Clustering experts in linguistic environment: A hybrid method

2.1 The 2TL representation model

Let S = {s _i |i = 0, 1, 2, . . . , t} be a linguistic term set with odd cardinality. Any label s _i represents a possible value for a linguistic variable, and it should satisfy the following characteristics: (a) the set is ordered: s _i  > s _j , if i > j; (b) max operator: max(s _i , s _j ) = s _i , if s _i  ≥ s _j ; (c) min operator: min(s _i , s _j ) = s _i , if s _i  ≤ s _j ; (d) there is a negation operator: Neg (s _i ) = s _j , where j = t–i [13]. For example, S can be defined as S={s₀= extremely poor, s₁= very poor, s₂= poor, s₃= medium, s₄= good, s₅= very good, s₆= extremely good}.

A 2TL representation model based on the concept of symbolic translation is used to represent the linguistic evaluation information by means of a 2-tuple (s _i , α _i ). Where, s _i is a linguistic term in predefined linguistic term set S; α _i is the value of symbolic translation and α _i  ∈ [-0.5, 0.5).

Definition 1 [13]. Suppose β ∈ [0, t] is the aggregation result of a set of linguistic terms from S, i = round (β) and α = β - i are two values, where i ∈ [0, t] and α ∈ [-0.5, 0.5). α is the symbolic translation and the round (·) is the usual rounding operation.

Definition 2 [13]. Suppose β ∈ [0, t] is the aggregation result of a set of linguistic terms from S. Then, the 2-tuple, which indicates the equivalent information to β, could be obtained by the following function: Δ : [ 0 , t ] → S × [ - 0.5 , 0.5 ) ,(1) Δ ( β ) = { s i , i = round ( β ) α = β - i , α ∈ [ - 0.5 , 0.5 )(2)

Definition 3 [13]. Let S = {s _i |i = 0, 1, 2, . . . , t} be a linguistic term set and (s _i , α _i ) be a 2-tuple. There is a Δ^-1 function that restores the 2-tuple to its equivalent numerical value β ∈ [0, t] ⊂ R, where Δ - 1 : S × [ - 0.5 , 0.5 ) → [ 0 , t ] ,(3) Δ - 1 ( s i , α ) = i + α = β .(4)

It is obvious that the transformation of a linguistic term into a 2TL representation consists of adding a value zero as a symbolic translation, i.e., Δ^-1 (s _i , 0) = i.

Definition 4 [31]. Let x = {(s₁, α₁) , (s₂, α₂) , . . . . , (s _n , α _n )} be a set of 2-tuples. Then, the 2-tuple arithmetic mean is defined as ( s ¯ , α ¯ ) = TAM ( ( s 1 , a 1 ) , ( s 2 , a 2 ) , . . . , ( s n , a n ) ) = Δ ( 1 n ∑ j = 1 n Δ - 1 ( s j , a j ) ) , s ¯ ∈ S , α ¯ ∈ [ - 0.5 , 0.5 ) .(5)

Definition 5 [32]. Let (s _i , α _i ) and (s _j , α _j ) be two arbitrary 2-tuples derived from one linguistic term set. Then, the 2-tuple deviation value between (s _i , α _i ) and (s _j , α _j ) is defined as d ( ( s i , α i ) , ( s j , α j ) ) = Δ - 1 ( s i , α i ) - Δ - 1 ( s j , α j ) .(6)

According to Definition 5, the properties of the 2-tuple deviation measure can be inferred as follows:

Theorem 1 [11]. Let (s _i , α _i ) , (s _j , α _j ) , (s _k , α _k ) be three 2-tuples. Then,

d ((s _i , α _i ) , (s _j , α _j )) ∈ R. In particular, d ((s _i , α _i ) , (s _j , α _j )) =0 if Δ^-1 (s _i , α _i ) = Δ^-1 (s _j , α _j );

Based on the 2-tuple deviation measure, the 2-tuple covariance and the 2-tuple standard deviation are introduced respectively, which will lay the foundation for the optimization model under the interval-valued linguistic environment in Section 3.

