# Grey Relational Bidirectional Projection Method for Multicriteria Decision Making with Hesitant Intuitionistic Fuzzy Linguistic Information

Authors: Yuqi Zang; Xiaodong Zhao; Shiyong Li; Adnan Nazir
Journal: Mathematical Problems in Engineering
 (2018)

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## Abstract

We investigate a novel approach for multicriteria decision making (MCDM) with hesitant intuitionistic fuzzy linguistic information. To compare the hesitant intuitionistic fuzzy linguistic term sets (HIFLTSs), we propose a comparison method of HIFLTSs. A family of distance measures of HIFLTSs is developed. After that, we propose the grey relational bidirectional projection method based on the proposed comparison method and distance measures of IVHFLNs for dealing with MCDM problems. Furthermore, we establish a nonlinear optimization model to obtain the weight vector of criteria. Finally, an illustrative example is given to demonstrate the effectiveness and flexibility of the proposed approach.


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## Body

## 1. Introduction

Since 1975, Zadeh [1] first proposed the fuzzy linguistic term set which can depict the qualitative fuzzy information. It has become the beginning of qualitative decision making research. Qualitative decision making has been widely concerned by scholars and successfully applied to many areas [2–10]. However, in many practical qualitative decision making cases, it is often difficult for decision makers to express preferences by using a single linguistic term. The hesitant fuzzy linguistic term set (HFLTS) was proposed by Rodriguez et al. [11], which permits decision makers to use several possible linguistic terms to express preferences. Because the hesitant fuzzy linguistic term set can adequately express vague and imprecise information, more close to the decision maker’s qualitative thinking cognition, which is regarded as a powerful tool for dealing with qualitative decision making problems, scholars had an in-depth study of hesitant fuzzy linguistic term sets and made some improvements.In the aspect of the comparison method, Rodriguez et al. [11] first proposed the possibility degree for comparing HFLTSs; the definition of the possibility degree is based on the interval value which is constructed by the HFLTSs’ envelopes. Wei et al. [12] pointed out that Rodriguez et al.’s method may not accord with common sense, and they constructed the new suitable possibility degree formula of HFLTSs by using the possibility degree theory. Lee and Chen [13] noted that the Rodriguez et al.’s and Wei et al.’s methods were defined by using the maximum and minimum operators, which can not rank the preference orders in some cases. Thus, they proposed a novel comparison method of HFLTSs based on the likelihood. Tian et al. [14] also from the perspective of the likelihood proposed a qualitative flexible multiple criteria method, which can measure the consistency and inconsistency of the preference order under the hesitant fuzzy linguistic environment.As to the distance measure, Liao et al. [15] defined a family of basic distance measures of HFLTs, then they proposed the distance measures between two collections of HFLTs in continuous and discrete cases and applied the measure method to the evaluation of the quality of movies. Liao and Xu [16] proposed a series of cosine distance measures of HFLTs from the geometric point of view, then the proposed measures were applied to the selection of ERP systems. Wang et al. [17] presented the hesitant fuzzy linguistic Hausdorff distance measure, which is not necessary to add and rearrange any linguistic terms in HFLTs and applied the distance measure to the TOPSIS and TODIM methods. Wang et al. [18] proposed the Euclidean distance measure based on the function of the position index for HFLTSs. Meng and Chen [19] considered the ordered positions and the internal interactions, developed the generalized hesitant fuzzy linguistic 2-additive Choquet weighted distance measure and the generalized hesitant fuzzy linguistic 2-additive Shapley Choquet weighted distance measure, and then applied the proposed measure to evaluate investment.There are also some extensions of classical MCDM methods. Liao and Xu [16] explored the TOPSIS and VIKOR methods based on the cosine distance measure for solving the hesitant fuzzy linguistic multicriteria decision making. Liao et al. [20] used the VIKOR method for solving the hesitant fuzzy linguistic multicriteria decision making in which the criteria conflict with each other. Wei et al. [21] considered the decision maker’s psychological behavior in the MCDM and proposed a hesitant fuzzy linguistic TODIM method. They applied the proposed method to evaluate the telecommunications service providers. Wang et al. [18] proposed the ELECTRE I method to deal with the MCDM problems based on the distance measure for HFLTSs. Liao et al. [22] defined the correlation coefficient of HFLTSs, then the traditional Chinese medical diagnosis was given to illustrate the applicability of the proposed method. Lin et al. [23] derived a family of hesitant fuzzy linguistic aggregation operators and applied the proposed operators to multiattribute decision making. Farhadinia [24] solved the hesitant fuzzy linguistic multiple criteria decision making problems with completely unknown weights by entropy measure; Gou et al. [25] proposed a hesitant fuzzy linguistic alternative queuing method based on the entropy and cross-entropy measures for the hesitant fuzzy linguistic term sets and applied the proposed measure to the tertiary hospital management.However, while using the hesitant fuzzy linguistic term sets, only the linguistic terms in the membership function are used to express the degree of certainty of the property, and the importance of the uncertainty is ignored. In this case, Beg and Rashid [26] proposed the hesitant intuitionistic fuzzy linguistic term sets (HIFLTSs) which take into account both membership and nonmembership. The hesitant intuitionistic fuzzy linguistic term sets can depict the hesitation more comprehensively when faced with qualitative decision making problems. They used the TOPSIS method to deal with the hesitant intuitionistic fuzzy linguistic multicriteria decision making problem. Liu et al. [27] extended the WA and OWA operators to the hesitant intuitionistic fuzzy linguistic environment and developed some hesitant intuitionistic fuzzy linguistic aggregation operators and applied the HIFLWA operator to the MCDM. Rashid et al. [28] constructed an ELECTRE-based outranking method to deal with MCDM. Faizi et al. [29] proposed an outranking method for hesitant intuitionistic fuzzy linguistic group decision making based on the support function, the risk function, and the credibility function.As mentioned previously, since HIFLTS was proposed in 2014, the study of MCDM methods based on hesitant intuitionistic fuzzy linguistic information is still at the initial stage, only a few studies are involved, and there is still much work we need to do for improving the research. First of all, the possibility degree for comparing HIFLTSs has not been studied. With respect to the shortcomings of the existing possibility degree of HFLTSs [11–13], we propose an improved probability degree of HFLTSs, then on the basis of the above, we define the possibility degree of HIFLTSs. This is the first motivation of our work. Then, on the study of distance measure, Beg and Rashid [26] defined the distance measure of HIFLTSs; however, the proposed distance measure was based on the envelope of HIFLTSs. If two HIFLTSs have the same envelope, they will obtain the same distance, which is unreasonable. Rashid et al. [28] only considered the case that the number of elements in the membership degree and nonmembership degree is equal. The distance measure should be improved to overcome the drawback of the existing distance measures. This is the second motivation of our work. In addition, there are many famous methods for solving the MCDM problems with hesitant fuzzy linguistic information or hesitant intuitionistic fuzzy linguistic information, but none of studies have used grey relational projection method to handle the MCDM problems with hesitant fuzzy linguistic information or hesitant intuitionistic fuzzy linguistic information. Grey relational projection technology is one of the effective methods to deal with MCDM problems [30–32]. We improved the traditional grey relational projection method and proposed the grey relational bidirectional projection method to deal with the MCDM problems under the hesitant intuitionistic fuzzy linguistic environment. This is the third motivation of our study.The rest of this paper is organized as follows: In Section2, we present some definitions of the LTSs, HFLTSs, and HIFLTSs. In Section 3, we propose the comparison method and the distance measures of HFLTSs. Section 4 denotes studying the hesitant intuitionistic fuzzy linguistic MCDM based on the grey relational bidirectional projection. In Section 5, we present a numerical example to demonstrate the effectiveness and practicality of the proposed method and also discuss the advantages of the proposed method. In Section 6, we briefly conclude the paper.


