A group decision making methodbased on double hesitant linguistic preference relations

(School of Economics and Management, Southeast University, Nanjing 211189, China)

Abstract：A simple decision method is proposed to solve the group decision making problems in which the weights of decision organizations are unknown and the

references： for alternatives are provided by double hesitant linguistic preference relations. First, double hesitant linguistic elements are defined as representing the uncertain assessment information in the process of group decision making accurately and comprehensively, and the double hesitant linguistic weighted averaging operator is developed based on the defined operational laws for double hesitant linguistic elements. Then, double hesitant linguistic preference relations are defined and a means to objectively determine the weights of decision organizations is put forward using the standard deviation of scores of preferences provided by the individual decision organization for alternatives. Finally the correlation coefficient between the scores of preferences and the scores of preferences are provided by the other decision organizations. Accordingly, a group decision method based on double hesitant linguistic preference relations is proposed, and a practical example of the Jiudianxia reservoir operation alternative selection is used to illustrate the practicability and validity of the method. Finally, the proposed method is compared with the existing methods. Comparative results show that the proposed method can deal with the double hesitant linguistic preference information directly, does not need any information transformation, and can thus reduce the loss of original decision information.

Key words：group decision making; double hesitant linguistic elements; double hesitant linguistic preference relations; double hesitant linguistic weighted averaging operator

In reality, people are used to using natural languages to assess the qualitative aspects of problems. For example, when assessing the environmental quality of a city, experts prefer to use natural languages, such as “very good”, “good” and “poor”, etc. The fuzzy linguistic approach is a technique to deal with the qualitative information[1]. So far, many linguistic models have been developed to extend and improve the fuzzy linguistic approaches in information modeling and computing processes, such as the semantic model[2], the symbolic model[3] and the 2-tuple linguistic model[4], etc. Among them, the symbolic model implements direct computations on linguistic labels, and thus possesses the merits of simple computational processes and high interpretability, which has been widely applied to many fields, such as decision making[5], information retrieval[6], supply chain management[7], marking[8], sustainable energy management[9], etc.

In the above-mentioned linguistic models, an expert can only use a single linguistic term to express his/her assessment for an alternative under a criterion, which indicates that the degree of the alternative to the linguistic term under the criterion being 1. Nevertheless, sometimes a single linguistic term is inadequate to exactly express the expert’s assessment for the alternative under the criterion because there may be some ambiguities when he/she provides the linguistic term as the assessment of the alternative under the criterion. For coping with such cases, Wang and Li[10] proposed the concept of intuitionistic linguistic sets, by which an alternative can be assessed by a linguistic term with a membership degree and a non-membership degree, both of which are values in [0,1]. Liu et al.[11] defined hesitant intuitionistic fuzzy linguistic sets, in which the membership and non-membership degrees of an element to a linguistic term are denoted by a set of intuitionistic fuzzy numbers. It is worthwhile mentioning that in the process of group decision making or anonymous assessment, different decision makers or evaluators may give different linguistic terms with different membership and non-membership degrees, which are subintervals of [0,1]. For example, a decision organization composed of several experts is invited to assess the environmental quality of a city in terms of the linguistic term set: S={s-3: extremely poor, s-2: very poor, s-1: poor, s0: fair, s1: good, s2: very good, s3: extremely good}. Suppose two experts think that the environmental quality of the city is surely not “good”. One of them deems the degree of the environmental quality of the city to be “good” and the degree of it not belonging to “good” as 0.6 and 0.3, respectively, and the other deems it as [0.7,0.8] and [0.1,0.2], respectively. The rest of the experts think that the environmental quality of the city is doubtlessly “very good”. Assume that they all insist on their own viewpoints and cannot persuade each other. To represent such assessments accurately, we will develop a new linguistic presentation model: double hesitant linguistic elements. By our model, we can use {〈s1,(0.6,0.3),([0.7,0.8],[0.1,0.2])〉,〈s2,(1,0)〉} to express the assessment information of the decision organization. Double hesitant linguistic elements consider people’s ambiguities when providing assessments, and encompass much more decision information; which is a very useful means to collect the decision makers’ assessment information in large group decision making or anonymous assessment problems accurately and comprehensively.