Definition 6 [11]. Let B = (b _ij ) _m×n be the 2-tuple matrix, where b _ij  = (s _ij , α _ij ), s _ij  ∈ S and α _ij  ∈ [-0.5, 0.5). Then, Cov ( b j , b k ) = 1 m ∑ i = 1 m [ d ( b ij , b j ¯ ) × d ( b ik , b k ¯ ) ] ( j , k = 1 , 2 , . . . , n ) = 1 m ∑ i = 1 m [ d ( ( s ij , α ij ) , ( s j ¯ , α j ¯ ) ) × d ( ( s ik , α ik ) , ( s k ¯ , α k ¯ ) ) ] = 1 m ∑ i = 1 m [ ( Δ - 1 ( s ij , α ij ) - Δ - 1 ( Δ ( 1 m ∑ i = 1 m Δ - 1 ( s ij , α ij ) ) ) ) × ( Δ - 1 ( s ik , α ik ) - Δ - 1 ( Δ ( 1 m ∑ i = 1 m Δ - 1 ( s ik , α ik ) ) ) ) ] = 1 m ∑ i = 1 m [ ( β ij - 1 m ∑ i = 1 m β ij ) × ( β ik - 1 m ∑ i = 1 m β ik ) ] ,(7) where, b _j and b _k are two arbitrary columns of 2-tuples; Cov (b _j , b _k ) is the covariance of b _j and b _k ; ( s j ¯ , α j ¯ ) = Δ ( 1 m ∑ i = 1 m Δ - 1 ( s ij , α ij ) ) is the mean of 2-tuples in the jth column; d ( ( s ij , α ij ) , ( s j ¯ , α j ¯ ) ) is the deviation value of the 2-tuple in the ith row of the jth column and the mean of the 2-tuples in the jth column; b k ¯ and d ( b ik , b k ¯ ) are the similar meanings as b j ¯ and d ( b ij , b j ¯ ) , respectively.

Specifically, when j = k, the 2-tuple variances can be obtained as Definition 7.

Definition 7 [11]. Let B = (b _ij ) _m×n be the 2-tuple matrix, where b _ij  = (s _ij , α _ij ), s _ij  ∈ S and α _ij  ∈ [-0.5, 0.5). Then, δ j 2 = 1 m ∑ i = 1 m [ d ( b i j , b j ¯ ) ] 2 ( j = 1 , 2 , ... , n ) = 1 m ∑ i = 1 m [ d ( ( s i j , α i j ) , ( s j ¯ , α j ¯ ) ) ] 2 = 1 m ∑ i = 1 m [ Δ - 1 ( s i j , α i j ) − Δ − 1 ( Δ ( 1 m ∑ i = 1 m Δ - 1 ( s i j , α i j ) ) ) ] 2 = 1 m ∑ i = 1 m [ β i j − 1 m ∑ i = 1 m β i j ] 2 ,(8) where, δ j 2 and δ j are the variance and the standard deviation of the jth column of the 2-tuple matrix; the meanings of b j ¯ and d ( b ij , b j ¯ ) are the same as Definition 6.

Since the opinions of experts are largely restricted by some factors (e.g., their knowledge, environment, and working experience), the attribute descriptions provided by experts may fall between the two linguistic terms (linguistic intervals). To solve this problem, we introduce the concept of the IV2TL representation model [11, 15]. To ensure the aggregation results of these linguistic intervals could be explained without a loss of information, we assume that all the experts share the same linguistic term set.

