇ1,hˇ2,…,hˇn=⨁j=1nwjhˇσj=⋃aσi∈hˇσis∑j=1nwjθσj,⋃μσi,νσi∈lhsθσi1-∏j=1n1-μσjwj,∏j=1nνσjwj.The GLHIFHGM operator becomes the linguistic hesitant intuitionistic fuzzy ordered weighted geometric mean (LHIFOWGM) operator:(34)LHIFOWGMhˇ1,hˇ2,…,hˇn=⋃aσi∈hˇσis∏j=1nθσjwj,⋃μσi,νσi∈lhsθσi∏j=1nμσjwj,1-∏j=1n1-νσjwj.
Ifλ=0 and ωhˇj=1/n(j=1,2,…,n), the GLHIFHWA operator becomes the following linguistic hesitant intuitionistic fuzzy ordered weighted aggregation operator: (35)GLHIFHWAhˇ1,hˇ2,…,hˇn=⨂j=1nhˇσjwj=⋃aσi∈hˇσis∏j=1nθσjwj,⋃μσi,νσi∈lhsθσie∏j=1nln⁡μσjwj,1-e∏j=1nln⁡1-νσjwj,and the GLHIFHGM operator reduces to the following operator:(36)GLHIFHGMhˇ1,hˇ2,…,hˇn=⋃aσi∈hˇσis∏j=1nθσjwj,⋃μσi,νσi∈lhsθσi1-e∏j=1nln⁡1-μσjwj,e∏k=1nln⁡νσjwj.


## 4. New Multiple Attribute Decision Making Method Based on Linguistic Hesitant Intuitionistic Fuzzy Information

In this section, we present a new multiple attribute decision making approach based on the new aggregation operators. Let{A1,A2,…,Am} be the set of alternatives and let {C1,C2,…,Cn} be the set of attributes. The decision makers evaluate the alternatives with respect to the attributes using linguistic terms and intuitionistic fuzzy memberships. Then, the linguistic hesitant intuitionistic fuzzy information is gotten and the matrix is formed as D=(hˇij)m×n(i=1,2,…,m, j=1,2,…,n). The algorithm can be described as follows.Step 1.
The experts evaluate the alternatives with linguistic terms and intuitionistic fuzzy memberships and the decision matrix is formed asD=(hˇij)m×n.Step 2.
Determine the position weight vectorw=(w1,w2,…,wn) and the argument weight vector ω=(ω1,ω2,…,ωn).Step 3.
Utilize the GLHIFHWA (or GLHIFHGM) operator to calculate the collective expected resultshˇi:(37)hˇi=GLHIFHWAhˇi1,hˇi2,…,hˇin=⨁j=1nwjωσjhˇiσjλ∑j=1nwjωσjλ1/λ=⋃aiσj∈hˇiσjs∑j=1nwjωσjθiσjλ/∑j=1nwjωσjλ1/λ,⋃μiσj,νiσj∈lhsθiσj1-∏j=1n1-μiσjλwjωσjλ/∑j=1nwjωσjλ1/λ,1-1-∏j=1n1-1-νiσjλwjωσjλ/∑j=1nwjωσjλ1/λ,hˇi=GLHIFHGMhˇi1,hˇi2,…,hˇin=⋃aiσj∈hˇiσjs∏j=1nθiσjwjωσj/∑j=1nwjωσj,⋃μiσj,νiσj∈lhsθiσj1-1-∏j=1n1-1-μiσjλwjωσj/∑j=1nwjωσj1/λ,1-∏j=1n1-νiσjλwjωσj/∑j=1nwjωσj1/λ.Step 4.
Calculate the score functionS(hˇi) and accuracy function A(hˇi) of each collective evaluation value hˇi(38)Shˇi=1lhˇi∑j=1lhˇiθijglhsθij∑αij,βij∈lhsθijαij-βij,Ahˇ=1lhˇi∑j=1lhˇiθijglhsθij∑αij,βij∈lhsθijαij+βij,and here lhˇi is the number of LIFEs in hˇi, lhsθij is the number of intuitionistic fuzzy memberships in lh(sθ(ij)), and g is the number of linguistic terms in linguistic term set S.
Rankhˇi according the method given in Definition 6 and rank the alternatives accordingly.Step 5.
End.