In the process of decision making, the decision makers are often required to provide their preferences by comparing each pair of alternatives and construct preference relations. Up to now, there have been many different kinds of preference relations, such as fuzzy preference relations[12], intuitionistic preference relations[13], interval-valued intuitionistic preference relations[14], and linguistic preference relations[15]. It is worth noting that none of the existing preference relations permit the decision makers to provide all possible linguistic terms with the membership and non-membership degrees, which are the subintervals of [0,1], as their preferences for a pair of alternatives. So another aim of this paper is to define double hesitant linguistic preference relations to overcome this limitation, and investigate their applications in group decision making.

1　Preliminaries

In this paper, we consider a subscript symmetric additive linguistic term set

α=-τ,…,-1,0,1,…,τ}, where τ is a positive integer. In general, S is required to satisfy the following conditions[15]:

1) There is a negation operator neg(si)=s-i, especially, neg(s0)=s0;

2) The set is ordered, si≤sj ⟺ i≤j.

To facilitate the calculation of linguistic terms, Xu[15] extended the discrete linguistic term set S to a continuous set

, and defined two operational laws as follows:

Usually, we call sα an original linguistic term if sα∈S; otherwise, we call sα a virtual linguistic term.

In order to express the membership and non-membership degrees of an element to a linguistic term, Wang and Li[10] defined intuitionistic linguistic sets.

Definition 3[10]　Let

α=-τ,…,-1,0,1,…,τ} be a linguistic term set,

be its extended linguistic term set, and X be a given domain. Then an intuitionistic linguistic set (ILS) in X is defined as

Afterwards, Liu et al.[11] defined hesitant intuitionistic fuzzy linguistic sets, in which the membership and non-membership degrees of an element to a linguistic term are denoted by a set of intuitionistic fuzzy numbers.

Definition 4[11]　Let

α=-τ,…,-1,0,1,…,τ} be a linguistic term set,

be its extended linguistic term set, and X be a given domain. Then a hesitant intuitionistic fuzzy linguistic set (HIFLS) in X is defined by

2　Double Hesitant Linguistic Elements and Basic Operations

In this section, we will define double hesitant linguistic elements and some basic operational laws.

Definition 5　Let

α=-τ,…,-1,0,1…,τ} be a linguistic term set,

be its extended linguistic term set, and X be a given domain. Then a double hesitant linguistic set (DHLS) in X is defined as

For convenience, we call d(x)={〈sθi(x),Mi(x)〉

). Clearly, the DHLS D can be written as D={{〈sθi(x),Mi(x)〉

}. Thus, DHLEs are the basic unites of a DHLS.

For comparing DHLEs, the following concepts are proposed.

Definition 6　Let d(x)={〈sθi(x),Mi(x)〉

Definition 7　Let d={〈sθi(d),Mi(d)〉

}. Then the score function of d is defined as

Then, based on Definitions 6 and 7, the following comparison rules are introduced.

Definition 8　Let d1={〈sθi(d1),Mi(d1)〉

{〈sθj(d2),Mj(d2)〉

1) If F(d1)>F(d2), then d1 is superior to d2, denoted by d1≻d2;

2) If F(d1)=F(d2), then

If H(d1)>H(d2), then d1 is superior to d2, denoted by d1≻d2;

If H(d1)=H(d2), then d1 is equivalent to d2, denoted by d1～d2.

In what follows, an example is provided to illustrate the comparison method shown in Definition 8.

