A novel uncertain information modeling method based on cosine similarity and cross entropy under spherical uncertain linguistic fuzzy set

1.1 Motivation and Contribution

The method proposed is motivated by five attractive branches of the literature, that is, SFSs, ULSs, fuzzy information entropy, cosine similarity measure and fuzzy information integration operators. Their combination results in novel techniques, named SULSs, spherical uncertain linguistic fuzzy cosine similarity measure, spherical uncertain linguistic fuzzy cross entropy and spherical uncertain linguistic fuzzy weight operators, which allow us to perform appropriate evaluations of decision in MAGDM. The main motivations of this paper can be described as follows:

In the evaluation of decision information, it is expected to capture more information. In the expression of assessment information, a novel concept, named spherical uncertain linguistic fuzzy set which can capture the subjective and objective information by expressing the satisfaction, abstinence and dissatisfaction nature of human decisions, is proposed.

The main contributions of the proposed method are as follows.

Modeling expert evaluation information using SULSs. This is because SULSs not only relax the limitations of experts expressing uncertain and fuzzy evaluation information but also integrate the expression ability of quantitative and qualitative information, making them suitable for more practical analysis scenarios.

1.2 Novelty

This paper introduces a new MAGDM method, in which the proposed SULS uses multiple scenarios to capture uncertain information, which can solve scenarios that existing uncertain information modeling tools cannot capture. Combining the inherent characteristics of integrating multiple experts and attributes in MAGDM problems, CSM and CEM in a spherical uncertain semantic environment are proposed to objectively obtain expert and attribute weights. The proposed method is used simultaneously for MAGDM problems with unknown and incompletely known attribute weights, which are not executable in many existing methods.

The rest of this paper is shown as follows: Section 2 reviews some basic concepts and proposes some new concepts by correcting the operator laws of the SFSs. In Section 3 and Section 4, we propose four SULHM operators and the MAGDM approach based on the proposed operators, respectively. Section 5 conducts a numerical experiment and some comparative analyses to illustrate the validity of the proposed MAGDM approach, and Section 6 summarizes this paper.

2.1 Spherical fuzzy sets and uncertain linguistic sets

Definition 1 [9] Let X be an ordinary fixed set, a spherical fuzzy set A defined on X is given by A = { 〈 x , μ A ( x ) , η A ( x ) , ν A ( x ) 〉 | x ∈ X }(1)

where μ _A (x), η _A (x) and ν _A (x) represent the positive membership degree, the neutral membership degree, and the negative membership degree respectively, satisfying the condition that 0≤μ _A (x), η _A (x), ν _A (x)≤1 and 0≤μ _A (x)²+η _A (x)²+ν _A (x)²≤1. The refusal degree of A is expressed as π _A (x) = (1-μ _A (x)²-η _A (x)²-ν _A (x)²)^(1/2). For convenience, (μ _A (x), η _A (x), ν _A (x)) is called a spherical fuzzy number (SFN), which can be simply denoted by α = (μ, η, ν).

Definition 2 [41] Suppose α₁=(μ₁, η₁, ν₁), α₂=(μ₂, η₂, ν₂) and α=(μ, η, ν) are three SFNs, and λ is a positive real number, then: ( 1 ) α 1 ⊕ α 2 = 〈 1 - ( 1 - μ 1 2 ) ( 1 - μ 2 2 ) , η 1 η 2 , ( η 1 2 + ν 1 2 ) ( η 2 2 + ν 2 2 ) - η 1 2 η 2 2 〉 , ( 2 ) α 1 ⊗ α 2 = 〈 ( μ 1 2 + η 1 2 ) ( μ 2 2 + η 2 2 ) - η 1 2 η 2 2 , η 1 η 2 1 - ( 1 - ν 1 2 ) ( 1 - ν 2 2 ) 〉 , ( 3 ) λ α = 〈 1 - ( 1 - μ 2 ) λ , η λ , ( η 2 + ν 2 ) λ - ( η 2 ) λ 〉 , ( 4 ) α λ = 〈 ( μ 2 + η 2 ) λ - ( η 2 ) λ , η λ , 1 - ( 1 - ν 2 ) λ 〉 .

Definition 4 [12] Let S ={ s _i |i = 0, 1, …, l - 1 } is a discrete linguistic term set, which denotes an evaluation value of the linguistic variable. For any two linguistic values s _i and s _k , the following characters should be satisfied:

If i > k, then s _i  > s _k ,

In general, l can be set to 5, 7, 9, etc. For example, when l = 7, a set S can be represented as follows: S = { s 0 , s 1 , s 2 , s 3 , s 4 , s 5 , s 6 } = { terrible , bad , poor , neutral , good , well , excellent }

However, the assessments of attributes cannot be described by a single LV accurately. Xu [13] proposed the concept of the uncertain linguistic number to save all given information.

