Two-layer weight large group decision-making method based on multi-granularity attributes

2.1 2-tuple semantics and interval 2-tuple linguistic variables

Taking into account the ambiguity of decision-making information and human thinking, DMs are inclined to use language to express their preferences. At present, the most commonly used models for the information processing of language-based decision-making are the 2-tuple semantic and interval 2-tuple semantic models. The 2-tuple semantic model is represented by a 2-tuple (s,  α), where s represents a language phrase in a predetermined set of languages, and α represents a symbolic transfer value. In order to overcome the limitations of the 2-tuple semantic model proposed by Herrera, Tai and Chen proposed a multi-granularity generalized 2-tuple semantic model [29], presented below.

Definition 1. Let S ={ s₀,  s₁, …,  s _g  } be a set of linguistic terms, and β ∈ [0, 1] be the integrated operational results. The 2-tuple will be converted by the integrated transfer function Δ into the real value of β:

Δ : [ 0 , 1 ] → S × [ - 1 2 g , 1 2 g ) Δ ( β ) = ( s i , α ) , with { s i , i = round ( β · g ) α = β - i g , α ∈ [ - 1 2 g , 1 2 g )(1)

Where round assigns to β the integer number i that is closest to β.

Remark 1. Δ is an objective function and Δ - 1 ( s i , α ) = i g + α = β .

Remark 2. The conversion of the linguistic term into a linguistic 2-tuple consists of adding the value 0 as a symbolic translation, s _i  ∈ S ⇒ (s _i ,  0).

Definition 2. Let S ={ s₀,  s₁, …,  s _g  } be a set of linguistic terms, then [(s _k ,  α _k ) ,   (s _l ,  α _l )] represents an interval of 2-tuple semantic variables, where (s _k ,  α _k ) ≤ (s _l ,  α _l ). The interval 2-tuple will be converted by the integrated transfer function Δ into an interval value of [β₁,  β₂] (β₁,  β₂ ∈ [0, 1] ,  β₁ ≤ β₂):

Δ [ β 1 , β 2 ] = [ ( s k , α k ) , ( s l , α l ) ] , with { s k , k = round ( β 1 · g ) s l , l = round ( β 2 · g ) α k = β 1 - k g , α k ∈ [ - 1 2 g , 1 2 g ) α l = β 2 - l g , α l ∈ [ - 1 2 g , 1 2 g )(2)

The objective function Δ^-1 is defined as. Δ - 1 [ ( s k , α k ) , ( s l , α l ) ] = [ k g + α k , l g + α l ]

In particular, if s _k  = s _l and α _k  = α _l , then the interval 2-tuple variables are equivalent to the 2-tuple semantic variables.

The consistency of the DM’s preference is important in large group decision-making. The decision preference is clustered to simplify the decision-making process, and the clusters’ consensus has an important influence on the choice of the scheme. In this paper, an extension of the possible degree formula is used to measure the consistency of the interval preference matrix.

Definition 3. [30] Let A = [a^-,  a⁺] and B = [b^-,  b⁺] be two intervals. The possible degree of the interval A relative to the interval B is defined as:

P ( A ≥ B ) = min { 1 , max { a + - b - ( a + - a - ) + ( b + - b - ) , 0 } }(3)

Definition 4. [31]: Let A = [a^-,  a⁺] and B = [b^-,  b⁺] be two intervals. The diversity degree of the interval A relative to the interval B is defined as: D AB = | P ( A ≥ B ) - P ( B ≥ A ) |(4)

Based on the diversity degree of any two intervals, the similarity degree of any two vectors is presented and the consensus measure for interval preference relations in group decision-making is discussed.