## 2. Preliminaries

### 2.1. Linguistic Term Sets

LetS=si∣i=0,1,…,g be a finite linguistic term set with odd cardinality, where si represents a possible value for a linguistic variable, and the following characteristics should be satisfied [33]:(1)
The set is ordered:sα≥sβ⇔α≥β.(2)
There is a negation operator:negsα=sg-α.(3)
Ifsα>sβ, then max⁡sα,sβ=sα, min⁡sα,sβ=sβ.


### 2.2. Hesitant Fuzzy Linguistic Term Sets

Definition 1 (see [11]).
LetS=s0,s1,…,sg be a linguistic term set; let HS be an HFLTS which is an ordered finite subset of the consecutive linguistic terms of S.Definition 2 (see [11]).
LetS=s0,s1,…,sg be a linguistic term set, let HS be an HFLTS, and the operational laws are defined as(1)
the upper bound:HS+=max⁡si=sj,si∈HS and si≤sj∀i;(2)
the lower bound:HS-=min⁡si=sj,si∈HS and si≤sj∀i;(3)
the envelope:envHS=HS-,HS+;(4)
the complement:HSc=S-HS=si,si∈Sandsi∉HS.


### 2.3. Hesitant Intuitionistic Fuzzy Linguistic Term Sets

Definition 3 (see [26]).
LetX be fixed; HIFLTSs on X are functions Hx and Gx that when applied to X return ordered finite subsets of the consecutive linguistic term set, S=s0,s1,…,sg; the mathematical symbol is defined as(1)ES=x,HSx,GSx∣x∈X,where Hx and Gx are the subsets of the consecutive linguistic terms of S, denoting the possible membership degrees and nonmembership degrees of the element x∈X to the set ES, respectively, with the conditions: max⁡HSx+min⁡GSx≤sg; min⁡HSx+max⁡GSx≤sg.
For convenience, the pairHSx,GSx is called the hesitant intuitionistic fuzzy linguistic term element (HIFLTE), denoted as HS,GS.Definition 4 (see [26]).
LetES=HS,GS be an HIFLTS; the upper bound and lower bound are defined as(2)HS+=max⁡si=sj,si∈HS,si≤sj∀i;HS-=min⁡si=sj,si∈HS,si≥sj∀i;GS+=max⁡si=sj,si∈GS,si≤sj∀i;GS-=min⁡si=sj,si∈GS,si≥sj∀i.Definition 5 (see [26]).
The envelope of the HIFLTS,env(ES), is defined as(3)envES=HS-,HS+,GS-,GS+.Definition 6.
The complement of HIFLTS,ES, is defined as(4)ESc=S-HS,S-GS=si,si∈S,si∉HS,si,si∈S,si∉GS.


## 2.1. Linguistic Term Sets

LetS=si∣i=0,1,…,g be a finite linguistic term set with odd cardinality, where si represents a possible value for a linguistic variable, and the following characteristics should be satisfied [33]:(1)
The set is ordered:sα≥sβ⇔α≥β.(2)
There is a negation operator:negsα=sg-α.(3)
Ifsα>sβ, then max⁡sα,sβ=sα, min⁡sα,sβ=sβ.


## 2.2. Hesitant Fuzzy Linguistic Term Sets

Definition 1 (see [11]).
LetS=s0,s1,…,sg be a linguistic term set; let HS be an HFLTS which is an ordered finite subset of the consecutive linguistic terms of S.Definition 2 (see [11]).
LetS=s0,s1,…,sg be a linguistic term set, let HS be an HFLTS, and the operational laws are defined as(1)
the upper bound:HS+=max⁡si=sj,si∈HS and si≤sj∀i;(2)
the lower bound:HS-=min⁡si=sj,si∈HS and si≤sj∀i;(3)
the envelope:envHS=HS-,HS+;(4)
the complement:HSc=S-HS=si,si∈Sandsi∉HS.


## 2.3. Hesitant Intuitionistic Fuzzy Linguistic Term Sets

Definition 3 (see [26]).
LetX be fixed; HIFLTSs on X are functions Hx and Gx that when applied to X return ordered finite subsets of the consecutive linguistic term set, S=s0,s1,…,sg; the mathematical symbol is defined as(1)ES=x,HSx,GSx∣x∈X,where Hx and Gx are the subsets of the consecutive linguistic terms of S, denoting the possible membership degrees and nonmembership degrees of the element x∈X to the set ES, respectively, with the conditions: max⁡HSx+min⁡GSx≤sg; min⁡HSx+max⁡GSx≤sg.
For convenience, the pairHSx,GSx is called the hesitant intuitionistic fuzzy linguistic term element (HIFLTE), denoted as HS,GS.Definition 4 (see [26]).
LetES=HS,GS be an HIFLTS; the upper bound and lower bound are defined as(2)HS+=max⁡si=sj,si∈HS,si≤sj∀i;HS-=min⁡si=sj,si∈HS,si≥sj∀i;GS+=max⁡si=sj,si∈GS,si≤sj∀i;GS-=min⁡si=sj,si∈GS,si≥sj∀i.Definition 5 (see [26]).
The envelope of the HIFLTS,env(ES), is defined as(3)envES=HS-,HS+,GS-,GS+.Definition 6.
The complement of HIFLTS,ES, is defined as(4)ESc=S-HS,S-GS=si,si∈S,si∉HS,si,si∈S,si∉GS.


## 3. A Comparison Method and Distance Measures of HIFLTSs

### 3.1. A Comparison Method of HIFLTSs

Possibility degree is one of the most appropriate comparison methods to rank the preference order of different arguments. We have already pointed out in Introduction that some scholars have studied the possibility formula of HFLTSs; however, they have the drawback that they cannot compare HFLTSs in some cases. For instance, when the hesitant fuzzy linguistic term sets reduce to only one element, the possibility formula denominator is zero by using Lee and Chen’s method [13]; obviously, that is unreasonable. Wei et al.’s method [12] may be a complex formula, but they gave us a good inspiration to develop the possibility degree of HIFLTSs.In this paper, we combine the ideas of methods in [12, 13] and propose the improved possibility formula of HFLTSs, which can rank the preference order more efficiently. The possibility degree formula of HFLTSs is given as(5)PHS1≥HS2=max⁡1-max⁡IndHS2+-IndHS1-+1#HS1+#HS2,0,0,where #HS1, #HS2 are the number of elements in HS1, HS2. Indsi=i (it provides the index associated with the label).As with the possibility degree axiom of the interval number, the possibility degree ofpHS1≥HS2 satisfies the following properties:(1)
0≤pHS1≥HS2≤1;(2)
pHS1≥HS1=0.5;(3)
ifHS1≥HS2, then pHS1≥HS2=1;ifHS1≤HS2, then pHS1≥HS2=0;(4)
pHS1≥HS2+pHS2≥HS1=1;(5)
ifHS1=HS2, then pHS1≥HS2=pHS2≥HS1=0.5.Similar to the definition of the possibility degree for HFLTS, we give the definition for HIFFLSs.Definition 7.
LetS=s0,s1,…,sg be a linguistic term set, let ES1=HS1,GS1 and ES2=HS2,GS2 be two HIFLTSs on S, and the possibility degree of pES1≥ES2 is denoted as(6)pES1≥ES2=12pHS1≥HS2+pGS2≥GS1=12max⁡1-max⁡IndHS2+-IndHS1-+1#HS1+#HS2,0,0+max⁡1-max⁡IndGS1+-IndGS2-+1#GS1+#GS2,0,0,where #HS1, #HS2 are the number of elements in HS1, HS2 and #GS1, #GS2 are the number of elements in GS1, GS2, respectively. Indsi=i (it provides the index associated with the label).
The possibility degree ofpES1≥ES2 satisfies the following properties:(1)
0≤pES1≥ES2≤1;(2)
pES1≥ES1=0.5;(3)
ifES1≥ES2, then pES1≥ES2=1;ifES1≤ES2, then pES1≥ES2=0;(4)
pES1≥ES2+pES2≥ES1=1;(5)
ifES1=ES2, then pES1≥ES2=pES2≥ES1=0.5.