Definition 5 [13] Let s = [s _a , s _b ] ,  s _a , s _b  ∈ S,   and 0≤a≤b_, S ={ s _i |s₀ ⩽ s _i  ⩽ s_l-1 i = ∈ [0,  l - 1] } is a continuous term set, s _a and s _b are lower limit and upper limit of S, then the LS is called ULS.

Definition 6 [13]. Let s₁ = [s _{θ 1} , s _{τ 1} ], s₂ = [s _{θ 2} , s _{τ 2} ] and s = [s _θ , s _τ ] are three ULNs, and λ is a positive real number, then:

s₁ ⊕ s₂ = [s_θ1+θ2, s_τ1+τ2] ,

It is obvious that the SFS only considers DMs’ quantitative information and loses sight of qualitative information and the ULS only expresses DMs’ qualitative information and loses sight of quantitative information. However, DMs usually express their evaluations both quantitatively and qualitatively. Moreover, in actual MAGDM problems, quantitative information and qualitative information should be taken into account to ensure the integrity of information. So the IULSs [14] and PFULSs [15] are developed whereas they inherit the drawback of IFSs and PFSs. To overcome this limitation, we propose the concept of SULS to more roundly express DMs’ information.

Definition 7. Let X be an ordinary fixed set, an SULS defined on X can be given as A = { 〈 [ s θ ( x ) , s τ ( x ) ] , ( μ A ( x ) , η A ( x ) , ν A ( x ) ) 〉 | x ∈ X }(2) where [s _θ (x),sτ(x)]∈S is a ULN, and μ _A (x), η _A (x) and ν _A (x) represent the positive membership degree, the neutral membership degree, and the negative membership degree, satisfying the conditions that 0 ≤ μ _A  (x) , η _A  (x) , ν _A  (x) ≤1 and 0 ≤ μ _A  (x) ² + η _A  (x) ² + ν _A  (x) ² ≤ 1. Then for each x ∈ X, the refusal degree of x to [s_θ(x), s_τ(x)] is π _A  (x) = (1 - μ _A  (x) ² - η _A  (x) ² - ν _A  (x) ²) ^(1/2). For convenience, we call α =< [s_θ(x), s_τ(x)] , (μ _A  (x) , η _A  (x) , ν _A  (x)) >a spherical uncertain linguistic number (SULN).

Based on the operational rules of SFNs and ULNs, we propose the operational rules of SULNs.

Definition 8. Suppose α₁ = 〈 [s _{θ 1} , s _{τ 1} ] , (μ₁, η₁, ν₁) 〉, α₂ = 〈 [s _{θ 2} , s _{τ 2} ] , (μ₂, η₂, ν₂) 〉 and α = 〈 [s _θ , s _τ ] , (μ, η, ν) 〉 are three SULNs, and λ is a positive real number, then: ( 1 ) α 1 ⊕ α 2 = 〈 [ s θ 1 + θ 2 , s τ 1 + τ 2 ] , ( 1 - ( 1 - μ 1 2 ) ( 1 - μ 2 2 ) , η 1 η 2 , ( η 1 2 + ν 1 2 ) ( η 2 2 + ν 2 2 ) - η 1 2 η 2 2 ) 〉 , ( 2 ) α 1 ⊗ α 2 = 〈 [ s θ 1 × θ 2 , s τ 1 × τ 2 ] , ( ( μ 1 2 + η 1 2 ) ( μ 2 2 + η 2 2 ) - η 1 2 η 2 2 , η 1 η 2 1 - ( 1 - ν 1 2 ) ( 1 - ν 2 2 ) ) 〉 , ( 3 ) λ α = 〈 [ s λ × θ , s λ × τ ] , ( 1 - ( 1 - μ 2 ) λ , η λ , ( η 2 + ν 2 ) λ - ( η 2 ) λ ) 〉 , ( 4 ) α λ = 〈 [ s θ λ , s τ λ ] , ( ( μ 2 + η 2 ) λ - ( η 2 ) λ , η λ , 1 - ( 1 - ν 2 ) λ ) 〉 .

Theorem 1. LetS = {s _i |s₀ ⩽ s _i  ⩽ s_l-1, i ∈ [0, l - 1]} is a continuous term set based on S, and Ω = {a =  〈 [s _θ , s _τ ] , (μ, η, ν) 〉 |s _θ , s _τ  ∈ S, μ, η, ν ∈ [0, 1] , 0 ⩽ μ² + η² + ν² ⩽ 1} is the set of all SULNs based on S. We suppose a, b ∈ Ω and λ is a positive real number, then new operations of SULNs are closed.

a ⊕ b ∈ Ω

The proof of Theorem 1 is provided in Appendix .