Let θ⁽ⁱ⁾ stand for the ith row vector of the interval preference matrix A = (a _ij ) _m×n i.e., θ⁽ⁱ⁾ = (a_i1,  a_i2, …,  a _in ), which can be denoted as θ _ ( i ) = min { a ij | a ij ∈ θ ( i ) } θ ¯ ( i ) = max { a ij | a ij ∈ θ ( i ) }

Definition 5. Let θ⁽ⁱ⁾ and θ ^j be two vectors of the interval preference matrix A = (a _ij ) _m×n. The dominance degree between θ⁽ⁱ⁾ and θ^(j) is defined as: P ( θ ( i ) ≥ θ ( j ) ) = min { 1 , max { θ ¯ ( i ) - θ _ ( j ) ( θ ¯ ( i ) - θ _ ( i ) ) + ( θ ¯ ( j ) - θ _ ( j ) ) , 0 } }(5)

According to the properties of the possible degree, it can be shown that:

0 ≤ P_{(θ⁽ⁱ⁾≥θ^(j))} ≤ 1 . when θ ¯ ( i ) = θ _ ( j ) , P ( θ ( i ) ≥ θ ( j ) ) = 0 , θ ¯ ( j ) = θ _ ( i ) , P ( θ ( i ) ≥ θ ( j ) ) = 1

Definition 6. Let θ⁽ⁱ⁾ and θ^(j) be two vectors of the interval preference matrix A = (a _ij ) _m×n. The similarity degree between θ⁽ⁱ⁾ and θ ^j is defined as: ρ ( θ ( i ) , θ ( j ) ) = 1 - | P ( θ ( i ) ≥ θ ( j ) ) - P ( θ ( j ) ≥ θ ( i ) ) |(6)

The smaller the value of ρ (θ⁽ⁱ⁾,  θ^(j)), the smaller the similarity degree of the interval preference vectors θ⁽ⁱ⁾ and θ^(j) of the interval preference matrix A = (a _ij ) _m×n.

Similarly, let A = (a _ij ) _m×n and B = (b _ij ) _m×n be two interval preference matrices, and θ⁽ⁱ⁾ = (a_i1,  a_i2, …,  a _in ) and ϑ⁽ⁱ⁾ = (b_i1,  b_i2, …,  b _in ) be the ith vector of the interval preference matrices A and B, respectively. The similarity degree between A and B is then defined as: ρ ( A , B ) = 1 m ∑ i = 1 m ρ ( θ ( i ) , ϑ ( i ) )(7)

Where 0 ≤ ρ (A,  B) ≤ 1, and ρ (A,  B) = ρ (B,  A).

The weight of the DM is determined by the reliability and certainty of the information. The more vague and uncertain the information is, the lower the degree of understanding of the problem and the smaller the weight of the DM. In order to measure the fuzzy degree of the decision-making information, the fuzzy entropy and fuzzy relative entropy are introduced.

Definition 7. [32] Let X ={ x₁,  x₂, …,  x _n  } be an infinite set. A ={ 〈 x,  u _A  (x) ,  v _A  (x) 〉 |x ∈ X  } is an intuitionistic fuzzy set. The fuzzy entropy of A is defined as: E ( A ) = min [ u A ( x ) , v A ( x ) ] + π A ( x ) max [ u A ( x ) , v A ( x ) ] + π A ( x )(8)

Where u _A  (x) and v _A  (x) are the membership and non-membership functions of the fuzzy set A respectively. u _A  (x) ,  v _A  (x) ∈ [0, 1]. π _A  (x) = 1 - u _A  (x) - v _A  (x) represents the hesitation or uncertainty of the fuzzy set A.

Property 1. 0 ≤ E (A) ≤ 1

E (A) = 0, when and only when A is a non-fuzzy set.

Definition 8. [33] Let X ={ x₁,  x₂, …,  x _n  } be an infinite set. The intuitionistic fuzzy relative entropy between B and A is defined as:

I ( A , B ) = A ( x i ) log 2 A ( x i ) 1 2 [ A ( x i ) + B ( x i ) ] + ( 1 - A ( x i ) ) log 2 1 - A ( x i ) 1 - 1 2 [ A ( x i ) + B ( x i ) ](9)

Where A ( x i ) = u A ( x ) + 1 - v A ( x ) 2 and B ( x i ) = u B ( x ) + 1 - v B ( x ) 2 . Since the relative entropy I (A,  B) is non-symmetric, the relative entropy distance between the two intuitionistic fuzzy sets A and B is defined as D (A,  B) = I (A,  B) + I (B,  A), which satisfies 0 ≤ D (A,  B) ≤ 1 and D (A,  B) = D (B,  A).