### 3.2. Distance Measure of HIFLTSs

Xu [34] first proposed the distance measure of the linguistic term sets as follows.Definition 8 (see [34]).
LetS=s0,s1,…,sg be a linguistic term set, let sα, sβ be two linguistic terms, and then the deviation degree between sα and sβ is(7)dsα,sβ=α-βg+1,where g+1 is the number of linguistic terms in the set S.
Motivated by the definition of the distance measure for LTSs, we define the distance measure of HIFLTSsES1=HS1,GS1 and ES2=HS2,GS2.(8)ES1=HS1,GS1=⋃σl1∈HS1sσl1∣l=1,…,#HS1,⋃δk1∈GS1sδk1∣k=1,…,#GS1,ES2=HS2,GS2=⋃σl2∈HS2sσl2∣l=1,…,#HS2,⋃δk2∈GS2sδk2∣k=1,…,#GS2,where #HS1, #HS2 are the number of linguistic terms in HS1, HS2 and #GS1, #GS2 are the number of linguistic terms in GS1, GS2, respectively. Where #HS1=#HS2=L and #GS1=#GS2=K, we need to make them equivalently by adding some linguistic terms to the shorter HIFLTS, according to the following principles: pessimistic principle, the lower bound will be added; optimistic principle, the upper bound will be added.
The distance measures for HIFLTSsES1=HS1,GS1, ES2=HS2,GS2 are defined as follows.
The hesitant intuitionistic fuzzy linguistic Hamming distance is as follows:(9)dhiflhdES1,ES2=121L∑l=1Lσl1-σl2g+1+1K∑k=1Kδk1-δk2g+1.
The hesitant intuitionistic fuzzy linguistic Euclidean distance is as follows:(10)dhifledES1,ES2=121L∑l=1Lσl1-σl2g+12+1K∑k=1Kδk1-δk2g+121/2.
With the generalization of (9) and (10), the generalized hesitant intuitionistic fuzzy linguistic distance can be obtained:(11)dghifldES1,ES2=121L∑l=1Lσl1-σl2g+1λ+1K∑k=1Kδk1-δk2g+1λ1/λ.
Based on the Hausdorff distance measure, the generalized hesitant intuitionistic fuzzy linguistic Hausdorff distance is expressed as(12)dghiglhdES1,ES2=max⁡maxlσl1-σl2g+1λ,maxkδk1-δk2g+1λ1/λ.
With the combination of (11) and (12), we get the generalized hybrid hesitant intuitionistic fuzzy linguistic distance:(13)dghhigldES1,ES2=12121L∑l=1Lσl1-σl2g+1λ+1K∑k=1Kδk1-δk2g+1λ+max⁡maxlσl1-σl2g+1λ,maxkδk1-δk2g+1λ1/λ,where σl1 and σl2 are the lth largest linguistic term in HS1 and HS2 and δk1 and δk2 are the kth largest linguistic term in GS1 and GS2.Definition 9.
The distance measuredES1,ES2 between ES1 and ES2 satisfies the following properties:(1)
0≤dES1,ES2≤1;(2)
dES1,ES2=0 if and only if ES1=ES2;(3)
dES1,ES2=dES2,ES1.


## 3.1. A Comparison Method of HIFLTSs

Possibility degree is one of the most appropriate comparison methods to rank the preference order of different arguments. We have already pointed out in Introduction that some scholars have studied the possibility formula of HFLTSs; however, they have the drawback that they cannot compare HFLTSs in some cases. For instance, when the hesitant fuzzy linguistic term sets reduce to only one element, the possibility formula denominator is zero by using Lee and Chen’s method [13]; obviously, that is unreasonable. Wei et al.’s method [12] may be a complex formula, but they gave us a good inspiration to develop the possibility degree of HIFLTSs.In this paper, we combine the ideas of methods in [12, 13] and propose the improved possibility formula of HFLTSs, which can rank the preference order more efficiently. The possibility degree formula of HFLTSs is given as(5)PHS1≥HS2=max⁡1-max⁡IndHS2+-IndHS1-+1#HS1+#HS2,0,0,where #HS1, #HS2 are the number of elements in HS1, HS2. Indsi=i (it provides the index associated with the label).As with the possibility degree axiom of the interval number, the possibility degree ofpHS1≥HS2 satisfies the following properties:(1)
0≤pHS1≥HS2≤1;(2)
pHS1≥HS1=0.5;(3)
ifHS1≥HS2, then pHS1≥HS2=1;ifHS1≤HS2, then pHS1≥HS2=0;(4)
pHS1≥HS2+pHS2≥HS1=1;(5)
ifHS1=HS2, then pHS1≥HS2=pHS2≥HS1=0.5.Similar to the definition of the possibility degree for HFLTS, we give the definition for HIFFLSs.Definition 7.
LetS=s0,s1,…,sg be a linguistic term set, let ES1=HS1,GS1 and ES2=HS2,GS2 be two HIFLTSs on S, and the possibility degree of pES1≥ES2 is denoted as(6)pES1≥ES2=12pHS1≥HS2+pGS2≥GS1=12max⁡1-max⁡IndHS2+-IndHS1-+1#HS1+#HS2,0,0+max⁡1-max⁡IndGS1+-IndGS2-+1#GS1+#GS2,0,0,where #HS1, #HS2 are the number of elements in HS1, HS2 and #GS1, #GS2 are the number of elements in GS1, GS2, respectively. Indsi=i (it provides the index associated with the label).
The possibility degree ofpES1≥ES2 satisfies the following properties:(1)
0≤pES1≥ES2≤1;(2)
pES1≥ES1=0.5;(3)
ifES1≥ES2, then pES1≥ES2=1;ifES1≤ES2, then pES1≥ES2=0;(4)
pES1≥ES2+pES2≥ES1=1;(5)
ifES1=ES2, then pES1≥ES2=pES2≥ES1=0.5.