Definition 8 [11, 15]. Let S = {s _i |i = 0, 1, 2, . . . , t} be a linguistic term set and [β _i , β _j ] be the aggregation result of a set of linguistic intervals, where β _i , β _j  ∈ [0, t] , β _i  < β _j . The interval-valued 2-tuple can be derived as: Δ ( [ β i , β j ] ) = [ ( s i , α i ) , ( s j , α j ) ] , with { s i , i = round ( β i ) s j , j = round ( β j ) α i = β i - i , α i ∈ [ - 0.5 , 0.5 ) α j = β j - j , α j ∈ [ - 0.5 , 0.5 )(9)

Definition 9 [11]. Let S = {s _i |i = 0, 1, 2, . . . , t} be a linguistic term set and [(s _i , α _i ) , (s _j , α _j )] be an interval-valued 2-tuple. There is a Δ^-1 function that restores the interval-valued 2-tuple to its equivalent interval value [β _i , β _j ] , β _i , β _j  ∈ [0, t] , β _i  < β _j , where Δ - 1 [ ( s i , α i ) , ( s j , α j ) ] = [ i + α i , j + α j ] = [ β i , β j ] .(10)

Similarly, a linguistic interval [s _i , s _j ] can be transformed into an IV2TL representation [(s _i , 0) , (s _j , 0)]. Besides, the 2-tuple could also be considered as a special interval-valued 2-tuple in the form of (s _i , α _i ) = [(s _i , α _i ) , (s _i , α _i )].

Example 1. Suppose an interval-valued 2-tuple assessed in S = {s₀, s₁, s₂, s₃, s₄, s₅, s₆} obtains its result β _i  = 3.1 and β _j  = 3.8. Then, the representation of this information with an interval-valued 2-tuple will be Δ ([3 . 1, 3 . 8]) = [(s₃, 0.1) , (s₄, - 0.2)] as shown in Fig. 1.

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Fig. 1

Example of an IV2TL representation model.

3.1 The LGDM problem under the interval-valued linguistic environment

In the LGDM problem, experts are invited to give their evaluations to the attributes of some alternatives. According to four features of the LGDM problem described in the introduction, suppose that E = {e₁, e₂, . . . , e _N } (N ≥ 20) is the expert set, X = {x₁, x₂, . . . , x _m } (m ≥ 2) is the alternative set, and U = {u₁, u₂, . . . , u _n } (n ≥ 2) is the attribute set, then each expert e _k  ∈ E should give the opinion to each attribute u _j  ∈ U belonging to each alternative x _i  ∈ X. Let B ˆ k = ( b ˜ ij k ) m × n be the IV2TL decision matrix of expert e _k , where b ˜ ij k represents the evaluation value of expert e _k towards the attribute u _j of the alternative x _i . We assume that the experts give the evaluation b ˜ ij k in form of the IV2TL representation model like [(s _i , α _i ) , (s _j , α _j )] derived from the linguistic term set S = {s _i |i = 0, 1, 2, . . . , t}. Then, the IV2TL decision information in the LGDM problem could be represented by the following matrix: B ˆ = [ B ˆ 1 ⋮ B ˆ N ] = ( [ b ˜ 11 1 ⋯ b ˜ 1 n 1 ⋮ ⋱ ⋮ b ˜ m 1 1 ⋯ b ˜ mn 1 ] ⋯ [ b ˜ 11 N ⋯ b ˜ 1 n N ⋮ ⋱ ⋮ b ˜ m 1 N ⋯ b ˜ mn N ] ) T

Thus, expert clustering in the LGDM problem under the interval-valued linguistic environment could be defined as the process of obtaining expert clusters based on the given decision matrix.

3.2 Framework of the hybrid method for clustering experts

Considering there is lack of mature standards to provide an exact number or a range for the proper threshold in the clustering issue, outside knowledge about experts (e.g., career, occupation, and age) can be regarded as the basis for pre-classification, and the attribute selection mainly depends on the actual decision-making issue. However, even if in the same group, there also exist some members with different decision attitudes. Based on this, the paper proposes a hybrid method for clustering experts in the LGDM problem combining outside knowledge and practical preference information. In other words, we firstly divide experts into several groups based on outside knowledge, and then allocate the individuals with different attitudes in each group into a more suitable group by means of the proposed method. Compared with traditional clustering methods, the method in this paper is applied under the precondition that partial information about experts is known. A framework of the hybrid method for clustering experts in the LGDM problem under the interval-valued linguistic environment is displayed in Fig. 2.