## 5. 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: (A1) computer company, (A2) real estate company, (A3) food company, (A4) car company, and (A5) communication company. Multiple experts are invited to evaluate alternatives with respect to the following possible situations for the economic environment: (C1) negative growth rate, (C2) low growth rate, (C3) medium growth rate, and (C4) 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 Table1.Table 1
Linguistic hesitant intuitionistic fuzzy decision matrix.
C
1
C
2
C
3
C
4
A
1
{
(
s
6
,
(
0.5,0.4
)
)
,
(
s
7
,
(
0.6,0.3
)
)
}
{
(
s
3
,
(
0.9,0.1
)
,
(
0.8,0.1
)
)
}
{
(
s
7
,
(
0.7,0.3
)
)
,
(
s
8
,
(
0.6,0.2
)
)
}
{
(
s
5
,
(
0.6,0.2
)
)
}
A
2
{
(
s
3
,
(
0.5,0.2
)
,
(
0.5,0.3
)
)
}
{
(
s
7
,
(
0.6,0.1
)
)
,
(
s
8
,
(
0.8,0.2
)
)
}
{
(
s
6
,
(
0.6,0.3
)
,
(
0.7,0.2
)
)
}
{
(
s
4
,
(
0.7,0.2
)
)
,
(
s
5
,
(
0.5,0.3
)
,
(
0.6,0.4
)
)
}
A
3
{
(
s
4
,
(
0.6,0.1
)
)
}
{
(
s
5
,
(
0.7,0.3
)
,
(
0.5,0.4
)
)
}
{
(
s
2
,
(
0.5,0.4
)
)
,
(
s
3
,
(
0.5,0.2
)
)
}
{
(
s
8
,
(
0.6,0.1
)
,
(
0.7,0.2
)
)
}
A
4
{
(
s
2
,
(
0.6,0.2
)
,
(
0.5,0.3
)
)
}
{
(
s
6
,
(
0.8,0.2
)
)
}
{
(
s
3
,
(
0.7,0.2
)
,
(
0.6,0.3
)
)
}
{
(
s
6
,
(
0.6,0.2
)
)
,
(
s
7
,
(
0.7,0.1
)
)
}
A
5
{
(
s
5
,
(
0.7,0.3
)
)
,
(
s
6
,
(
0.6,0.2
)
)
}
{
(
s
4
,
(
0.9,0.1
)
,
(
0.8,0.2
)
)
}
{
(
s
7
,
(
0.8,0.2
)
)
,
(
s
8
,
(
0.6,0.4
)
)
}
{
(
s
4
,
(
0.5,0.2
)
)
}Step 2.
The position weight vector is determined asw=(0.20,0.30,0.35,0.15) and the argument weight vector is determined as ω=(0.10,0.40,0.15,0.35).Step 3.
The GLHIFHSWA operator is used to aggregate the alternatives’ evaluation values into collective ones,hˇi(i=1,2,…,5). For example, in order to calculate hˇ1 for λ=0.1, we first calculate ω1hˇ11={(s0.6,(0.0670,0.9124)), (s0.7,(0.0876,0.8866))}, ω2hˇ12={(s0.8,(0.3069,0.5253), (0.3822,0.5253))}, ω3hˇ13={(s1.05,(0.1672,0.8348)), (s1.2,(0.1284,0.7855))}, and ω4hˇ14={(s1.75,(0.2744,0.5693))}. The scores of ωjhˇ1j(j=1,…,4) can be calculated as S(ω1hˇ11)=-0.0593, S(ω2hˇ12)=-0.0161, S(ω3hˇ13)=-0.0829, and S(ω4hˇ14)=-0.0574. Then, ω2hˇ12>ω4hˇ14>ω1hˇ11>ω3hˇ13. Let ϖj=wjωσ(j)0.1/∑j=1nwjωσ(j)0.1 and ϖ1=w1ωσ(1)0.1/∑j=1nwjωσ(j)0.1=(0.2∗0.40.1)/(0.2∗0.40.1+0.3∗0.350.1+0.35∗0.10.1+0.15∗0.350.1)=0.2135. We can get ϖ2=0.3160, ϖ3=0.3253, and ϖ4=0.1452, similarly. By using the GLHIFHSWA0.1 operator, we can get(39)hˇ1=GLHIFHWAhˇ11,hˇ12,…,hˇ14=⋃a1σj∈hˇ1σjs0.2135∗θ1σ10.1+0.3160∗θ1σ20.1+0.3253∗θ1σ30.1+0.1452∗θ1σ40.11/0.1,1-1-ν1σj0.10.14521/λ⋃μ1σj,ν1σj∈lhsθ1σj1-1-μ1σ10.10.21351-μ1σ20.10.31601-1-ν1σj0.10.14521/λ·1-μ1σ30.10.32531-μ1σ40.10.14521-1-ν1σj0.10.14521/λ1/0.1,1-1-1-1-ν1σj0.10.21351-1-ν1σj0.10.3160·1-1-ν1σj0.10.32531-1-ν1σj0.10.14521/λ=s4.2989,0.5689,0.2683,0.5933,0.2683,s4.4945,0.5689,0.2531,0.5933,0.2531,s4.4002,0.6000,0.428,0.6000,0.2496,s4.5999,0.6000,0.2288,0.6231,0.2288.Similarly, we can get other collective alternative evaluation values as