Example 1　Let S={s-3: extremely poor, s-2: very poor, s-1: poor, s0: fair, s1: good, s2: very good, s3: extremely good} be a linguistic term set, d1={〈s-3,(0.6,0.3),([0.8,0.9],[0,0.1])〉,〈s2,(0.8,0.2)〉} and d2={〈s0,([0.6,0.7],[0.1,0.2])〉, 〈s-2,(0.15,0.8),([0.2,0.4],[0.5,0.6])〉} be two DHLVs, then by Eq.(1), we obtain

After giving the comparison rules of DHLEs, the next thing we need to do is to introduce their operational laws.

Definition 9　Let d1={〈sθi(d1),Mi(d1)〉

{〈sθj(d2),Mj(d2)〉

1) neg(d1)={〈s-θi(d1),Mi(d1)〉

2) d1⨁d2={〈sθi(d1)+θj(d2),Mij(d1⨁d2)〉

⨁

3) λd1={〈sλθi(d1),Mi(λd1)〉

Example 2　Let S={s-3: extremely poor, s-2: very poor, s-1: poor, s0: fair, s1: good, s2: very good, s3: extremely good} be a linguistic term set, d1={〈s1,(0.6,0.2),([0.7,0.9],[0,0.1])〉,〈s3,(0.7,0.1)〉} and d2={〈s0,([0.6,0.8],[0.1,0.2])〉,〈s2,(0.8,0.2)〉,〈s3,(0.5,0.4),([0.6,0.7],[0.2,0.3])〉} be two DHLEs. Then according to Definition 9, we obtain

1) neg(d1)={〈s-3,(0.7,0.1)〉,〈s-1,(0.6,0.2),([0.7,0.9],[0,0.1])〉};

2) 0.4d1={〈s0.4,(0.306 9,0.525 3),([0.382 2,0.601 9],[0,0.398 1])〉,〈s1.2,(0.382 2,0.398 1)〉};

3) d1⨁d2={〈s1,([0.84,0.92],[0.02,0.04]),([0.88,0.98],[0,0.02])〉,〈s3,(0.92,0.04),([0.94,0.98],[0,0.02]),([0.88,0.94],[0.01,0.02])〉,〈s4,(0.8,0.08),([0.84,0.88],[0.04,0.06]),([0.85,0.95],[0,0.04]),([0.88,0.97],[0,0.03])〉,〈s5,(0.94,0.02)〉,〈s6,(0.85,0.04),([0.88,0.91],[0.02,0.03])〉}.

For the operations of DHLEs in Definition 9, the following desirable properties are satisfied.

Property　Let d1={〈sθi(d1),Mi(d1)〉

{〈sθj(d2),Mj(d2)〉

{〈sθk(d3),Mk(d3)〉

1) λneg(d1)=neg(λd1);

2) neg(d1)⨁neg(d2)=neg(d1⨁d2);

4) (d1⨁d2)⨁d3=d1⨁(d2⨁d3);

5) λ(d1⨁d2)=λd1⨁λd2.

1) λneg(d1)=λ{〈s-θi(d1),Mi(d1)〉

{〈s-λθi(d1),Mi(λd1)〉

〉

2) neg(d1)⨁neg(d2)={〈s-θi(d1),Mi(d1)〉

⨁{〈s-θj(d2),Mj(d2)〉

⨁d2))〉

({〈sθi(d1)+θj(d2),Mij(d1⨁d2)〉

⨁d2).

3) The result is easily obtained by 2) in Definition 9.

4) (d1⨁d2)⨁d3={〈sθi(d1)+θj(d2),Mij(d1⨁d2)〉

⨁{〈sθk(d3),Mk(d3)〉

{〈sθi(d1)+θj(d2)+θk(d3),Mijk((d1⨁d2)⨁d3)〉

⨁d2)⨁

⨁

⊕

⨁

, then we obtain

Similarly, we obtain

Therefore, we prove (d1⨁d2)⨁d3=d1⨁(d2⨁d3).

5) λ(d1⨁d2)=λ{〈sθi(d1)+θj(d2), Mij(d1⨁d2)〉

{〈sλ(θi(d1)+θj(d2)),Mij(λ(d1⨁d2))〉

⨁

…,t1;j=1,2,…,t2}={〈sλ(θi(d1)+θj(d2)),Mij(λd1⨁

⨁

⨁λd2)=Mij(λ(d1⨁d2)). Thus, we prove λ(d1⨁d2)=λd1⨁λd2.