## 3.2. Distance Measure of HIFLTSs

Xu [34] first proposed the distance measure of the linguistic term sets as follows.Definition 8 (see [34]).
LetS=s0,s1,…,sg be a linguistic term set, let sα, sβ be two linguistic terms, and then the deviation degree between sα and sβ is(7)dsα,sβ=α-βg+1,where g+1 is the number of linguistic terms in the set S.
Motivated by the definition of the distance measure for LTSs, we define the distance measure of HIFLTSsES1=HS1,GS1 and ES2=HS2,GS2.(8)ES1=HS1,GS1=⋃σl1∈HS1sσl1∣l=1,…,#HS1,⋃δk1∈GS1sδk1∣k=1,…,#GS1,ES2=HS2,GS2=⋃σl2∈HS2sσl2∣l=1,…,#HS2,⋃δk2∈GS2sδk2∣k=1,…,#GS2,where #HS1, #HS2 are the number of linguistic terms in HS1, HS2 and #GS1, #GS2 are the number of linguistic terms in GS1, GS2, respectively. Where #HS1=#HS2=L and #GS1=#GS2=K, we need to make them equivalently by adding some linguistic terms to the shorter HIFLTS, according to the following principles: pessimistic principle, the lower bound will be added; optimistic principle, the upper bound will be added.
The distance measures for HIFLTSsES1=HS1,GS1, ES2=HS2,GS2 are defined as follows.
The hesitant intuitionistic fuzzy linguistic Hamming distance is as follows:(9)dhiflhdES1,ES2=121L∑l=1Lσl1-σl2g+1+1K∑k=1Kδk1-δk2g+1.
The hesitant intuitionistic fuzzy linguistic Euclidean distance is as follows:(10)dhifledES1,ES2=121L∑l=1Lσl1-σl2g+12+1K∑k=1Kδk1-δk2g+121/2.
With the generalization of (9) and (10), the generalized hesitant intuitionistic fuzzy linguistic distance can be obtained:(11)dghifldES1,ES2=121L∑l=1Lσl1-σl2g+1λ+1K∑k=1Kδk1-δk2g+1λ1/λ.
Based on the Hausdorff distance measure, the generalized hesitant intuitionistic fuzzy linguistic Hausdorff distance is expressed as(12)dghiglhdES1,ES2=max⁡maxlσl1-σl2g+1λ,maxkδk1-δk2g+1λ1/λ.
With the combination of (11) and (12), we get the generalized hybrid hesitant intuitionistic fuzzy linguistic distance:(13)dghhigldES1,ES2=12121L∑l=1Lσl1-σl2g+1λ+1K∑k=1Kδk1-δk2g+1λ+max⁡maxlσl1-σl2g+1λ,maxkδk1-δk2g+1λ1/λ,where σl1 and σl2 are the lth largest linguistic term in HS1 and HS2 and δk1 and δk2 are the kth largest linguistic term in GS1 and GS2.Definition 9.
The distance measuredES1,ES2 between ES1 and ES2 satisfies the following properties:(1)
0≤dES1,ES2≤1;(2)
dES1,ES2=0 if and only if ES1=ES2;(3)
dES1,ES2=dES2,ES1.


## 4. Proposed Method for MCDM

The MCDM problem with hesitant intuitionistic fuzzy linguistic information is shown as follows. Suppose that there arem alternatives A=A1,A2,…,Am and n criteria C=C1,C2,…,Cn; the weight vector of the criteria is W=w1,w2,…,wn, where wj≥0, ∑j=1n‍wj=1; assume that the evaluation values are taken in the form of HIFLTSs, where Eij is the evaluation value of alternative Ai respect to criteria Cj. Hence, we construct the hesitant intuitionistic fuzzy linguistic decision matrix E=Eijm×n.Note. This paper omits the process of transforming the linguistic information into HIFLTSs. The detailed process was shown in [11].


### 4.1. Normalized Decision Matrix

We should translate the decision matrixE=Eijm×n into the normalized decision matrix E~=E~ijm×n before calculating the grey relational bidirectional projection. In the MCDM system, there are usually two types of criteria, the benefit criteria and the cost criteria; the normalized E~ij is shown as(14)E~ij=Eij,Cj∈benefitcriteriaEijc,Cj∈costcriteria.


### 4.2. Grey Relational Bidirectional Projection

Grey relational projection method combines the advantages of the grey relational analysis [35–40] and the projection method [41–44], which is an effective method to deal with MCDM problems. However, the traditional projection method sometimes has some shortcomings, for example, during the projection of two vectors a and c onto the ideal solution b at the same vertical point, as shown in Figure 1. Their projection value is equal; that is, the alternatives can not be compared.Figure 1
The projection vector.Ye [45] proposed the bidirectional projection method, which can overcome the shortcomings of the traditional projection method; that is,(15)BPa,b=11+a·b/a-a·b/b=a·ba·b+a-b·a·b.Motivated by Ye’s method, we combine the grey relational analysis and the bidirectional projection method and propose the grey relational bidirectional projection method, which can rank the preference order of alternatives efficiently. Then, we present the grey relational bidirectional projection method and apply it to the hesitant intuitionistic fuzzy linguistic MCDM.Firstly, determine the hesitant intuitionistic fuzzy linguistic ideal solution. For a normalized hesitant intuitionistic fuzzy linguistic decision making matrix, the hesitant intuitionistic fuzzy linguistic positive ideal solution (HIFLPIS) and the hesitant intuitionistic fuzzy linguistic negative ideal solution (HIFLNIS) are expressed as(16)E~+=E~1+,E~2+,…,E~n+E~-=E~1-,E~2-,…,E~n-.Then calculate the generalized hybrid hesitant intuitionistic fuzzy linguistic distance betweenE~ij and E~+, and that between E~ij and E~- is shown as(17)dij+=12121L∑l=1Lσlij-σl+g+1λ+1K∑k=1Kδkij-δk+g+1λ+max⁡maxlσlij-σl+g+1λ,maxkδkij-δk+g+1λ1/λ,dij-=12121L∑l=1Lσlij-σl-g+1λ+1K∑k=1Kδkij-δk-g+1λ+max⁡maxlσlij-σl-g+1λ,maxkδkij-δk-g+1λ1/λ.Thus, the grey relational coefficient of each alternative from the HIFLPIS and HIFLNIS can be formulated as(18)rij+=miniminjdij++θmaximaxjdij+dij++θmaximaxjdij+,rij-=miniminjdij-+θmaximaxjdij-dij-+θmaximaxjdij-,where θ∈0,1 is the resolution coefficient which is decided by the decision maker; thus we can construct the grey relational coefficient matrices r+=rij+m×n and r-=rij-m×n.The grey relational coefficient between HIFLPIS and HIFLPIS and that between HIFLNIS and HIFLNIS are, respectively, shown as(19)r0+=r01+,r02+,…,r0n+=1,1,…,1︸n,r0-=r01-,r02-,…,r0n-=1,1,…,1︸n.We can get the weighted grey relational coefficient matricesR+=Rij+m×n=wjrij+m×n and R-=Rij-m×n=wjrij-m×n.The weighted grey relational coefficient between HIFLPIS and HIFLPIS and that between HIFLNIS and HIFLNIS are, respectively, shown as(20)R0+=R01+,R02+,…,R0n+=w1,w2,…,wn,R0-=R01-,R02-,…,R0n-=w1,w2,…,wn.Combining the weighted grey relational coefficient and the bidirectional projection method, we get the weighted grey relational bidirectional projection betweenRi+ and R0+, and that between Ri- and R0- is, respectively, shown as(21)BPi+=11+Ri+·R0+/Ri+-Ri+·R0+/R0+=Ri+·R0+Ri+·R0++Ri+-R0+·Ri+·R0+=∑j=1nRij+2·∑j=1nwj2∑j=1nRij+2·∑j=1nwj2+∑j=1nRij+2-∑j=1nwj2·∑j=1nRij+wj,BPi-=11+Ri-·R0-/Ri--Ri-·R0-/R0-=Ri-·R0-Ri-·R0-+Ri--R0-·Ri-·R0-=∑j=1nRij-2·∑j=1nwj2∑j=1nRij-2·∑j=1nwj2+∑j=1nRij-2-∑j=1nwj2·∑j=1nRij-wj.For the alternativeAi, the closer the value of BPi+ is to 1, the closer it is to HIFLPIS, the closer the value of BPi- is to 1, the closer it is to HIFLNIS, obviously, the larger the value of BPi+ is, the better Ai is, and the smaller the value of BPi- is, the better Ai is and vice versa. Thus we construct the relative closeness formula.(22)C~i=BPi+BPi++BPi-.In general, the largerC~i is, the better Ai is and vice versa.