3.1 Description of the problem

Let E ={ e₁,  e₂, …,  e _M  } be a group of DMs, X ={ x₁,  x₂, …,  x _P  } be a discrete set of P feasible alternatives, U ={ u₁,  u₂, …,  u _N  } be a finite set of attributes, and W = (w₁,  w₂, …,  w _N )  ^T be the weight vector of the attributes, where w j ≥ 0 , ∑ j = 1 N w j = 1 . The DM e _i  ∈ E compares these alternatives with respect to all the attributes U of the linguistic terms in the set S ={ s₀,  s₁, …,  s _g  }. All DMs use language value, interval linguistic value, or the empty set to express their preferences for the attributes of alternatives. The DMs construct the original preference matrix A i = ( a lj i ) P × N , 1 ≤ i ≤ M,  1 ≤ l ≤ P,  1 ≤ j ≤ N where the value of element a lj i represents the ith DM’s preference degree for alternative x _i  ∈ X with respect to the attribute u _j  ∈ U. The interval 2-tuple semantic model is used to transform the initial preference matrix into the interval 2-tuple linguistic preference matrix B ˜ i = ( b lj i ) P × N . Equation (2) is used to transform the interval 2-tuple linguistic preference matrix into a matrix of interval preferences V i = ( v lj i ) P × N .

In order to simplify the process of large group decision-making, the preference matrices of the DMs are clustered [34]. M DMs can be divided into K clusters, where C = { C¹,  C², …,  C ^K  } (K ≥ 1) is the cluster set, and the cluster C ^k  (k = 1,  2 … ,  K) consists of n _k DMs, where ∑ k = 1 K n k = M .

3.2 The solution of the two-layer weight

Although the DMs in the same cluster are similar, they are not identical; therefore the contribution of each DM is different. In order to reach a consensus, the DM’s weight is based on the degree of clusters. In addition, the weight of different preferences is based on the number of internal DMs, and also incorporates the effectiveness of the information provided by the various clusters. The solution of the two-layer weight and related definitions are presented below.

Definition 8. Let V kn η = ( v lj kn η ) P × N , V kn τ = ( v lj kn τ ) P × N be any two DMs’ interval preference matrix of C ^k . The consistency degree of the two DMs’ preference matrices is defined as follows: ρ ( V kn η , V kn τ ) = 1 P ∑ l = 1 P ρ ( θ ( l ) , ϑ ( l ) )(10)

Since the contribution of each DM in the same cluster is different, the weight of each decision maker λ _{kn k} is obtained from the degree of consistency among different DMs: λ kn k = ρ ( V kn η , V kn τ ) ∑ n η = 1 n k ∑ n τ = 1 n k ρ ( V kn η , V kn τ )(11)

From each DM’s weight λ _{kn k}  = (λ_k1,  λ_k2, …,  λ _{kn k} )  ^T in the cluster, the preference matrix G ^k of the cluster C ^k is calculated as follows: G k = ∑ t = 1 n k λ kn k V kn k(12)

The quantity and quality of information obtained in the decision-making process determine the accuracy and reliability of the decision. In order to express the quality of the clusters’ information, the interval-valued intuitionistic fuzzy sets are first transformed to the intuitionistic fuzzy sets, and the weights are then determined by fuzzy entropy.