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Fig. 2

A framework of the hybrid method for clustering experts in the LGDM problem.

It can be seen that the hybrid method consists of four elements, that is, the IV2TL decision information, the optimization model, the two-stage MIP DEA-DA model (in what follows, this model is abbreviated as the DEA-DA model) to deal with a two-cluster issue, and iterative clustering to generate multiple expert clusters. Where, the optimization model developed in this study is used to transform the IV2TL decision information into information represented by 2 tuples (or finally numerical values); the DEA-DA model as a non-parametric approach proposed by Sueyoshi [33] incorporated the discriminant capabilities of DEA into DA (an important evaluation and clustering analysis method), in order to avoid misclassification and overlap in DA.

Suppose that there exist N experts in the LGDM problem under an interval-valued linguistic environment and the total samples G to be clustered consist of all the experts. If we divide all the experts into h (h ≥ 2) groups or sample groups according to their outside knowledge, suppose there exist N _g experts in each sample group G _g  (g = 1, 2, . . . , h), and ∑ g = 1 h N g = N . This paper adopts the concept of iterative clustering to transform a h-group clustering problem into h–1 two-group clustering ones, in order to overcome the limitations of the DEA-DA model: model performance largely depends on the sample pre-classification and the method is applied to solve clustering issues with two groups. As such, in Fig. 2, there are h–1 phases for the entire iterative clustering and each phase has its own DEA-DA model labelled by a serial number. If we take phase 2 as an example, the DEA-DA model (i.e., DEA-DA₂) will split the cluster G _g that was generated at phase 1 into another two clusters (G₂ and G _g  (g ≠ 1, 2)) until clustering results at this phase remain stable. Similarly, the new cluster G _g at a certain phase will be sent to the next phase to repeat the steps and the iterative process will end until h groups of experts are generated at phase h–1. Furthermore, an example is given as Fig. 3 to visually mimic the results of iterative clustering.

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Fig. 3

Visualization of iterative clustering.

Figure 3(a) represents the discriminant result obtained from the clustering of the original sample, which indicates the result achieved from the first iteration of the DEA-DA model based on the IV2TL decision information at the first phase. The two parallel lines in each iteration are regarded as the two hyperplanes determined in the HO model (the second stage of the DEA-DA model). It can be observed from this iterative process that the number of overlapping samples between the two sample groups would decrease gradually with a convergent trend as the iteration continues. The whole process will end until no overlapping areas exist.

To make a relatively comprehensive description of the evaluation information from all experts regarding different alternatives, this paper regards the attributes of each alternative as different evaluation objects. Thus, the number of attributes that all the experts evaluate increases from n to l = m × n. If we collect each line of B ˆ k ( k = 1 , 2 , . . . , N ) as one group of evaluation values of attributes, the vector B ˆ k ′ = ( b ˜ ij k ) 1 × ( m × n ) is obtained. Thus, the original IV2TL decision matrix B ˆ can be transformed into the new IV2TL decision matrix C ˆ = ( c ˜ kj ) N × l , where l = m × n. Then, the corresponding interval value [ β _ kj , β ¯ kj ] of c ˜ kj could be obtained by Equation (10). Basically, [ β _ kj , β ¯ kj ] can be regarded as the number set with the same probability in a certain scope, where β _ kj and β ¯ kj represent the two endpoints of the interval value. Then, [ β _ kj , β ¯ kj ] can be represented by a segment β kj ′ = μ β _ kj + ( 1 - μ ) β ¯ kj ( 0 ≤ μ ≤ 1 ) . From this point, when variable μ is assigned to a specific value, the interval value [ β _ kj , β ¯ kj ] will be transformed into a general real-number. In other words, the attribute value will be converted from one segment to one point. Also, the linguistic meaning of the general real number can be translated based on Definition 2. However, different values of μ will lead to various distributions of all attribute values, and further influence clustering results of the experts. Therefore, the quality of clustering results is determined by confirming the value of μ reasonably and effectively. This paper considers that the optimal value of μ is to make the distance of intragroup samples as small as possible (a preferable homogeneity is equipped) and the distance of intergroup samples as large as possible (a preferable diversity is equipped). Take three groups as an example to show the expected clustering results, when the optimal μ is determined (see Fig. 4). We desire the values of d₁, d₂ and d₃ are as large as possible and distributions of samples in G₁, G₂ and G₃ are relatively concentrated.