Theorem 2. Let α₁ = 〈 [s _{θ 1} , s _{τ 1} ] , (μ₁, η₁, ν₁) 〉 and α₂ = 〈 [s _{θ 2} , s _{τ 2} ] , (μ₂, η₂, ν₂) 〉 are two SULNs, λ is a positive real number. Then:

α₁ ⊕ α₂ = α₂ ⊕ α₁

Definition 9. Let α =〈 [s _θ , s _τ ] , (μ, η, ν) 〉 is a SULN, then the expected function α can be defined as: E ( α ) = θ + τ 4 ( μ 2 - ν 2 + 2 )(3)

Definition 10. Let α =〈 [s _θ , s _τ ] , (μ, η, ν) 〉 is a SULN, then the accuracy function α can be defined as: H ( α ) = θ + τ 2 ( μ 2 + η 2 + ν 2 )(4)

Definition 11. Let α₁ =〈 [s _{θ 1} , s _{τ 1} ] , (μ₁, η₁, ν₁) 〉 and α₂ =〈 [s _{θ 2} , s _{τ 2} ] , (μ₂, η₂, ν₂) 〉 are two SULNs, E (α₁) and E (α₂) are expected function, H (α₁) and H (α₂) are accuracy function, then:

if E (α₁) > E (α₂), then α₁ ≻ α₂,

if E (α₁) = E (α₂), then:

  if H (α₁) > H (α₂), then α₁ ≻ α₂,

  if H (α₁) = H (α₂), then α₁ = α₂.

CEM and CSM play crucial roles in the measurement of ambiguous information and are widely used in different fields. Ye and Liu et al. proposed the CSM of the IFSs and IFLSs, respectively [22]. Ye and Li derived the CEM of the IVIFSs and PyFs, respectively [27, 40]. Based on the inspiration from Zhao et al., [43] and Sahin [44], we derive the CEM and CSM of SULNs.

Definition 12. Let α₁ = 〈 [s _{θ 1} , s _{τ 1} ] , (μ₁, η₁, ν₁) 〉, α₂ = 〈 [s _{θ 2} , s _{τ 2} ] , (μ₂, η₂, ν₂) 〉 are two SULNs, l + 1 is the cardinality of linguistic term set, then the CEM of α₁ and α₂ is defined as: CE * ( α 1 , α 2 ) = θ 1 + τ 1 2 l ln 2 ( θ 1 + τ 1 ) z + τ 1 - θ 1 l ln 2 ( θ 1 - τ 1 ) ( θ 1 - τ 1 ) + ( θ 2 - τ 2 ) + ( 1 - θ 1 + τ 1 2 l ) ln 4 l - 2 ( θ 1 + τ 1 ) 4 l - z + ( 1 + θ 1 - τ 1 l ) ln 2 ( l + ( θ 1 - τ 1 ) ) 2 l + ( θ 1 - τ 1 ) + ( θ 2 - τ 2 ) + ∑ x ∈ μ , η , ν ( x 1 2 ln x 1 2 1 2 ( x 1 2 + x 2 2 ) + ( 1 - x 1 2 ) ln 1 - x 1 2 1 - 1 2 ( x 1 2 + x 2 2 ) )(5) where z = θ₁ + τ₁ + θ₂ + τ₂.

Theorem 3. Let α₁ = 〈 [s _{θ 1} , s _{τ 1} ] , (μ₁, η₁, ν₁) 〉 and α₂ = 〈 [s _{θ 2} , s _{τ 2} ] , (μ₂, η₂, ν₂) 〉 are two SULNs, l + 1 is the cardinality of the linguistic term set, then the cross-entropy CE (α₁, α₂) satisfies the following properties:

(1) CE^* (α₁, α₂) ⩾ 0,

(2) CE^* (α₁, α₂)  = 0, if α₁ = α₂,

(3) CE^* (α₁, α₂)  = CE^* (α₂, α₁).

It can be easily proofed based on Definition 12.

Definition 13. Let X = {x₁, x₂, . . . , x _n } is a discrete universe of discourse. For any two SULSs, A = {(x _i , 〈 [s_{θ A (x i )}, s_{τ A (x i )}] , (μ _A  (x _i ) , η _A  (x _i ) , ν _A  (x _i )) 〉) |x _i  ∈ X} and B = {(x _i , 〈 [s_{θ B (x i )}, s_{τ B (x i )}] , (μ _B  (x _i ) , η _B  (x _i ) , ν _B  (x _i )) 〉) |x _i  ∈ X} in the universe of discourse X, the CSM of A and B can be defined as: C SULS ( A , B ) = M K × T(6) where C SULS ( A , B ) = M K × T , M = ∑ i = 1 n ∑ j = 1 m 1 2 l ∑ ξ ∈ μ , η , ν ψ = θ , τ ψ A ( x ij ) ξ A ( x ij ) 2 ψ B ( x ij ) ξ B ( x ij ) 2 , K = ∑ i = 1 n ∑ j = 1 m 1 2 l 2 ∑ ξ ∈ μ , η , ν ψ = θ , τ ( ψ A ( x ij ) ξ A ( x ij ) 2 ) 2 , T = ∑ i = 1 n ∑ j = 1 m 1 2 l 2 ∑ ξ ∈ μ , η , ν ψ = θ , τ ( ψ B ( x ij ) ξ B ( x ij ) 2 ) 2 .