In the process of decision making, the aggregation operators are usually used to incorporate the individual decision information into the collective one. In order to fuse the double hesitant linguistic information, the following basic aggregation operator is developed based on the operational laws in Definition 9.

s. Then a double hesitant linguistic weighted averaging (DHLWA) operator is a mapping DHLWA: Vn→V such that

Theorem 1　Let di={〈sθpi(di),Mpi(di)〉

}. Then the aggregation result derived from Eq.(3) is still a DHLE, and

Proof　According to Definitions 9 and 10, the theorem can be easily proven by mathematical induction.

3　A Method to Group Decision Making with Double Hesitant Linguistic Preference Relations

t. Then a double hesitant linguistic preference relation (DHLPR) on X1 is represented by a matrix R=(dij)n×n⊂X1×X1 with dij={〈sθpij(dij),Mpij(dij)〉

where sσ(l)(dij) is the l-th smallest linguistic term of sθ1(dij),sθ2(dij),…,sθtij(dij), and Mσ(l)(dij) is the set of possible membership and non-membership degrees to sσ(l)(dij). We call dij={〈sθpij(dij),Mpij(dij)〉

Below we consider the group decision making problem. Let G={G1,G2,…,Gn} be a set of alternatives, and O={O1,O2,…,Om} be a set of decision organizations. Suppose that each decision organization is composed of several experts and each expert provides his/her preferences for each pair of alternatives. To convey the preferences accurately, each expert independently offers his/her preferences by using a linguistic term in the linguistic term set

α=-τ,…,-1,0,1,…,τ}, the membership and non-membership degrees to the linguistic term. So the preference information given by each decision organization Ok(k=1,2,…,m) can construct a DHLPR

)n×n with

={〈

)〉

} being a DHLE, where

⨁

={〈s0,(1,0)〉},

for all i,j=1,2,…,n. Here

is the l-th smallest linguistic term of

, and

) is the set of possible membership and non-membership degrees to

To solve the above group decision making problem with DHLPRs, the following steps are given.

Step 1　For the decision organization Ok(k=1,2,…,m) and the alternative Gi(i=1,2,…,n), we aggregate all

(j=1,2,…,n,j≠i) to obtain the average double hesitant linguistic preference of the alternative Gi over the others by using the DHLA operator:

Step 2　Aggregate all

,k=1,2,…,m by the DHLWA operator to obtain the collective double hesitant linguistic preference of the alternative Gi over the others:

Step 3　Compare di, i=1,2,…,n by Definition 8, and obtain the priority of alternatives Gi, i=1,2,…,n by ranking di, i=1,2,…,n.

In Step 2, if the importance weights of decision organizations are unknown, the following method is used to determine the importance weight of each decision organization.

Assume that for the decision organization Ok(k=1,2,…,m), the average double hesitant linguistic preference of the alternative Gi(i=1,2,…,n) over the others derived by Eq.(8) is

={〈

)〉

}. Then by Eq.

Moreover, we calculate the correlation coefficient zkl between Fk and Fl:

where for k,l=1,2,…,m,

/n and

/n, respectively. Clearly, -1≤zkl≤1, and the larger zkl, the closer the numerical distribution of Fk is to that of Fl; in contrast, the smaller zkl, the more different the numerical distribution of Fk is from that of Fl. During the decision making process, it is expected that the difference between the preference information provided by one decision organization and those provided by the others is as small as possible. Thus, we may assign the weight to the decision organization Ok(k=1,2,…,m) by the following rules: the larger

the larger the weight allocated to the decision organization Ok; otherwise, the smaller weight is assigned.