### 4.3. Determine the Attribute Weight

In this paper we consider the case where the criteria weight information is partly known. As we know, there are many methods for deriving criteria weights, such as the deviation-based method [46, 47] and the entropy-based method [48, 49]. We combine the advantages of the deviation-based method and the entropy-based method and construct the combined optimization model.We first construct the optimization modelM1 according to the minimum deviation method. The grey relational coefficient deviation between the alternative Ai and HIFLPIS is 1-rij+, to eliminate the effects of symbols, we take the form of the squared, and totally, we have(23)M1:minC1w=∑i=1m∑j=1n1-rij+wj2s.t.0≤wjL≤wj≤wjU≤1∑j=1nwj=1.The principle of information entropy method is that, in all feasible solutions or possible solutions, the maximum entropy is chosen. The maximum entropy means that the amount of information obtained is the smallest. In the process of solving, the amount of information added is the least, so the entropy-based method is reasonable when the criteria weight information is partly known. We construct the optimization modelM2 as follows:(24)M2:maxC2w=-∑j=1nwjln⁡wjs.t.0≤wjL≤wj≤wjU≤1∑j=1nwj=1.CombineM1 and M2, then we have optimization model M3:(25)M3:minCw=α∑i=1m∑j=1n1-rij+wj2-1-α∑j=1nwjln⁡wjs.t.0≤wjL≤wj≤wjU≤1∑j=1nwj=1,where α represents the equilibrium coefficient; without losing generality, we let α=0.5. The criteria weights can be obtained by Matlab software.


### 4.4. The Procedure of the MCDM Method

The algorithm for the proposed method is shown as follows.Step 1.
Construct the decision matrixE=Eijm×n, then translate the decision matrix into the normalized decision matrix E~=E~ijm×n.Step 2.
Determine the hesitant intuitionistic fuzzy linguistic positive ideal solution (HIFLPIS)E~+and the hesitant intuitionistic fuzzy linguistic negative ideal solution (HIFLNIS) E~-, according to Definition 7.Step 3.
Calculate the generalized hybrid hesitant intuitionistic fuzzy linguistic distance betweenE~ij and E~+ and that between E~ij and E~-, shown in (17), based on the distance measure. We get the grey relational coefficient of each alternative from the HIFLPIS and HIFLNIS, according to (18), then the grey relational coefficient matrices r+=rij+m×n and r-=rij-m×n are constructed.Step 4.
Obtain the criteria weight by solving the optimization modelM3.Step 5.
Calculate the grey relational bidirectional projection between each alternative and the HIFLPIS and that between each alternative and the HIFLNIS, according to (21); thus, the relative closeness can be constructed in (22).Step 6.
Rank all alternatives according to the relative closeness.


## 4.1. Normalized Decision Matrix

We should translate the decision matrixE=Eijm×n into the normalized decision matrix E~=E~ijm×n before calculating the grey relational bidirectional projection. In the MCDM system, there are usually two types of criteria, the benefit criteria and the cost criteria; the normalized E~ij is shown as(14)E~ij=Eij,Cj∈benefitcriteriaEijc,Cj∈costcriteria.


## 4.2. Grey Relational Bidirectional Projection

Grey relational projection method combines the advantages of the grey relational analysis [35–40] and the projection method [41–44], which is an effective method to deal with MCDM problems. However, the traditional projection method sometimes has some shortcomings, for example, during the projection of two vectors a and c onto the ideal solution b at the same vertical point, as shown in Figure 1. Their projection value is equal; that is, the alternatives can not be compared.Figure 1
The projection vector.Ye [45] proposed the bidirectional projection method, which can overcome the shortcomings of the traditional projection method; that is,(15)BPa,b=11+a·b/a-a·b/b=a·ba·b+a-b·a·b.Motivated by Ye’s method, we combine the grey relational analysis and the bidirectional projection method and propose the grey relational bidirectional projection method, which can rank the preference order of alternatives efficiently. Then, we present the grey relational bidirectional projection method and apply it to the hesitant intuitionistic fuzzy linguistic MCDM.Firstly, determine the hesitant intuitionistic fuzzy linguistic ideal solution. For a normalized hesitant intuitionistic fuzzy linguistic decision making matrix, the hesitant intuitionistic fuzzy linguistic positive ideal solution (HIFLPIS) and the hesitant intuitionistic fuzzy linguistic negative ideal solution (HIFLNIS) are expressed as(16)E~+=E~1+,E~2+,…,E~n+E~-=E~1-,E~2-,…,E~n-.Then calculate the generalized hybrid hesitant intuitionistic fuzzy linguistic distance betweenE~ij and E~+, and that between E~ij and E~- is shown as(17)dij+=12121L∑l=1Lσlij-σl+g+1λ+1K∑k=1Kδkij-δk+g+1λ+max⁡maxlσlij-σl+g+1λ,maxkδkij-δk+g+1λ1/λ,dij-=12121L∑l=1Lσlij-σl-g+1λ+1K∑k=1Kδkij-δk-g+1λ+max⁡maxlσlij-σl-g+1λ,maxkδkij-δk-g+1λ1/λ.Thus, the grey relational coefficient of each alternative from the HIFLPIS and HIFLNIS can be formulated as(18)rij+=miniminjdij++θmaximaxjdij+dij++θmaximaxjdij+,rij-=miniminjdij-+θmaximaxjdij-dij-+θmaximaxjdij-,where θ∈0,1 is the resolution coefficient which is decided by the decision maker; thus we can construct the grey relational coefficient matrices r+=rij+m×n and r-=rij-m×n.The grey relational coefficient between HIFLPIS and HIFLPIS and that between HIFLNIS and HIFLNIS are, respectively, shown as(19)r0+=r01+,r02+,…,r0n+=1,1,…,1︸n,r0-=r01-,r02-,…,r0n-=1,1,…,1︸n.We can get the weighted grey relational coefficient matricesR+=Rij+m×n=wjrij+m×n and R-=Rij-m×n=wjrij-m×n.The weighted grey relational coefficient between HIFLPIS and HIFLPIS and that between HIFLNIS and HIFLNIS are, respectively, shown as(20)R0+=R01+,R02+,…,R0n+=w1,w2,…,wn,R0-=R01-,R02-,…,R0n-=w1,w2,…,wn.Combining the weighted grey relational coefficient and the bidirectional projection method, we get the weighted grey relational bidirectional projection betweenRi+ and R0+, and that between Ri- and R0- is, respectively, shown as(21)BPi+=11+Ri+·R0+/Ri+-Ri+·R0+/R0+=Ri+·R0+Ri+·R0++Ri+-R0+·Ri+·R0+=∑j=1nRij+2·∑j=1nwj2∑j=1nRij+2·∑j=1nwj2+∑j=1nRij+2-∑j=1nwj2·∑j=1nRij+wj,BPi-=11+Ri-·R0-/Ri--Ri-·R0-/R0-=Ri-·R0-Ri-·R0-+Ri--R0-·Ri-·R0-=∑j=1nRij-2·∑j=1nwj2∑j=1nRij-2·∑j=1nwj2+∑j=1nRij-2-∑j=1nwj2·∑j=1nRij-wj.For the alternativeAi, the closer the value of BPi+ is to 1, the closer it is to HIFLPIS, the closer the value of BPi- is to 1, the closer it is to HIFLNIS, obviously, the larger the value of BPi+ is, the better Ai is, and the smaller the value of BPi- is, the better Ai is and vice versa. Thus we construct the relative closeness formula.(22)C~i=BPi+BPi++BPi-.In general, the largerC~i is, the better Ai is and vice versa.