Theorem 4. Let X = {x₁, x₂, . . . , x _n } is a discrete universe of discourse, and let SULS(X) denote the set of all SULSs in X. For any two SULSs, A ={ (x _i , 〈 [s_{θ A (x i )}, s_{τ A (x i )}] , (μ _A  (x _i ) , η _A  (x _i ) , ν _A  (x _i )) 〉) |x _i  ∈ X } and B ={ (x _i , 〈 [s_{θ B (x i )}, s_{τ B (x i )}] , (μ _B  (x _i ) , η _B  (x _i ) , ν _B  (x _i )) 〉) |x _i  ∈ X } in the universe of discourse X, the cosine similarity measure satisfies the following properties:

0 ⩽ C _SULS  (A, B) ⩽ 1,

3.1 Hamy mean operator

The HM operator, introduced by Hara et al. [33], is an aggregation operator for nonnegative real numbers, which can capture the interrelationships among the multiple input arguments. The mathematical form is defined as follows:

Definition 14 [33] Let α _i  (i = 1, 2, . . . , n) is a collection of nonnegative real numbers, and k = 1, 2, . . . , n. If HM ( k ) ( α 1 , α 2 , . . . , α n ) = ∑ 1 ⩽ i 1 < i 2 . . . < i k ⩽ n ( ∏ j = 1 k α i j ) 1 / k c n k ,(7)

then HM^(k) is the HM operator, where (i₁i₂, . . . , i _n ) traverses all the k-tuple combination of (1, 2, . . . , n) and C n k is the binomial coefficient.

The HM is a function that has Schur convex and monotonic when aggregating numerical information. So a dual form of the HM that satisfies the characteristics of Schur convexity and monotonic is proposed by Wu et al. [45], which is called the DHM.

Definition 15 [45] Let α _i  (i = 1, 2, . . . , n) is a collection of nonnegative real numbers, and k = 1, 2, . . . , n. If DHM ( k ) ( α 1 , α 2 , . . . , α n ) = ( ∏ 1 ⩽ i 1 ≺ i 2 . . . ≺ i k ⩽ n ( ∑ j = 1 k α i j k ) ) 1 / c n k ,(8) then DHM^(k) is the dual Hamy mean, where (i₁i₂, . . . , i _n ) traverses all the k-tuple combination of (1, 2, . . . , n) and C n k is the binomial coefficient.

Based on Equation (7) and Equation (8), some definitions of spherical uncertain linguistic HM (SULHM), spherical uncertain linguistic DHM (SULDHM), spherical uncertain linguistic weighted HM (SULWHM) and spherical uncertain linguistic weighted DHM (SULWDHM) for aggregating SUL information are developed.

Definition 16. Let α _i  =  〈 [s _{θ i} , s _{τ i} ] (μ _i , η _i , ν _i ) 〉 (i = 1, 2, . . . , n) is a collection of SULNs, and k = 1, 2, . . . . , n, then the SULHM operator is defined as SULHM ( k ) ( α 1 , α 2 , . . . , α n ) = ∑ 1 ⩽ i 1 ≺ i 2 . . . ≺ i k ⩽ n ( ∏ j = 1 k α i j ) 1 / k c n k(9) where (i₁i₂, . . . , i _n ) traverses all the k-tuple combination of (1, 2, . . . , n) and C n k is the binomial coefficient. According to the operations of SULNs, the following theorem can be derived.