follows:(40)hˇ2=s5.1977,0.6145,0.2146,0.6145,0.2404,0.6299,0.2146,0.6299,0.2404,s5.4676,0.6649,0.2759,0.6649,0.2468,0.6787,0.2468,0.6787,0.2759,s5.3650,0.5371,0.2146,0.5371,0.2404,0.5544,0.2404,0.5544,0.2146,s5.6426,0.5943,0.2468,0.5943,0.2759,0.6102,0.2468,0.6102,0.2759,hˇ3=s5.6426,0.5943,0.2468,0.5943,0.2759,0.6102,0.2468,0.6102,0.2759,s3.7152,0.5994,0.1585,0.6392,0.2042,0.5665,0.1668,0.6085,0.1370,hˇ4=s4.2826,0.6856,0.1728,0.6630,0.1946,0.6526,0.1996,0.6283,0.2243,s4.4806,0.7002,0.1549,0.6784,0.1746,0.6683,0.1791,0.6448,0.2016,hˇ5=s4.1516,0.7211,0.1947,0.6811,0.2243,0.6918,0.2164,0.6486,0.2488,s4.3197,0.6543,0.2000,0.6970,0.1732,0.6657,0.1928,0.6198,0.2222.
For otherλ, we can get the results similarly and the results are omitted here for space limit.Step 4.
Calculate the scoresShˇi(i=1,2,…,5) of the comprehensive evaluation values hˇi(i=1,2,…,5) and rank the alternatives according to the ranking of Shˇi(i=1,2,…,5). For λ=0.1, we can calculate the scores as S(hˇ1)=0.1666, S(hˇ2)=0.2202, S(hˇ3)=0.1915, S(hˇ4)=0.2327, and S(hˇ5)=0.2181. The scores can be ranked as S(hˇ4)>S(hˇ2)>S(hˇ5)>S(hˇ3)>S(hˇ1) and the alternatives can be ranked accordingly as A4>A2>A5>A3>A1. For other λ, we can calculate the scores and rank the alternatives similarly and the results are shown in Table 2.Table 2
Ranking results based on the GLHIFHSWA operator with differentλ.
S
(
h
ˇ
1
)
S
(
h
ˇ
2
)
S
(
h
ˇ
3
)
S
(
h
ˇ
4
)
S
(
h
ˇ
5
)
Ranking of alternatives
λ
=
0.1
0.1666
0.2202
0.1915
0.2327
0.2181
A
4
>
A
2
>
A
5
>
A
3
>
A
1
λ
=
0.2
0.1750
0.2239
0.1918
0.2336
0.2192
A
4
>
A
2
>
A
5
>
A
3
>
A
1
λ
=
0.5
0.2025
0.2352
0.1939
0.2357
0.2618
A
5
>
A
4
>
A
2
>
A
1
>
A
3
λ
=
1.0
0.2377
0.2538
0.2018
0.2365
0.2684
A
5
>
A
2
>
A
1
>
A
4
>
A
3
λ
=
2.0
0.2717
0.2864
0.2300
0.2297
0.2853
A
2
>
A
5
>
A
1
>
A
3
>
A
4
λ
=
5.0
0.2875
0.3453
0.3033
0.2065
0.3386
A
2
>
A
5
>
A
3
>
A
1
>
A
4
λ
=
10.0
0.2931
0.3896
0.3497
0.1996
0.3908
A
5
>
A
2
>
A
3
>
A
1
>
A
4
λ
=
20.0
0.2956
0.4132
0.3736
0.1989
0.4212
A
5
>
A
2
>
A
3
>
A
1
>
A
4Step 5.
End.From the above results, we can see that the ranking of alternatives may be different due to different aggregation operators.A4 is the optimal alternative if λ=0.1 or λ=0.2, since the smallest evaluation values are relatively large in A4. The best alternative is A2 if the GLHIFHSWA2 operator and the GLHIFHSWA5 operator are used, since these operators focus on the values with the largest and second largest ωj. A5 become the best alternative if the following linguistic hesitant intuitionistic fuzzy aggregation operators are used: the GLHIFHSWA0.5 operator, the LHIFHSWA operator, the GLHIFHSWA10 operator, and the GLHIFHSWA20 operator. The GLHIFHSWA0.5 operator and the LHIFHSWA operator focus on the average values of the evaluation values. The GLHIFHSWA10 operator and the GLHIFHSWA20 operator focus on the value with the largest weight ωj. With the increasing λ, the largest weights ωj play more and more important role in aggregation. λ 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.


## 6. 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.



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*Source: 983628-2015-08-06.xml*