## 4.3. Determine the Attribute Weight

In this paper we consider the case where the criteria weight information is partly known. As we know, there are many methods for deriving criteria weights, such as the deviation-based method [46, 47] and the entropy-based method [48, 49]. We combine the advantages of the deviation-based method and the entropy-based method and construct the combined optimization model.We first construct the optimization modelM1 according to the minimum deviation method. The grey relational coefficient deviation between the alternative Ai and HIFLPIS is 1-rij+, to eliminate the effects of symbols, we take the form of the squared, and totally, we have(23)M1:minC1w=∑i=1m∑j=1n1-rij+wj2s.t.0≤wjL≤wj≤wjU≤1∑j=1nwj=1.The principle of information entropy method is that, in all feasible solutions or possible solutions, the maximum entropy is chosen. The maximum entropy means that the amount of information obtained is the smallest. In the process of solving, the amount of information added is the least, so the entropy-based method is reasonable when the criteria weight information is partly known. We construct the optimization modelM2 as follows:(24)M2:maxC2w=-∑j=1nwjln⁡wjs.t.0≤wjL≤wj≤wjU≤1∑j=1nwj=1.CombineM1 and M2, then we have optimization model M3:(25)M3:minCw=α∑i=1m∑j=1n1-rij+wj2-1-α∑j=1nwjln⁡wjs.t.0≤wjL≤wj≤wjU≤1∑j=1nwj=1,where α represents the equilibrium coefficient; without losing generality, we let α=0.5. The criteria weights can be obtained by Matlab software.


## 4.4. The Procedure of the MCDM Method

The algorithm for the proposed method is shown as follows.Step 1.
Construct the decision matrixE=Eijm×n, then translate the decision matrix into the normalized decision matrix E~=E~ijm×n.Step 2.
Determine the hesitant intuitionistic fuzzy linguistic positive ideal solution (HIFLPIS)E~+and the hesitant intuitionistic fuzzy linguistic negative ideal solution (HIFLNIS) E~-, according to Definition 7.Step 3.
Calculate the generalized hybrid hesitant intuitionistic fuzzy linguistic distance betweenE~ij and E~+ and that between E~ij and E~-, shown in (17), based on the distance measure. We get the grey relational coefficient of each alternative from the HIFLPIS and HIFLNIS, according to (18), then the grey relational coefficient matrices r+=rij+m×n and r-=rij-m×n are constructed.Step 4.
Obtain the criteria weight by solving the optimization modelM3.Step 5.
Calculate the grey relational bidirectional projection between each alternative and the HIFLPIS and that between each alternative and the HIFLNIS, according to (21); thus, the relative closeness can be constructed in (22).Step 6.
Rank all alternatives according to the relative closeness.