Additionally, it is required to compute the standard deviation σk of the preference score vector Fk of the decision organization Ok:

where

/n for k=1,2,…,m. Clearly, the larger σk, the greater difference among the preference information given to alternatives by the decision organization Ok, which indicates the greater the influence of the decision organization Ok on decision making. In this case, a larger weight should be assigned to the decision organization Ok. Alternately, a smaller weight could be assigned.

According to the above analysis, the importance weight vk of the decision organization Ok can be computed by the following formula:

4　Case Studies

We below use the example of the Jiudianxia Reservoir operation alternative selection[16-17] to illustrate the application and implementation process of the proposed method, and then make a comparative analysis to show its advantages.

4.1　Decision background

The Jiudianxia Reservoir is designed for many purposes, such as power generation, irrigation, total water supply for industry, agriculture, residents and the environment. Now four reservoir operation alternatives G1, G2, G3 and G4 are proposed because there are different requirements for the partition of the amount of water.

G1: The maximum plant output, sufficient supply of water used in the Tao River basin, larger and lower supply for society and economy.

G2: The maximum plant output, sufficient supply of water used in the Tao River basin, larger and lower supply for society and economy, lower supply for the ecosystem.

G3: The maximum plant output, sufficient supply of water used in the Tao River basin, larger and lower supply for society and economy, total supply for the ecosystem and environment, of which 90% is used for flushing out sands during a low water period.

G4: The maximum plant output, sufficient supply of water used in the Tao River basin, larger and lower supply for society and economy, total supply for the ecosystem and environment, of which 50% is used for flushing out sands during a low water period.

In order to select the optimal alternative, the government invites three organizations O1, O2, O3 from different fields to assess the four alternatives. Suppose that each expert in each organization independently gives his/her preferences for each pair of alternatives in the form of a linguistic term in the linguistic term set S={s-3: extremely poor, s-2: very poor, s-1: poor, s0: fair,s1: good, s2: very good, s3: extremely good}and the membership and non-membership degrees to the linguistic term. For instance, when three experts in the organization O3 compare the alternative G1 with the alternative G2, they give the following preferences: 〈s0,([0.8,0.9],[0,0.1])〉, 〈s2,(0.2,0.6)〉 and 〈s1,(0.7,0.3)〉. Since the experts in the organization O3 are independent, we may think of the preferences given by the organization O3 as the DHLE {〈s0,([0.8,0.9],[0,0.1])〉,〈s1,(0.7,0.3)〉,〈s2,(0.2,0.6)〉}. The preferences provided by the three organizations for the four alternatives are presented in matrices R1 to R3, all of which are DHLPRs. For convenience, we denote the matrix Rk(k=1,2,3) by

)4×4. Due to limited space, we merely list the DHLEs in the upper triangular part of each matrix and the rest can be correspondingly obtained according to Definition 11.

{〈s1,([0.8,0.9],[0.05,0.1])〉,〈s2,(0.2,0.5)〉},

{〈s2,(0.7,0.2)〉,〈s3,([0.7,0.8],[0.1,0.2]),(0.9,0.1)〉},

〈s0,(0.7,0.2),(0.6,0)〉},

{〈s0,([0.6,0.8],[0.1,0.2]),(1,0)〉},

{〈s1,(0.9,0.1)〉, 〈s2,(0.7,0.2)〉},

〉,〈s-1,(0.7,0.1)〉}.

{〈s-3,([0.1,0.3],[0.6,0.7])〉,

{〈s1,([0.7,0.8],[0.1,0.2])〉,〈s2,(0.6,0.2)〉,〈s3,(0.9,0.1)〉},

{〈s-2,(1,0)〉,〈s-1,(0.9,0.1),(0.6,0.3)〉},

{〈s0,(0.9,0.1)〉,〈

{〈s0,(0.8,0.1)〉,〈s1,(0.6,0.2)〉,〈s2,([0.1,0.5],[0.3,0.4])〉},

{〈s1,([0.7,0.9],[0,0.1])〉,〈s2,(0.9,0.1)〉}.