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Fig. 4

The expected clustering results.

To achieve this goal, variances are used to measure the deviation between samples and their sample means. There are numerous examples concerning the applications of variances [34–37]. Based on this, we use the concept of variance to establish the optimization model for clustering experts, which lays the foundation to construct the DEA-DA model under the interval-valued linguistic environment. If we divide all the experts into h (h ≥ 2) sample groups, there exist N _g experts in each sample group G _g  (g = 1, 2, . . . , h), and ∑ g = 1 h N g = N . Since the expected clustering results are to maximize the variances of intergroup samples and to minimize the variances of intragroup samples, multi-objective functions should be established.

Step 1. Establish the objective function f (μ) for the maximum intergroup variance.

Based on Equation (5), in sample group G _g , the mean c gj ′ of N _g experts’ evaluation values for attribute u _j is c gj ′ = TAM ( Δ ( μ β _ 1 j + ( 1 - μ ) β ¯ 1 j ) , Δ ( μ g β _ 2 j + ( 1 - μ ) β ¯ 2 j ) , . . . , Δ ( μ β _ N g j + ( 1 - μ ) β ¯ N g j ) ) = Δ ( 1 N g ∑ k = 1 N g ( μ β _ kj + ( 1 - μ ) β ¯ kj ) ) ( g = 1 , 2 , . . . , h ; j = 1 , 2 , . . . , l ) .(11)

According to Equations (5) and (8), the variance σ j 2 , which represents variances of all sample groups’ evaluation values of attribute u _j , is σ j 2 = 1 h ∑ g = 1 h [ d ( c gj ′ , c ¯ j ′ ) ] 2 ( j = 1 , 2 , . . . , l ) = 1 h ∑ g = 1 h [ d ( Δ ( 1 N g ∑ k = 1 N g ( μ β _ kj + ( 1 - μ ) β ¯ kj ) ) , TAM ( Δ ( 1 N 1 ∑ k = 1 N 1 ( μ β _ kj + ( 1 - μ ) β ¯ kj ) ) , Δ ( 1 N 2 ∑ k = 1 N 2 ( μ β _ kj + ( 1 - μ ) β ¯ kj ) ) , . . . , Δ ( 1 N h ∑ k = 1 N h ( μ β _ kj + ( 1 - μ ) β ¯ kj ) ) ) ) ] 2 = 1 h ∑ g = 1 h [ 1 N g ∑ k = 1 N g ( μ β _ kj + ( 1 - μ ) β ¯ kj ) - 1 h ∑ g = 1 h ( 1 N g ∑ k = 1 N g ( μ β _ kj + ( 1 - μ ) β ¯ kj ) ) ) ] 2 ,(12) where, c ¯ j ′ is the mean of evaluation values of all the sample groups.

As clustering experts in the LGDM problem is based on the experts’ evaluations of each attribute of all the alternatives, this paper regards the importance degree of all attributes as equal to determine the value of μ more objectively. In this case, the variance weights of each attribute are all assigned to one. Then, the objective function with the maximum intergroup variance could be represented by max f ( μ ) = ∑ j = 1 l σ j 2 .