Theorem 5. Let α _i  =  〈 [s _{θ i} , s _{τ i} ] (μ _i , η _i , ν _i ) 〉 (i = 1, 2, . . . , n) is a collection of SULNs, k = 1, 2, . . . , n, (i₁, i₂, . . . , i _n ) traverses all the k-tuple combinations of (1, 2, . . . , n) and C n k is the binomial coefficient. Then the aggregated value by the SULHM operator is also the SULN and SULHM ( k ) ( α 1 , α 2 , . . . , α n ) = 〈 [ s ∑ 1 ⩽ i 1 ≺ i 2 . . . ≺ i k ⩽ n ( ∏ j = 1 k θ i j ) 1 / k c n k , s ∑ 1 ⩽ i 1 ≺ i 2 . . . ≺ i k ⩽ n ( ∏ j = 1 k τ i j ) 1 / k c n k ] , ( 1 - ( ∏ 1 ⩽ i 1 ≺ i 2 . . . ≺ i k ⩽ n ( 1 - ( ( ∏ j = 1 k ( μ i j 2 + η i j 2 ) ) 1 / k - ( ∏ j = 1 k ( η i j 2 ) ) 1 / k ) ) ) 1 / c n k , ( ∏ 1 ⩽ i 1 ≺ i 2 . . . ≺ i k ⩽ n ( ∏ j = 1 k ( η i j ) ) 1 / k ) 1 / c n k , ( ∏ 1 ⩽ i 1 … < i k ⩽ n ( ( ∏ j = 1 k ( η i j 2 ) ) 1 / k + 1 - ( ∏ j = 1 k ( 1 - ν i j 2 ) ) 1 / k ) ) 1 / C n k - ( ∏ 1 ⩽ i 1 … < i k ⩽ n ( ∏ j = 1 k ( η i j 2 ) ) 1 k ) 1 / C n k ) 〉(10)

The proof of Theorem 5 is provided in Appendix .

By adjusting different values of the parameter k, some special cases can be obtained as follows.

Case 1: When k = 1, the SULHM operator can reduce to the spherical uncertain linguistic averaging operator as follows: SULHM ( k ) ( α 1 , α 2 , … α n ) = 1 n ∑ i = 1 n α i(11)

Case 2: When k = n, the SULHM operator can be reduced to the spherical uncertain linguistic geometric averaging operator as follows: SULHM ( k ) ( α 1 , α 2 , … α n ) = ( ∏ i = 1 n α i ) 1 n(12)

From the above two cases, it can be seen that when k = 1 and n, the SULHM operator can be reduced into ordinary arithmetic mean and geometric mean operators, without considering the relationships between internal inputs.

Theorem 6 (Idempotency). Let α _i  =  〈 [s _{θ i} , s _{τ i} ] (μ _i , η _i , ν _i ) 〉 (i = 1, 2, . . . , n) is a collection of SULNs, and k = 1, 2, . . . , n. If all the SULNs are equal, i.e. α₁ = α₂ = ⋯ = α _n  = α = 〈 [s _θ , s _τ ] , μ, η, ν 〉, then we can get SULHM ( k ) ( α 1 , α 2 , . . . , α n ) = α(13)

The proof of Theorem 6 is provided in Appendix .

Theorem 7 (Boundedness). Let α _i  = 〈 [s _{θ i} , s _{τ i} ] (μ _i , η _i , ν _i ) 〉 (i = 1, 2, . . . , n) is a collection of SULNs, and k = 1, 2, . . . , n, if μ * = min 1 ⩽ i 1 < i 2 ⋯ < i k ⩽ n 1 ⩽ j ⩽ k ( μ i j ) , η * = min 1 ⩽ i 1 < i 2 ⋯ < i k ⩽ n 1 ⩽ j ⩽ k ( η i j ) , ν_* = 1 - μ_* - η_*, μ * = max 1 ⩽ i 1 < i 2 ⋯ < i k ⩽ n 1 ⩽ j ⩽ k ( μ i j ) , η * = max 1 ⩽ i 1 < i 2 ⋯ < i k ⩽ n 1 ⩽ j ⩽ k ( η i j ) , ν^* = 1 - μ^* - η^* and α_* = 〈 [s _θ , s _τ ] , (μ_*, η_*, ν^*) 〉 , α^* =〈 [s _θ , s _τ ] ,  (μ^*, η^*, ν_*) 〉

and α * ⩽ SULHM ( k ) ( α 1 , α 2 , … , α n ) ⩽ α *(14)

The proof of Theorem 7 is provided in Appendix .

Theorem 8 (Monotonicity). Let α _i  = 〈 [s _{θ i} , s _{τ i} ] (μ _i , η _i , ν _i ) 〉 and β _t  =〈 [s _{θ t} , s _{τ t} ] , (μ _t , η _t , ν _t ) 〉 (1 ⩽ i, t ⩽ n) are two SULNs, for every i and t. If μ _i  ⩽ μ _t , η _i  ⩽ η _t we have SULHM ( k ) ( α 1 , α 2 , … α n ) ⩽ SULHM ( k ) ( β 1 , β 2 , … β n )(15)

The proof of Theorem 8 is provided in Appendix .

The SULHM operator does not consider the importance of the input arguments. Nevertheless, in many real-world problems, the influence of each argument is different in the aggregation process. To take account of the weight of input arguments, the SULWHM operator is developed.