{〈s0,([0.8,0.9],[0,0.1])〉,〈s1,(0.7,0.3)〉,〈s2,(0.2,0.6)〉},

{〈s1,([0.7,0.8],[0.1,0.2]),(0.9,0.1)〉},

〈s1,([0.1,0.3],[0.5,0.7])〉},

{〈s0,([0.7,1],[0,0]),(0.8,0.1),(0.7,0.3)〉},

{〈s1,([0.7,0.9],[0,0.1])〉,〈s2,([0.3,0.6],[0.1,0.2]),(0.7,0.1)〉},

{〈s-1,([0.6,0.8],[0.1,0.2])〉,〈s1,(0.9,0.1)〉,〈s3,(0.2,0.4)〉}.

4.2　Decision model

In the following, we apply the proposed method to solve the above group decision making problem with DHLPRs. The solution process and computation results are summarized as follows:

Step 1　For the organizations Ok(k=1,2,3) and the alternative Gi (i=1,2,3,4), we aggregate all

(j=1,2,3,4, j≠i) to obtain the average double hesitant linguistic preference of the alternative Gi over the others by using the DHLA operator:

For example, for the organization O1 and the alternative, we obtain

⨁

{〈s-1.666 7,([0.636 6,0.748],[0.126,0.2]),([0.711 6,0.841 3],[0,0.158 7]),([0.748,0.8],[0.126,0.158 7]),([0.8,0.874],[0,0.126]),(1,0)〉,〈s-1.333 3,([0.636 6,0.711 6],[0.158 7,0.2]),([0.669 8,0.771 1],[0.1,0.158 7]),([0.711 6,0.818 3],[0,0.158 7]),([0.771 1,0.818 3],[0.1,0.126]),(1,0)〉,〈s-1,([0.669 8,0.737 9],[0.126,0.158 7]),(1,0)〉}

Similarly, other overall results can be obtained. Due to limited space, we here do not list them one by one.

Step 2　According to Eq.

reference s：btain the preference score vector Fk of the organization Ok (k=1,2,3) as

Moreover, by Eqs.(9) and (10), we obtain the following correlation coefficient matrix Z=(zkl)3×3 and the standard deviation of the preference score vector Fk of the organization Ok(k=1,2,3), respectively:

By Eq.(11), we obtain the importance weight vk of the organization Ok(k=1,2,3) as

Step 3　For the alternatives Gi (i=1,2,3,4), we aggregate all

(k=1,2,3) by the DHLWA operator to obtain the collective double hesitant linguistic preference of the alternative Gi over the others:

Due to limited space, here we merely show how to obtain one element in d1. Since 〈s0.666 7, ([0.689 3,0.753 4],[0.171,0.215 4])〉

〈s0,(0.8,0.144 2),

〈s0.333 3,([0.818 3,0.874],[0,0.126]),([0.874,0.9],[0,0.1])〉

q.(14), we obtain 〈s0.357 2,([0.766 3,0.806 2],[0,0.164 2]),([0.787,0.817 3],[0,0.154 8]),([0.8,0.834 2],[0,0.145 1]),([0.817 8,0.843 6],[0,0.136 9])〉∈d1.

Step 4　Compute the score Fi of di (i=1,2,3,4) by Eq.(1) as follows:

by which we obtain the rankings d2≻d1≻d4≻d3. Thus, the priority of alternatives is G2≻G1≻G4≻G3, and the alternative G2 is the best among the four alternatives.

4.3　Comparative analysis

In this subsection, we conduct a specific comparison of our method with the method based on interval-valued intuitionistic trapezoidal fuzzy numbers (IITFNs). To hold the same known information in the comparison process, we here assign the same importance weights to organizations as those obtained in Subsection 4.2, that is, the importance weight vector of organizations is v={0.408 9,0.337 4,0.253 7}T. The detailed comparison process is shown as follows.

Step 1　Transform the DHLEs in matrices R1 to R3 into the corresponding IITFNs.