Step 2. Establish the objective function f _g  (μ) for the minimum intragroup variance.

According to Equations (5) and (8), in the sample group G _g , the variance σ gj 2 of the N _g experts’ evaluations of attribute μ _j could be represented by σ gj 2 = 1 N g ∑ k = 1 N g [ d ( c kj ′ , c ¯ j ″ ) ] 2 ( j = 1 , 2 , . . . , l ) = 1 N g ∑ k = 1 N g [ d ( Δ ( μ β _ kj + ( 1 - μ ) β ¯ kj ) , TAM ( Δ ( μ β _ 1 j + ( 1 - μ ) β ¯ 1 j ) , Δ ( μ β _ 2 j + ( 1 - μ ) β ¯ 2 j ) , . . . , Δ ( μ β _ N g j + ( 1 - μ ) β ¯ N g j ) ) ] 2 = 1 N g ∑ k = 1 N g [ ( μ β _ kj + ( 1 - μ ) β ¯ kj ) - 1 N g ∑ k = 1 N g ( μ β _ kj + ( 1 - μ ) β ¯ kj ) ] 2 ,(13) where, c ¯ j ″ is the mean of N _g experts’ 2-tuple evaluation value c kj ′ of attribute u _j in the sample group G _g .

Similar to the disposal of variance weights of intergroup attributes, the objective function with the minimum intragroup variance could be represented by min f g ( μ ) = ∑ j = 1 l σ gj 2 , g = 1 , 2 , . . . , h .

Step 3. Based on the concept of linear weighting methods in the multi-objective programming, an optimization model is established as Equation (14). Notably, the objective function with the maximum intergroup variance max f (μ) should be transformed into min - f ( μ ) = - ∑ j = 1 n σ j 2 to ensure that the obtained value of the target function is the minimum. min ∑ g = 1 h λ g f g ( μ ) - λ h + 1 f ( μ ) s . t . 0 ≤ μ ≤ 1 , ∑ g = 1 h + 1 λ g = 1 , λ g ≥ 0 .(14)

If place the expressions f (μ) and f _g  (μ) into the above formula, then the formula could be further delivered as min ∑ g = 1 h ( λ g N g ∑ j = 1 l ∑ k = 1 N g [ ( μ β _ kj + ( 1 - μ ) β ¯ kj ) - 1 N g ∑ k = 1 N g ( μ β _ kj + ( 1 - μ ) β ¯ kj ) ] 2 ) - λ h + 1 h ∑ j = 1 l ∑ g = 1 h [ 1 N g ∑ k = 1 N g ( μ β _ kj + ( 1 - μ ) β ¯ kj ) - 1 h ∑ g = 1 h ( 1 N g ∑ k = 1 N g ( μ β _ kj + ( 1 - μ ) β ¯ kj ) ) ] 2 s . t . 0 ≤ μ ≤ 1 , ∑ g = 1 h + 1 λ g = 1 , λ g ≥ 0 ,(15) where, λ _g is the importance degree of each objective. The determination methods of their values include Delphi, the analytic hierarchy process and the αmethod. When we adopt the DEA-DA model to cluster experts, assume that the importance degree of each objective should be equal. In other words, the value of λ _g should be 1/(h + 1). Furthermore, the objective function can be finally simplified in form of a quadratic function with μ under the constraint μ ∈ [0, 1]. If the quadratic function opens upward, the variable μ^* only has one optimal solution; otherwise, one or two optimal solution(s) may exist.

The DEA-DA model (basically, the mixed integer linear programming) can produce a locally optimal solution for weight estimates of discriminant functions, further generating evaluation scores to determine group membership [33]. Suppose there are two sample groups G₁ and G₂ in G during phase g (g = 1, 2, . . . , h - 1), where there exist N₁ and N₂ experts, respectively and N₁ + N₂ ≤ N. We could obtain the DEA-DA model based on the general real-numbers β kj ′ = μ * β _ kj + ( 1 - μ * ) β ¯ kj ( 0 ≤ μ * ≤ 1 ) , where μ^* is calculated by Equation (15). A more detailed introduction for the DEA-DA model can refer to the study of Sueyoshi [33].