## 5. Illustrative Example

### 5.1. Example

In this section, an illustrative example about courses evaluation for a MCDM problem adopted from [28] is given to show the method proposed in this paper. There are three courses we need to evaluate, A1, A2, and A3, and four criteria, C1, C2, C3, and C4. The criteria weights information is partly known, assuming 0.38≤w1≤0.42, 0.30≤w2≤0.35, 0.18≤w3≤0.21, and 0.07≤w4≤0.09. The linguistic term set S=s0,s1,s2,s3,s4,s5,s6={s0= very poor, s1= poor, s2= medium, s3= fair, s4= medium good, s5= good, s6= very good}, the linguistic information. The evaluation information takes the form of HIFLTSs, where Eij is the evaluation value of the alternative Ai on the criteria Cj, and then the decision matrix is as follows:(26)E=s2,s3,s0,s4,s5,s6,s0,s0,s1,s2,s4,s4,s5,s0,s1s3,s2,s3,s2,s3,s4,s0,s1,s2,s3,s3,s4,s6,s0s3,s4,s0,s1,s3,s4,s0,s2,s4,s0,s2,s0,s1,s2,s3,s3.The procedure to obtain the most desirable alternative is as follows.Step 1.
Transform the decision matrixE=Eij3×4 into the normalized decision matrix E~=E~ij3×4. We get(27)E~=s2,s3,s0,s0,s4,s5,s6,s0,s0,s0,s1,s2,s4,s4,s4,s4,s4,s5,s0,s1s3,s3,s2,s3,s2,s3,s4,s0,s1,s2,s2,s3,s3,s4,s6,s6,s6,s6,s0,s0s3,s4,s0,s1,s3,s3,s4,s0,s2,s4,s4,s4,s0,s2,s0,s1,s2,s3,s3,s3.Step 2.
According to Definition7, we determine HIFLPIS and HIFLNIS and have(28)E~+=s3,s4,s0,s1,s4,s5,s6,s0,s0,s4,s4,s4,s0,s2,s6,s6,s6,s6,s0,s0,E~-=s3,s3,s2,s3,s2,s3,s4,s0,s1,s0,s1,s2,s4,s4,s0,s1,s2,s3,s3,s3.Step 3.
Calculate the generalized hybrid hesitant intuitionistic fuzzy linguistic distancesdij+ and dij-, based on (17), let λ=2, and then we construct the distance matrix as follows:(29)D+=0.13360.00000.52730.27200.25250.26730.37120.00000.00000.25250.00000.7660,D-=0.35710.26730.00000.49740.00000.00000.24220.76600.25250.11290.52730.0000.
According to (18), we calculate the grey relational coefficient of each alternative from HIFLPIS and HIFLNIS, without losing generality; let θ=0.5. We have(30)r+=0.74131.00000.42070.58470.60260.58900.50791.00001.00000.60261.00000.3333,r-=0.51750.58901.00000.43501.00001.00000.61260.33330.60260.77230.42071.0000.Step 4.
Solve the optimization modelM3 by using the fmincon function in Matlab software. We get (31)W=0.3837,0.3163,0.2100,0.0900T.Step 5.
Calculate the grey relational bidirectional projection between each alternative and the HIFLPIS and the grey relational bidirectional projection between each alternative and the HIFLNIS, according to (21). We have(32)BP1+=0.9040,BP2+=0.8216,BP3+=0.9374,BP1-=0.8376,BP2-=0.9691,BP3-=0.8445.
Calculate the relative closeness, shown in (22). We obtain(33)C~1=0.5191,C~2=0.4588,C~3=0.5261.Step 6.
Rank all the alternatives(34)A3≻A1≻A2.Furthermore, for comparative analysis, we apply the weight vector of the criteria in [26], then the result is as follows.Step  5 ∗. Calculate the grey relational bidirectional projection between each alternative and the HIFLPIS and the grey relational bidirectional projection between each alternative and the HIFLNIS, according to (21). We have(35)BP1+=0.8671,BP2+=0.8270,BP3+=0.9702,BP1-=0.8690,BP2-=0.9433,BP3-=0.8319.Calculate the relative closeness, shown in (22). We obtain(36)C~1=0.4995,C~2=0.4671,C~3=0.5384.Step  6 ∗. Rank all the alternatives(37)A3≻A1≻A2.The results are the same as those in [26], so the proposed method in this paper is effective.From Tables1–4, we note that when the parameters θ, λ change, the relative closeness changes trends: when λ is fixed, the change θ has obvious effect on the relative closeness, but when θ is fixed, the change λ has no obvious effect on the relative closeness. All ranking results are A3≻A1≻A2, which means that A3 is the best alternative. Moreover, the trends of the relative closeness of three alternatives can be shown directly in Figures 2–5, and we also present the relative closeness for the three alternatives when λ and θ change simultaneously, which is shown in Figures 6–8; the results show that A3 is always the best alternative. Thus, the choice of the parameter θ can reflect the decision maker’s risk attitude, if the decision maker is risk averse, let θ be a larger value, and vice versa.Table 1
The relative closeness with respect toλ=2.
θ A 1 A 2 A 3 Rankings 0.2 0.4922 0.4506 0.5550 A3≻A1≻A2 0.4 0.4980 0.4611 0.5445 A3≻A1≻A2 0.6 0.4999 0.4684 0.5370 A3≻A1≻A2 0.8 0.5005 0.4734 0.5317 A3≻A1≻A2 1.0 0.5007 0.4771 0.5277 A3≻A1≻A2Table 2
The relative closeness with respect toλ=5.
θ A1 A2 A3 Rankings 0.2 0.4923 0.4504 0.5558 A3≻A1≻A2 0.4 0.4984 0.4608 0.5452 A3≻A1≻A2 0.6 0.5005 0.4681 0.5376 A3≻A1≻A2 0.8 0.5012 0.4731 0.5322 A3≻A1≻A2 1.0 0.5014 0.4768 0.5282 A3≻A1≻A2Table 3
The relative closeness with respect toθ=0.2.
λ A 1 A 2 A 3 Rankings 2 0.4922 0.4506 0.5550 A 3 ≻ A 1 ≻ A 2 4 0.4922 0.4504 0.5557 A 3 ≻ A 1 ≻ A 2 6 0.4923 0.4504 0.5559 A 3 ≻ A 1 ≻ A 2 8 0.4924 0.4504 0.5559 A 3 ≻ A 1 ≻ A 2 10 0.4924 0.4504 0.5559 A 3 ≻ A 1 ≻ A 2Table 4
The relative closeness with respect toθ=0.5.
λ A 1 A 2 A 3 Rankings 2 0.4992 0.4651 0.5404 A3≻A1≻A2 4 0.4996 0.4648 0.5410 A3≻A1≻A2 6 0.4998 0.4648 0.5411 A3≻A1≻A2 8 0.5000 0.4648 0.5411 A3≻A1≻A2 10 0.5000 0.4649 0.5411 A3≻A1≻A2Figure 2
Relative closeness with respect toθ when λ=2.Figure 3
Relative closeness with respect toθ when λ=5.Figure 4
Relative closeness with respect toλ when θ=0.2.Figure 5
Relative closeness with respect toλ when θ=0.5.Figure 6
Relative closeness forA1λ∈0,10,θ∈0,1.Figure 7
Relative closeness forA2λ∈0,10,θ∈0,1.Figure 8
Relative closeness forA3λ∈0,10,θ∈0,1.


### 5.2. Advantages of the Proposed Method

(1)
Compared with the possibility degree of HFLTSs studied in [11–13], the possibility degree formula proposed in this paper can overcome the drawback of the previous work. We derived the possibility degree of HIFLTSs based on the possibility degree of HFLTSs, which can compare the HIFLTSs more effectively. Moreover, the possibility degree and the distance measure which are proposed in this paper are not only the foundation of the proposed grey relational bidirectional projection method but also foundation of many classical decision making methods including TOPSIS, PROMETHEE, and VIKOR. (2)
In contrast to the studies in [15, 22], the grey relational bidirectional projection method combines the distance measure and the correlation coefficient, which can integrate the influence of the whole criteria space and avoid the unidirectional deviation. (3)
The criteria weights are determined by the nonlinear optimization model, which consists of the minimum deviation model [46, 47] and the information entropy model [48, 49]. The optimization model combines the advantages of both methods, which is based on the evaluation and the weights information. Therefore, we can reduce the impact of subjective human factors and obtain the reasonable criteria weights. (4)
The proposed grey relational bidirectional projection method integrates the grey relational analysis into the bidirectional projection. The method can overcome the drawbacks of the traditional grey relational projection [30–32], which cannot rank the preference order of the alternative in some situations.