{〈s2,(0.7,0.2)〉,〈s3,([0.7,0.8],[0.1,0.2]),(0.9,0.1)〉} for an example. First, according to Ref.

So the trapezoidal fuzzy number corresponding to s2 is [0.692 3,0.769 2,0.846 2,0.923 1], and the one corresponding to s3 is [0.846 2,0.923 1,1,1]. Then the

{〈[0.692 3,0.769 2,0.846 2,0.923 1],(0.7,0.2)〉,〈[0.846 2,0.923 1,1,1],([0.7,0.8],[0.1,0.2]),(0.9,0.1)〉}. Furthermore, we average the interval-valued intuitionistic fuzzy numbers in 〈[0.846 2,0.923 1,1,1],([0.7,0.8],[0.1,0.2]),(0.9,0.1)〉 by the interval-valued intuitionistic fuzzy arithmetic averaging operator[19]:

([0.826 8,0.858 6],[0.1,0.141 4])

{〈[0.692 3,0.769 2,0.846 2,0.923 1],(0.7,0.2)〉,〈[0.846 2,0.923 1,1,1], ([0.826 8,0.858 6],[0.1,0.141 4])〉}. Finally, we average the IITFN 〈[0.692 3,0.769 2,0.846 2,0.923 1],(0.7,0.2)〉 and the IITFN 〈[0.846 2,0.923 1,1,1],([0.826 8,0.858 6],[0.1,0.141 4])〉 by using the interval-valued intuitionistic trapezoidal fuzzy arithmetic averaging operator[20]:

〈[0.692 3,0.769 2,0.846 2,0.923 1],(0.7,0.2)〉

〈[0.846 2,0.923 1,1,1],([0.826 8,0.858 6],[0.1,0.141 4])〉=〈[0.769 3,0.846 2,0.923 1,0.961 6],([0.772 1,0.794 0],[0.141 4,0.168 2])〉

{〈s2,(0.7,0.2)〉,〈s3,([0.7,0.8],[0.1,0.2]),(0.9,0.1)〉} into the IITFN 〈[0.769 3,0.846 2,0.923 1,0.961 6],([0.772 1,0.794],[0.141 4,0.168 2])〉. In a similar way, we transform the rest of DHLEs in matrices R1 to R3 into the corresponding IITFNs. Due to limited space, we here omit their transformed results.

Step 2　For convenience, we denote

)4×4(k=1,2,3), and for the organizations Ok(k=1,2,3), aggregate

(j=1,2,3,4,j≠i) to obtain the averaged

of the alternatives Gi(i=1,2,3,4) over the others by the interval-valued intuitionistic trapezoidal fuzzy arithmetic averaging operator[20]:

{〈[0.564 1,0.64 1,0.718,0.782 1],([0.664,0.710 6],[0,0])〉},

{〈[0.384 6,0.461 5,0.538 5,0.615 4],(1,0)〉},

{〈[0.192 3,0.256 4,0.333 3,0.410 3],(1,0)〉},

{〈[0.410 3,0.487 2,0.564 1,0.641],([0.724 1,0.739 6],[0,0])〉}.

{〈[0.294 9,0.35 9,0.435 9,0.504 3],(1,0)〉},

{〈[0.615 4,0.693 2,0.769 3,0.833 3],([0.672,0.719 1],[0.164 5,0.195 6])〉},

,0.848 7],[0,0.130 9])〉},

{〈[0.333 3,0.410 2,0.487 2,0.564 1],(1,0)〉}.

{〈[0.512 8,0.589 7,0.666 7,0.743 6],([0.733 7,0.779],[0,0.214])〉},

{〈[0.410 3,0.487 2,0.564 1 ,0.641],([0.671 9,1],[0,0])〉},

{〈[0.384 6,0.461 5,0.538 5,0.606 8],([0.756 6,1],[0,0])〉},

{〈[0.239 3,0.307 7,0.384 6,0.461 5],([0.671 9,0.768 5],[0,0.184 6])〉}.