Stage 1. Build the classification and overlap identification (COI) model min s s . t . ∑ j = 1 l ( λ j + - λ j - ) [ μ * β _ kj + ( 1 - μ * ) β ¯ kj ] - d + s ≥ 0 , k ∈ G 1 , ∑ j = 1 l ( λ j + - λ j - ) [ μ * β _ kj + ( 1 - μ * ) β ¯ kj ] - d - s ≤ 0 , k ∈ G 2 , ∑ j = 1 l ( λ j + + λ j - ) = 1 , ζ j + ≥ λ j + ≥ ɛ ζ j + and ζ j - ≥ λ j - ≥ ɛ ζ j - , j = 1 , 2 , . . . , l , ζ j + + ζ j - ≤ 1 , j = 1 , 2 , . . . , l , ∑ j = 1 l ( ζ j + + ζ j - ) = l , d , s : unrestricted , ζ j + = 0 / 1 , ζ j - = 0 / 1 , and all other variables ≥ 0 ,(16) where, d is a discriminant score; s is a slack that represents an absolute distance between the discriminant score (d) and the discriminant function.

Assume λ j + * , λ j - * , d^* and s^* are the optimal solutions of the COI model, the corresponding discriminant rule is: if s^* < 0, then there is no overlap within sample groups; if s^* ≥ 0: (a) if ∑ j = 1 l ( λ j + * - λ j - * ) [ μ * β _ kj + ( 1 - μ * ) β ¯ kj ] > d * + s * , then sample k is allocated to G₁; (b) if ∑ j = 1 l ( λ j + * - λ j - * ) [ μ * β _ kj + ( 1 - μ * ) β ¯ kj ] < d * - s * , then sample k is allocated to G₂; (c) if d * - s * ≤ ∑ j = 1 l ( λ j + * - λ j - * ) [ μ * β _ kj + ( 1 - μ * ) β ¯ kj ] ≤ d * + s * , then sample k belongs to G₁ ∩ G₂. According to this rule, G can be divided into the following subsets:

R 1 = { k ∈ G 1 | ∑ j = 1 l ( λ j + * - λ j - * ) [ μ * β _ kj + ( 1 - μ * ) β ¯ kj ] > d * + s * } ,

Stage 2. Build the handling overlap (HO) model min ∑ k ∈ D 1 y k + ∑ k ∈ D 2 y k s . t . ∑ j = 1 l ( λ j + - λ j - ) [ μ * β _ kj + ( 1 - μ * ) β ¯ kj ] - c + My k ≥ 0 , k ∈ D 1 , ∑ j = 1 l ( λ j + - λ j - ) [ μ * β _ kj + ( 1 - μ * ) β ¯ kj ] - c - My k ≤ - ɛ , k ∈ D 2 , ∑ j = 1 l ( λ j + + λ j - ) = 1 , ζ j + ≥ λ j + ≥ ɛ ζ j + and ζ j - ≥ λ j - ≥ ɛ ζ j - , ζ j + + ζ j - ≤ 1 , j = 1 , 2 , . . . , l , ∑ j = 1 l ( ζ j + + ζ j - ) = p , c : unrestricted , ζ j + = 0 / 1 , ζ j - = 0 / 1 , y k = 0 / 1 and all other variables ≥ 0 ,(17) where, M is a given large number; y _k , a binary variable, is used to count the number of misclassified observations; c (k ∈ G₁) and c - ɛ (k ∈ G₂) are the discriminant scores. If the number of overlapping or misclassified samples exceeds the number of observations l, then p = l; otherwise, p is a positive integer satisfying p < l.