Definition 17. Let α _i  =  〈 [s _{θ i} , s _{τ i} ] (μ _i , η _i , ν _i ) 〉 (i = 1, 2, . . . , n) is a collection of SULWHM and k = 1, 2, . . . , n. Let w = (w₁, w₂, . . . , w _n )  ^T is the weight vector of α _i  (i = 1, 2, . . . , n), satisfying w _i  ∈ [0, 1] and ∑ i = 1 n w i = 1 , then SULWHM operator is defined as SULWHM ( k ) ( α 1 , α 2 , . . . , α n ) = ∑ 1 ⩽ i 1 < i 2 ⋯ < i k ⩽ n ( ∏ j = 1 k α i j w i j ) 1 / k c n k(16)

where (i₁, i₂, . . . , i _k ) traverses all the k-tuple combinations of (1, 2, . . . , n), and C n k is the binomial coefficient.

According to the operations of SULNs, the following theorem can be derived.

Theorem 9. Let α _i  =  〈 [s _{θ i} , s _{τ i} ] (μ _i , η _i , ν _i ) 〉 (i = 1, 2, . . . , n) is a collection of SULWHM and k = 1, 2, . . . , n. (i₁, i₂, . . . , i _k ) traverses all the k-tuple combination of (1, 2, . . . , n) and C n k is the binomial coefficient. Let w = (w₁, w₂, . . . , w _n )  ^T is the weight vector of α _i (i = 1, 2, . . . , n), satisfying w _i  ∈ [0, 1] and ∑ i = 1 n w i = 1 , then the aggregated value by the SULWHM operator is also a SULN and SULWHM ( k ) ( α 1 , α 2 , … , α n ) = 〈 [ S ∑ 1 ⩽ i 1 < i 2 ⋯ < i k ⩽ n ( ∏ j = 1 k θ i j w i j ) 1 / k C n k , S ∑ 1 ⩽ i 1 < i 2 ⋯ < i k ⩽ n ( ∏ j = 1 k τ i j w i j ) 1 / k C n k ] , ( 1 - ∏ 1 ⩽ i 1 < i 2 ⋯ < i k ⩽ n ( 1 - ( ( ∏ j = 1 k ( ( μ i j 2 + η i j 2 ) w i j ) ) 1 / k - ( ∏ j = 1 k ( ( η i j 2 ) w i j ) ) 1 / k ) ) 1 / C n k , ( ∏ 1 ⩽ i 1 < i 2 ⋯ < i k ⩽ n ( ∏ j = 1 k ( ( η i j ) w i j ) ) 1 / k ) 1 / C n k , ( ∏ 1 ⩽ i 1 < i 2 ⋯ < i k ⩽ n ( ( ∏ j = 1 k ( ( η i j 2 ) w i j ) ) 1 / k + 1 - ( ∏ j = 1 k ( ( 1 - ν i j 2 ) w i j ) ) 1 / k ) ) 1 / C n k - ( ∏ 1 ⩽ i 1 < i 2 ⋯ < i k ⩽ n ( ( ∏ j = 1 k ( η i j 2 ) w i j ) 1 k ) ) 1 / C n k ) 〉(17)

The proof of Theorem 9 is provided in Appendix .

It is easy to prove that the SULWHM has properties of monotonicity and boundedness, but doesn’t have the property of idempotency.

The SULHM and SULWHM operators can obtained the integration results of input arguments by first performing a geometric mean operation on any k inputs and then an arithmetic mean operation. Next, it is necessary to explore the effectiveness of dual HM in information integration by first calculating the arithmetic mean operator and then calculating the geometric mean operator.

According to Equation (7), the definition of the SULDHM based on the rules of SULNs is proposed.

Definition 18. Let α _i  = 〈 [s _{θ i} , s _{τ i} ] (μ _i , η _i , ν _i ) 〉 (i = 1, 2, . . . , n) is a collection of SULNs and k = 1, 2, . . . , n, then the SULDHM operator is defined as SULDHM ( k ) ( α 1 , α 2 , . . . , α n ) = ( ∏ 1 ⩽ i 1 ≺ i 2 . . . ≺ i k ⩽ n ( ∑ j = 1 k α i j k ) ) 1 / c n k(18) where (i₁, i₂, . . . , i _k ) traverses all the k-tuple combinations of (1, 2, . . . , n), and C n k is the binomial coefficient.

According to Definition 8 and Definition 15, the following theorem can be derived.