## 5.1. Example

In this section, an illustrative example about courses evaluation for a MCDM problem adopted from [28] is given to show the method proposed in this paper. There are three courses we need to evaluate, A1, A2, and A3, and four criteria, C1, C2, C3, and C4. The criteria weights information is partly known, assuming 0.38≤w1≤0.42, 0.30≤w2≤0.35, 0.18≤w3≤0.21, and 0.07≤w4≤0.09. The linguistic term set S=s0,s1,s2,s3,s4,s5,s6={s0= very poor, s1= poor, s2= medium, s3= fair, s4= medium good, s5= good, s6= very good}, the linguistic information. The evaluation information takes the form of HIFLTSs, where Eij is the evaluation value of the alternative Ai on the criteria Cj, and then the decision matrix is as follows:(26)E=s2,s3,s0,s4,s5,s6,s0,s0,s1,s2,s4,s4,s5,s0,s1s3,s2,s3,s2,s3,s4,s0,s1,s2,s3,s3,s4,s6,s0s3,s4,s0,s1,s3,s4,s0,s2,s4,s0,s2,s0,s1,s2,s3,s3.The procedure to obtain the most desirable alternative is as follows.Step 1.
Transform the decision matrixE=Eij3×4 into the normalized decision matrix E~=E~ij3×4. We get(27)E~=s2,s3,s0,s0,s4,s5,s6,s0,s0,s0,s1,s2,s4,s4,s4,s4,s4,s5,s0,s1s3,s3,s2,s3,s2,s3,s4,s0,s1,s2,s2,s3,s3,s4,s6,s6,s6,s6,s0,s0s3,s4,s0,s1,s3,s3,s4,s0,s2,s4,s4,s4,s0,s2,s0,s1,s2,s3,s3,s3.Step 2.
According to Definition7, we determine HIFLPIS and HIFLNIS and have(28)E~+=s3,s4,s0,s1,s4,s5,s6,s0,s0,s4,s4,s4,s0,s2,s6,s6,s6,s6,s0,s0,E~-=s3,s3,s2,s3,s2,s3,s4,s0,s1,s0,s1,s2,s4,s4,s0,s1,s2,s3,s3,s3.Step 3.
Calculate the generalized hybrid hesitant intuitionistic fuzzy linguistic distancesdij+ and dij-, based on (17), let λ=2, and then we construct the distance matrix as follows:(29)D+=0.13360.00000.52730.27200.25250.26730.37120.00000.00000.25250.00000.7660,D-=0.35710.26730.00000.49740.00000.00000.24220.76600.25250.11290.52730.0000.
According to (18), we calculate the grey relational coefficient of each alternative from HIFLPIS and HIFLNIS, without losing generality; let θ=0.5. We have(30)r+=0.74131.00000.42070.58470.60260.58900.50791.00001.00000.60261.00000.3333,r-=0.51750.58901.00000.43501.00001.00000.61260.33330.60260.77230.42071.0000.Step 4.
Solve the optimization modelM3 by using the fmincon function in Matlab software. We get (31)W=0.3837,0.3163,0.2100,0.0900T.Step 5.
Calculate the grey relational bidirectional projection between each alternative and the HIFLPIS and the grey relational bidirectional projection between each alternative and the HIFLNIS, according to (21). We have(32)BP1+=0.9040,BP2+=0.8216,BP3+=0.9374,BP1-=0.8376,BP2-=0.9691,BP3-=0.8445.
Calculate the relative closeness, shown in (22). We obtain(33)C~1=0.5191,C~2=0.4588,C~3=0.5261.Step 6.
Rank all the alternatives(34)A3≻A1≻A2.Furthermore, for comparative analysis, we apply the weight vector of the criteria in [26], then the result is as follows.Step  5 ∗. Calculate the grey relational bidirectional projection between each alternative and the HIFLPIS and the grey relational bidirectional projection between each alternative and the HIFLNIS, according to (21). We have(35)BP1+=0.8671,BP2+=0.8270,BP3+=0.9702,BP1-=0.8690,BP2-=0.9433,BP3-=0.8319.Calculate the relative closeness, shown in (22). We obtain(36)C~1=0.4995,C~2=0.4671,C~3=0.5384.Step  6 ∗. Rank all the alternatives(37)A3≻A1≻A2.The results are the same as those in [26], so the proposed method in this paper is effective.From Tables1–4, we note that when the parameters θ, λ change, the relative closeness changes trends: when λ is fixed, the change θ has obvious effect on the relative closeness, but when θ is fixed, the change λ has no obvious effect on the relative closeness. All ranking results are A3≻A1≻A2, which means that A3 is the best alternative. Moreover, the trends of the relative closeness of three alternatives can be shown directly in Figures 2–5, and we also present the relative closeness for the three alternatives when λ and θ change simultaneously, which is shown in Figures 6–8; the results show that A3 is always the best alternative. Thus, the choice of the parameter θ can reflect the decision maker’s risk attitude, if the decision maker is risk averse, let θ be a larger value, and vice versa.Table 1
The relative closeness with respect toλ=2.
θ A 1 A 2 A 3 Rankings 0.2 0.4922 0.4506 0.5550 A3≻A1≻A2 0.4 0.4980 0.4611 0.5445 A3≻A1≻A2 0.6 0.4999 0.4684 0.5370 A3≻A1≻A2 0.8 0.5005 0.4734 0.5317 A3≻A1≻A2 1.0 0.5007 0.4771 0.5277 A3≻A1≻A2Table 2
The relative closeness with respect toλ=5.
θ A1 A2 A3 Rankings 0.2 0.4923 0.4504 0.5558 A3≻A1≻A2 0.4 0.4984 0.4608 0.5452 A3≻A1≻A2 0.6 0.5005 0.4681 0.5376 A3≻A1≻A2 0.8 0.5012 0.4731 0.5322 A3≻A1≻A2 1.0 0.5014 0.4768 0.5282 A3≻A1≻A2Table 3
The relative closeness with respect toθ=0.2.
λ A 1 A 2 A 3 Rankings 2 0.4922 0.4506 0.5550 A 3 ≻ A 1 ≻ A 2 4 0.4922 0.4504 0.5557 A 3 ≻ A 1 ≻ A 2 6 0.4923 0.4504 0.5559 A 3 ≻ A 1 ≻ A 2 8 0.4924 0.4504 0.5559 A 3 ≻ A 1 ≻ A 2 10 0.4924 0.4504 0.5559 A 3 ≻ A 1 ≻ A 2Table 4
The relative closeness with respect toθ=0.5.
λ A 1 A 2 A 3 Rankings 2 0.4992 0.4651 0.5404 A3≻A1≻A2 4 0.4996 0.4648 0.5410 A3≻A1≻A2 6 0.4998 0.4648 0.5411 A3≻A1≻A2 8 0.5000 0.4648 0.5411 A3≻A1≻A2 10 0.5000 0.4649 0.5411 A3≻A1≻A2Figure 2
Relative closeness with respect toθ when λ=2.Figure 3
Relative closeness with respect toθ when λ=5.Figure 4
Relative closeness with respect toλ when θ=0.2.Figure 5
Relative closeness with respect toλ when θ=0.5.Figure 6
Relative closeness forA1λ∈0,10,θ∈0,1.Figure 7
Relative closeness forA2λ∈0,10,θ∈0,1.Figure 8
Relative closeness forA3λ∈0,10,θ∈0,1.


## 5.2. Advantages of the Proposed Method

(1)
Compared with the possibility degree of HFLTSs studied in [11–13], the possibility degree formula proposed in this paper can overcome the drawback of the previous work. We derived the possibility degree of HIFLTSs based on the possibility degree of HFLTSs, which can compare the HIFLTSs more effectively. Moreover, the possibility degree and the distance measure which are proposed in this paper are not only the foundation of the proposed grey relational bidirectional projection method but also foundation of many classical decision making methods including TOPSIS, PROMETHEE, and VIKOR. (2)
In contrast to the studies in [15, 22], the grey relational bidirectional projection method combines the distance measure and the correlation coefficient, which can integrate the influence of the whole criteria space and avoid the unidirectional deviation. (3)
The criteria weights are determined by the nonlinear optimization model, which consists of the minimum deviation model [46, 47] and the information entropy model [48, 49]. The optimization model combines the advantages of both methods, which is based on the evaluation and the weights information. Therefore, we can reduce the impact of subjective human factors and obtain the reasonable criteria weights. (4)
The proposed grey relational bidirectional projection method integrates the grey relational analysis into the bidirectional projection. The method can overcome the drawbacks of the traditional grey relational projection [30–32], which cannot rank the preference order of the alternative in some situations.


## 6. Conclusion

The hesitant intuitionistic fuzzy linguistic term sets have a good advantage in the expression of hesitation information, which is especially important in the background of qualitative decision making. In this paper, we have investigated a novel qualitative decision making method based on the grey relational bidirectional projection with the hesitant intuitionistic fuzzy linguistic information. We have derived the possibility degree for comparing the HIFLTSs. We have proposed a family of distance measure for the HIFLTSs. Based on the possibility degree and the distance measure, we have proposed the grey relational bidirectional projection method, which combines the grey relational analysis and the vector projection theory. The combined nonlinear optimization model has been used to determine the criteria weight, where the information on criteria weights is partly known. The numerical example shows that the proposed method is suitable for dealing with the MCDM. Moreover, the model has good practicability and can be further applied in wider fields.In future research, we will apply our possibility degree and distance measures to order decision making methods, such as ELECTRE and PROMETHEE. Furthermore, we will also focus on the extended hesitant intuitionistic fuzzy linguistic term sets theory, including the interval-valued hesitant intuitionistic fuzzy linguistic term sets and the hesitant intuitionistic fuzzy uncertain linguistic term sets.



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