Assume λ j + * , λ j - * and c^* are the optimal solutions of the HO model, the corresponding discriminant rule is: (a) if ∑ j = 1 l ( λ j + * - λ j - * ) [ μ * β _ kj + ( 1 - μ * ) β ¯ kj ] ≥ c * , then sample k is allocated to G₁; (b) if ∑ j = 1 l ( λ j + * - λ j - * ) [ μ * β _ kj + ( 1 - μ * ) β ¯ kj ] ≤ c * - ɛ , then sample k is allocated to G₂.

Then, we classify the two sample groups G₁ and G₂ on a iterative basis, until samples belonging to G₁ and G₂ remain stable. Based on the framework presented in Fig. 2, the integrated process of the hybrid method is presented in Algorithm 1. Based on the clusters derived from the DEA-DA model at a given iteration, the principles to select the new group for clustering in the next iteration are: (a) if the two clusters are significantly unbalanced, the larger-size group will be handled further; (b) if the two clusters are nearly balanced, the group with more different group members pre-classified by outside knowledge would be selected.

4.1 Example with the hybrid method

To reasonably determine the salary level of each professor, the university authority decides to enact a concrete salary reform plan. The financial department at the university invites 30 experts e _k  (k = 1, 2, . . . , 30), including administrative staff, research and teaching staff as well as retirees, to evaluate three candidate alternatives x _i  (i = 1, 2, 3). Assume that each alternative has three attributes (research contribution u₁, teaching quality u₂ and service assurance u₃) and all the experts adopt the linguistic term set S= {s₀= extremely poor, s₁= very poor, s₂ =poor, s₃ =medium, s₄ =good, s₅ =very good, s₆ =extremely good} to evaluate each attribute of all alternatives. Note that the number of experts, alternatives or attributes in reality may be much larger than that listed in the example.

Step 1. According to the occupation of each expert (i.e., administrative staff, research and teaching staff, and retirees), we divided the 30 experts into three groups. Suppose e₁-e₁₀ belonged to Group 1, e₁₁-e₂₀ belonged to Group 2, and e₂₁-e₃₀ belonged to Group 3.

Step 2. We gathered the evaluation values given by the experts with the IV2TL representation model in Table 1.

Step 3. We selected each line of B ˆ k ( k = 1 , 2 , . . . , 30 ) as one group of evaluation values of attributes and transformed the matrix B ˆ k into a vector. Then, matrix B ˆ was transformed into the new IV2TL decision matrix C ˆ = ( c ˜ kj ) 30 × 9 in Table 2.

Step 4. Based on Table 2, we transformed the IV2TL decision matrix C ˆ into the corresponding interval value [ β _ kj , β ¯ kj ] of c ˜ kj by Equation (10). Then, the optimal solution μ^* of the optimization model could be obtained by Equation (15), that is, μ^* is equal to 0.731. Thus, the matrix C ˆ was finally converted into the matrix R = (r _kj ) _30×9 with general real-numbers by β kj ′ = 0.731 β _ kj + 0.269 β ¯ kj (see Table 3), which could be delivered to the DEA-DA model for the latter processing. The linguistic meaning of the number in Table 3 could be translated by means of Definition 2.

Step 5. We first divided all the experts into two sample groups, where samples from Group 1 made up G₁, and samples from Group 2 and Group 3 constituted G₂. From Fig. 2, if the experts intend to be clustered into h groups, h-1 phases should be processed. In term of the salary reform problem, two phases need to be completed to realize the expert clustering.

Step 6. For phase 1, we adopted the DEA-DA model to conduct a two-group clustering of G₁ and G₂ through iterative clustering based on the matrix R. From Fig. 3, the DEA-DA model will undergo an iteration process, which means that the elements in G₁ and G₂ should always change until the process converges. To sum up, G₁ and G₂ could be simply regarded as two containers whose elements would vary during the iteration. The results of the first phase of expert clustering are summarized in Tables 4 and 5.