Theorem 10. Let α _i  =  〈 [s _{θ i} , s _{τ i} ] (μ _i , η _i , ν _i ) 〉 (i = 1, 2, . . . , n) is a collection of SULNs and k = 1, 2, . . . , n, and (i₁, i₂, . . . , i _k ) traverses all the k-tuple combination of (1, 2, . . . , n) and C n k is the binomial coefficient. then the aggregated value by the SULDHM operator is also the SULN and SULDHM ( k ) ( α 1 , α 2 , . . . , α n ) = 〈 [ S ( ∏ 1 ⩽ i 1 … < i k ⩽ n ( ∑ j = 1 k θ i j k ) ) 1 / - C n k , S ( ∏ 1 ⩽ i 1 … < i k ⩽ n ( ∑ j = 1 k τ i j k ) ) 1 / - C n k ] , ( ( ∏ 1 ⩽ i 1 … < i k ⩽ n ( 1 - ( ∏ j = 1 k ( 1 - μ i j 2 ) ) 1 / k + ( ∏ j = 1 k η i j 2 ) 1 / k ) ) 1 / C n k - ( ∏ 1 ⩽ i 1 … < i k ⩽ n ( ( ∏ j = 1 k η i j 2 ) 1 / k ) ) 1 / C n k , ( ∏ 1 ⩽ i 1 … < i k ⩽ n ( ( ∏ j = 1 k η i j ) 1 / k ) ) 1 / C n k , 1 - ( ∏ 1 ⩽ i 1 … < i k ⩽ n ( 1 - ( ( ∏ j = 1 k ( η i j 2 + ν i j 2 ) ) 1 / k - ( ∏ j = 1 k η i j 2 ) 1 / k ) ) ) 1 / C n k ) 〉(19)

The proof of Theorem 10 is provided in Appendix .

Definition 19. Let α _i  =  〈 [s _{θ i} , s _{τ i} ] (μ _i , η _i , ν _i ) 〉 (i = 1, 2, . . . , n) is a collection of SULNs, and w = (w₁, w₂, . . . , w _n )  ^T is the weight vector of α _i  (i = 1, 2, . . . , n), satisfying w _i  ∈ [0, 1] and ∑ i = 1 n w i = 1 , then the SULWDHM operator is defined as SULWDHM ( k ) ( α 1 , α 2 , . . . , α n ) = ( ∏ 1 ⩽ i 1 … < i k ⩽ n ( ∑ j = 1 k w i j α i j k ) ) 1 / c n k(20) where (i₁, i₂, . . . , i _k ) traverses all the k-tuple combinations of (1, 2, . . . , n), and C n k is the binomial coefficient.

Similarly, the following theorem can be acquired according to Definition 8.

Theorem 11. Let α _i  =  〈 [s _{θ i} , s _{τ i} ] (μ _i , η _i , ν _i ) 〉 (i = 1, 2, . . . , n) is a collection of SULNs, and (i₁, i₂, . . . , i _k ) traverses all the k-tuple combination of (1, 2, . . . , n) and C n k is the binomial coefficient and w = (w₁, w₂, . . . , w _n )  ^T is the weight vector of α _i  (i = 1, 2, . . . , n), satisfying w _i  ∈ [0, 1] and ∑ i = 1 n w i = 1 , then aggregated value by the SULWDHM operator is still a SULN which is shown in Equation (21). SULWDHM ( k ) ( α 1 , α 2 , . . . , α n ) = 〈 [ S ( ∏ 1 ⩽ i 1 … < i k ⩽ n ∑ j = 1 k w i j θ i j k ) 1 / C n k , S ( ∏ 1 ⩽ i 1 … < i k ⩽ n ∑ j = 1 k w i j τ i j k ) 1 / C n k ] ( ∏ 1 ⩽ i 1 … < i k ⩽ n ( 1 - ( ∏ j = 1 k ( 1 - μ i j 2 ) w i j ) 1 / k + ( ∏ j = 1 k ( η i j 2 ) w i j ) 1 / k ) 1 / C n k - ( ∏ 1 ⩽ i 1 … < i k ⩽ n ( ( ∏ j = 1 k ( η i j 2 ) w i j ) 1 / k ) ) 1 / C n k , ( ∏ 1 ⩽ i 1 … < i k ⩽ n ( ∏ j = 1 k η i j w i j ) 1 / k ) 1 / C n k 1 - ( ∏ 1 ⩽ i 1 … < i k ⩽ n ( 1 - ( ∏ j = 1 k ( ( ν i j 2 + η i j 2 ) w i j ) ) 1 / k + ( ∏ j = 1 k ( η i j 2 ) w i j ) 1 / k ) ) 1 / C n k ) 〉(21)

The proof of Theorem 11 is provided in Appendix .

It is also easy to prove that SULWDHM has properties of monotonicity and boundedness, but doesn’t have the property of idempotency. Similarly, the SULWDHM operator also has properties of idempotency, monotonicity and boundedness.