Step 3　For the alternatives Gi(i=1,2,3,4), by using the interval-valued intuitionistic trapezoidal fuzzy weighted arithmetic averaging operator[20] and the weight vector v={0.408 9,0.337 4,0.253 7}T, we aggregate all

(k=1,2,3) to obtain a collective Ii of the alternative Gi over the others:

Step 4　By ranking these IITFNs according to the method in Ref.[20], we obtain the rankings of these alternatives: G2≻G1≻G4≻G3.

It can be clearly seen that the rankings of the alternatives obtained by the two methods are the same, which demonstrates that our method is reasonable and effective. Furthermore, from the solution processes of the two methods, we can see that our method has some advantages, as shown below:

1) Our method can reduce the loss of information since it does not need a transformation of DHLEs into IITFNs and does not need to use the average interval-valued intuitionistic trapezoidal fuzzy information obtained by incorporating the preferences of the decision makers in one decision organization to represent the group’s preference; but uses the DHLEs to represent the group’s preferences directly, which is more intuitive and reasonable.

2) Our method provides a technique to objectively determine the weights of decision organizations, which is more rational.

5　Conclusion

A method is proposed to solve the group decision making problems in which the weights of decision organizations are unknown and the preferences for the alternatives are provided by double hesitant linguistic preference relations. In this method, the weights of decision organizations are determined objectively by using the standard deviation of scores of preferences provided by the individual decision organization and the correlation coefficient between the scores of preferences and those provided by the other decision organizations; then the preferences of decision organizations are aggregated by the double hesitant linguistic weighted averaging operator. Compared with the existing methods, the proposed method is much simpler, has less information loss and can deal with the group decision making problems with double hesitant linguistic preference relations in a more objective way.

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基于双重犹豫语言偏好关系的群决策方法

摘要：针对决策小组权重未知、方案的偏好信息以双重犹豫语言偏好关系形式给出的群决策问题, 提出了一种简单决策方法. 首先, 为准确、全面地描述群决策过程中的不确定评估信息, 定义双重犹豫语言数, 并根据定义的运算法则, 提出双重犹豫语言加权平均算子. 其次, 定义双重犹豫语言偏好关系, 并利用单个决策小组对方案偏好信息得分值的标准差和其偏好信息得分值与其他决策小组偏好信息得分值的相关系数, 提出一种客观确定决策小组权重的方法, 进而提出一种基于双重犹豫语言偏好关系的群决策方法. 同时, 通过九甸峡水库运行方案选择实例说明该方法的可行性和有效性. 最后, 将该方法与现有方法进行比较, 结果表明,所提出的方法能够直接处理双重犹豫语言偏好信息, 不需要进行信息转化, 从而可以减少原始决策信息的丢失.

关键词：群决策;双重犹豫语言数;双重犹豫语言偏好关系;双重犹豫语言加权平均算子

Biographies：Zhao Na (1988—), female, graduate; Xu Zeshui (corresponding < class="emphasis_italic">author

Foundation item：s：The National Natural Science Foundation of China (No.61273209, 71571123), the Scientific Research Foundation of Graduate School of Southeast University (No.YBJJ1527), the Scientific Innovation Research of College Graduates in Jiangsu Province (No. KYLX_0207).

Citation：Zhao Na, Xu Zeshui. A group decision making method based on double hesitant linguistic preference relations[J]．Journal of Southeast University (English Edition),2016,32(2):240-249．< class="emphasis_italic">doi

A group decision making methodbased on double hesitant linguistic preference relations

1 Preliminaries

2 Double Hesitant Linguistic Elements and Basic Operations

3 A Method to Group Decision Making with Double Hesitant Linguistic Preference Relations

4 Case Studies

5 Conclusion

基于双重犹豫语言偏好关系的群决策方法

1　Preliminaries

2　Double Hesitant Linguistic Elements and Basic Operations

3　A Method to Group Decision Making with Double Hesitant Linguistic Preference Relations

4　Case Studies

5　Conclusion