In order to protect the DMs’ privacy and avoid psychic contagion, Yu and colleagues [35–38] explored aggregation methods for hesitant fuzzy elements and interval-valued multiplicative intuitionistic fuzzy methods for personnel evaluation, and used them to evaluate network information system security. According to [39], the intuitionistic fuzzy number and interval number can be transformed into each other, which avoids the loss of useful information and makes the decision more obvious. When the interval value [ y ij - , y ij + ] is transformed into the intuitionistic fuzzy number r ˜ ij = ( u ij , v ij , π ij ) , then u ij = y ij - , v ij = 1 - y ij + , and π ij = y ij + - y ij - [32]. The clusters’ preference matrix G ^k can be transformed into the intuitionistic fuzzy preference matrix R k = ( r lj k ) P × N . The weight of the DM is determined by the reliability and certainty of the information. The more uncertain the information is, the lower the level of understanding of the problem, and the smaller the weight. The fuzziness and uncertainty of the information can be used to measure the fuzzy entropy. The weights can be acquired from the following equations: ω k = 1 - H k K - ∑ k = 1 K H k(13) and H k = ∑ j = 1 N w j [ 1 P ∑ l = 1 P E ( r lj k ) ] .

In summary, the decision-making method based on multi-granularity attributes of the two-layer weight is as follows:

Step 1. The DMs’ initial preference matrix A i = ( a lj i ) P × N is converted to the interval 2-tuple semantic decision matrix B ˜ i = ( b lj i ) P × N . S = { s 0 = verypoor , s 1 = poor , s 2 = medium , s 3 = good , s 4 = very } is the linguistic term the DMs use to provide their evaluation, which is then transformed into an interval 2-tuple semantic preference. The preference can be divided into one of three possibilities:

The preference information given by the DM is the language value s₁, which can be transformed into [(s₁,  0) ,   (s₁,  0)].

Step 2. Through the function Δ^-1, the interval 2-tuple decision matrix B i ∼ = ( b lj i ) P × N is converted into the interval value preference matrix V i = ( v lj i ) P × N .

Step 3. Using the method of [31], the preference matrix of the DM is clustered. Then using Equation (11), the weight of each DM in the cluster is obtained.

Step 4. The interval preference matrix G ^k of the cluster C ^k is transformed into the intuitionistic fuzzy preference matrix R k = ( r lj k ) P × N .

Step 5. Relative entropy is used to determine the clusters’ weights ω k = 1 - H k K - ∑ k = 1 K H k .

Step 6. Comprehensive ranking of the scheme.

➀ Positive and negative ideal solutions for each cluster are determined: R + = { 〈 u j , r j k + 〉 | u j ∈ U } = { 〈 u j , [ max l u lj k , min l v lj k ] 〉 | u j ∈ U } R - = { 〈 u j , r j k - 〉 | u j ∈ U } = { 〈 u j , [ min l u lj k , max l v lj k ] 〉 | u j ∈ U }

➁ The fuzzy relative entropy distance of the positive and negative ideal solutions are calculated, and the distance of the weighted fuzzy relative entropy is calculated: D l + = ∑ j = 1 N w j D ( r lj k , r lj k + ) D l - = ∑ j = 1 N w j D ( r lj k , r lj k - )

➂ Information about the decision-making is gathered:

D l + = ∑ k = 1 K ω k D l k + , D l - = ∑ k = 1 K ω k D l k -

➃ The closeness degree of each alternative is calculated, and the alternatives are sorted.

C l = D l - D l - + D l +

If D l - = 0 , C _l  = 0, and if D l + = 0 , C _l  = 1. The larger C _l is, the smaller the distance between x _l and the ideal solution, and the better the solution.

4.1 Numerical example

The proposed method is used to choose a rescue scheme for a large-scale flooding accident. In 2015, a coalmine flooding accident occurred in which there were more than 20 people trapped underground. In order to rescue the personnel, the relevant departments organized a decision group consisting of 20 experts E ={ e₁,  e₂, …,  e₂₀ } from the rescue group, hydrological analysis group, and the security measures expert group to assess the scene of the accident. Three emergency plans were initially developed based on the existing contingency plans: x₁ enables local blasting and mining machine rescue; x₂ uses a motor pump for drainage and rescue; x₃ enables local blasting and sending fire officers and soldiers to rescue the personnel. Three attributes are considered: the rescue cost, the rescue time, and the safety of the personnel. The attribute weight vector is W = (0.2,  0.3,  0.5)  ^T . Different language sets are given to the three attributes:

S 1 = { s 0 = extremelylow , s 1 = verylow , s 2 = low , s 3 = slightlylow , s 4 = medium , s 5 = slightlyhigh , s 6 = high , s 7 = veryhigh , s 8 = extremelyhigh } S 2 = { s 0 = veryshort , s 1 = short , s 2 = medium , s 3 = long , s 4 = verylong } S 3 = { s 0 = verypoor , s 1 = poor , s 2 = slightlypoor , s 3 = medium , s 4 = slightlygood , s 5 = good , s 6 = verygood }

It is assumed that there is no conflict of interest among the experts involved in the decision-making, and they are not personally involved in the problem itself. According to the different linguistic sets of the three attributes, the DMs give solution preference values according to their judgment of the problem, and the initial preference matrices of the DMs is shown in Table 1.

Step 1. The initial preference matrix of each DM is directly converted to the interval preference matrix using Definition 2, and is shown in Table 2.

Step 2. Using the method of [31], the DMs’ preferences are clustered, and the six clusters are shown in Table 3. Equation (11) is used to determine the weight of each member, which is shown in Table 4.

Step 3. After determining the weight of each member in the cluster, the clusters’ preference information is determined and is shown in Table 5. The result of the clusters’ interval preference information is converted to fuzzy preference information, which is shown in Table 6.

Step 4. Using Equation (13) the weight of the clusters is determined to be: ω 1 = 0.1467 , ω 2 = 0.1338 , ω 3 = 0.1954 ω 4 = 0.2315 , ω 5 = 0.1463 , ω 6 = 0.1463

Step 5. First, the positive and negative ideal solutions are obtained for each cluster, as shown in Table 7. The relative entropy distance between the positive and negative ideal solutions is then obtained. The results are shown in Table 8.

Finally, the closeness degree of each alternative is calculated and the results are sorted, as shown in Table 9. The results indicate that the preferred order of the problem solution is x₂ ≻ x₁ ≻ x₃, and the motor pump should be used for drainage and rescue.

In order to validate this method, the results are compared with the results of Xu and Li [34], which assumed the weight of each DM in the cluster is the same. At the same time, according to the majority principle, the clusters’ weights are proportional to the number of the internal DMs, so the weights of the DMs are as follows: λ ki = n k ∑ k = 1 K n k 2

λ _ki represents the Kth cluster’s weight of each DM. Therefore, the weight of each gathering decision member is shown in Table 10.

According to the weights of the DMs, the preference matrix of the decision-making group can be obtained:

G = [ [ 0.3924 , 0.7907 ] [ 0.5029 , 0.8081 ] [ 0.1142 , 0.7315 ] [ 0.3852 , 0.6192 ] [ 0.6657 , 0.9448 ] [ 0.5931 , 0.8544 ] [ 0.7151 , 0.9259 ] [ 0.5436 , 0.9157 ] [ 0.2790 , 0.5730 ] ]

The intuitionistic fuzzy preference matrix of the decision-making group is found to be: R = [ [ 0.3924 , 0.2093 ] [ 0.5029 , 0.1919 ] [ 0.1142 , 0.2685 ] [ 0.3852 , 0.3808 ] [ 0.6657 , 0.0052 ] [ 0.5931 , 0.1456 ] [ 0.7151 , 0.0741 ] [ 0.5436 , 0.0843 ] [ 0.2790 , 0.4267 ] ]

The closeness degree of each solution alternative and the sorted results are shown in Table 11.

The order of the scheme from both methods is identical, which indicates that the proposed method is effective. Although the results are generally consistent between the two methods, the closeness degree results are slightly different because the weights of the DMs were calculated slightly differently in the two methods. It is not realistic to assume the weight of all DMs in the same cluster are identical, because their decision preference may be similar but not identical. Similarly, the clusters’ weight should be based on the majority principle as well as the decision information.