4.1 Description and decision-making process of MAGDM problem

A typical MAGDM problem can be described as follows. Let X = {X₁,X₂,X₃, . . .  ,X _m } is a collection of alternatives and C = {C₁,C₂,C₃, . . .  ,C _n } is a collection of attributes. The weight vector of attributes is w = {w₁,w₂,w₃, . . .  ,w _n }, satisfying w_j ∈ [0, 1], ∑ j = 1 n w j = 1 ( i = 1 , 2 , . . . , n ) . Let D ={ D₁, D₂, D₃, ⋯ , D _z  } is a collection of DMs and the weight vector of DMs is λ = {λ₁, λ₂, λ₃, …, λ _z } which satisfies λ_d ∈ [0, 1] and ∑ d = 1 n λ d = 1 ( d = 1 , 2 , . . . . , z ) . The DMs are required to apply SULN α i j d = 〈 [ s θ , s τ ] , ( μ i j , η i j , ν i j ) 〉 to express their evaluation information. Then we can get a spherical uncertain linguistic decision matrix, which can be denoted as

R d = ( α i j d ) m × n ( d = 1 , 2 , . . . , z ) .

The method of tackling the MAGDM problem involves three key parts shown in Fig. 1. The weight matrices of DMs and attributed are obtained in the first and second parts and the ranking result is obtained in the third part. The detailed steps of the algorithm are given as follows.

OPEN IN VIEWER

Fig. 1

The flow chart of the decision-making process.

In this subsection, the algorithm for MAGDM is given and its detailed steps are shown as follows.

Step 1. Standardized the initial decision matrix. A decision-making problem usually includes two types, which are the benefit attribute and the cost attribute. The standardized method can be detonated as: α i j = { 〈 [ s θ i j , s τ i j ] , ( μ i j , η i j , ν i j ) 〉 G 1 〈 [ s θ i j , s τ i j ] , ( μ i j , η i j , ν i j ) 〉 G 2(22)

where G₁ represents the cost attribute, and G₂ represents the benefit attribute.

Step 2. Determine the ideal assessment matrix of DMs according to SULHM or SULDHM operators, which can be described as follows: α i = { SULHM k ( α 1 , α 2 , . . . . , α n ) SULDHM k ( α 1 , α 2 , . . . . , α n )(23)

Step 3. According to the ideal matrix obtained in Step  2, calculate the cosine similarity measure between each DM evaluation matrix and the ideal matrix by using Equation (6). The weight matrix of DMs can be obtained as follows: λ k = λ k / ∑ k = 1 k = z λ k(24)

Step 4. Integrate information of DMs according to SULWHM or SULWDHM operator, which can be described as follows. α i = { SULWHM k ( α 1 , α 2 , . . . . , α n ) SULWDHM k ( α 1 , α 2 , . . . . , α n )(25)

Step 5. Determine the weight type of attributes according to actual MAGDM problems.

The weight of attributes can be divided into two types, i.e., incompletely known and completely unknown, which depend on the practice problems.

Step 6. Compute the value of cross-entropy.

For the C _j , the deviation of X _i to the other alternatives can be defined as follows according to Equation (5): D ij = 1 m - 1 ∑ r = 1 m CE * ( α ij , α rj ) = 1 2 ( m - 1 ) ∑ r = 1 m ( CE ( α ij , α rj ) + CE ( α rj , α ij ) )(26)

Then the overall deviation of all alternatives to other alternatives for attribute C _j is represented as: D j = ∑ i = 1 m D ij ( C j ) = ∑ i = 1 m ∑ r = 1 m CE * ( α ij , α rj ) , j = 1 , 2 , . . . , n

Step 7. Construct the linear programming model.

Based on the above analysis, we have to choose the weight vector w to maximize the deviation values for all the attributes. That is, we can reasonably construct the following linear programming model: ( M - 1 ) { max D ( w ) = ∑ j = 1 n w j D j = ∑ j = 1 n ∑ i = 1 m ∑ r = 1 m w j CE * ( α ij , α rj ) subject to w ∈ H , ∑ j = 1 n w j = 1 , w j ⩾ 0 , j = 1 , 2 , . . . , n(27)

For other situations, if the information regarding the weights of attributes is completely unknown, we can construct a linear programming model as follows: ( M - 2 ) { max D ( w ) = ∑ j = 1 n w j D j = ∑ j = 1 n ∑ i = 1 m ∑ r = 1 m w j CE * ( α ij , α rj ) subject to ∑ j = 1 n w j = 1 , w j ⩾ 0 , j = 1 , 2 , . . . , n(28)

Step 8. Calculate the optimal weights.

By solving model (M-1) or (M-2), the optimal solution w ={ w₁, w₂, w₃, ⋯ w _n  } can be obtained.

Step 9. Integrate all attributes of every alternative using the SULWHM or SULWDHM operators Equation (25).

Step 10. Compute the score values according to Equation (3).

Step 11. Judge the score values.

If there are two same score values, Step 12 is executed, otherwise, Step 13 is executed.

Step 12. Compute the accuracy values according to Equation (4).

Step 13. Rank all alternatives.

According to their expected values based on the comparison rules from Definition 11, the best alternative can be selected.