Interval-valued intuitionistic pure linguistic entropy weight method and its application to group decision-making

3.1 Linguistic operators

Definition 1. Consider the finite discrete set H = { h _α |α = - ρ, - ρ + 1, ⋯ , ρ - 1, ρ } (ρ ∈ Z^*andρ ⩾ 1), and let the linguistic variable h _α satisfy the following properties.

(1) If α > β, then h _α  > h _β ;

(2) If α > β, then max { h _α , h _β  } = h _α ;

(3) If α > β, then min { h _α , h _β  } = h _β ;

(4) There is a negative operator neg (h _α ) = h_-α.

Consider the function φ : h _α  = φ (α), where φ (α) is a monotone increasing function of α. To facilitate calculation and avoid any loss of decision information, a continuous linguistic variable set is necessary. Let H ¯ = { h α | α ∈ [ - q , q ] } be an extended linguistic variable set, where q (q > ρ) is a sufficiently large positive number. In addition, the extended linguistic variable has the following properties.

( 1 ) h α ⊕ h β = h β ⊕ h α = h α + β , h α , h β ∈ H ¯ ;

(2) λh _α  = h _λα , λ > 0.

Definition 2 [44]. The linguistic variable operation is defined as follows.

Let x = θ , f ( x ) = tan π x 2 ρ + 2 , x ∈ ( - ρ - 1 , ρ + 1 ) and f - 1 ( x ) = ( 2 ρ + 2 ) arctan ( x ) π , x ∈ R , ∀ l ¯ θ , l ¯ γ ∈ L ¯ , λ ∈ R : l ¯ θ ⊕ l ¯ γ = l ¯ f - 1 [ f ( θ ) + f ( γ ) ] .(1) λ l ¯ θ = l ¯ f - 1 [ λ f ( θ ) ] .(2)

Let g : L ¯ → ( - ∞ , + ∞ ) and g ( l ¯ θ ) = tan ( π θ 2 ρ + 2 ) , l ¯ θ ∈ L ¯ .

Then ∀ l ¯ θ , l ¯ γ ∈ L ¯ , λ ∈ R : l ¯ θ ⊕ l ¯ γ = g - 1 ( g ( l ¯ θ ) + g ( l ¯ γ ) ) .(3) λ l ¯ θ = g - 1 ( λ g ( l ¯ θ ) ) .(4)

Definition 3 [44]. Let {l _{α 1} , l _{α 2} , ⋯ , l _{α m}  } be the linguistic data set to be aggregated. Then, for the mapping of the linguistic weighted arithmetic average LWAA : L ¯ m → L ¯ :

LWAA { l α 1 , l α 2 , ⋯ , l α m } = ω 1 l α 1 ⊕ ω 2 l α 2 ⊕ ⋯ ⊕ ω m l α m = l α ,(5) α = f - 1 [ ∑ j = 1 m ω j f ( α j ) ] .(6)

Here, ω _j  = (ω₁, ω₂, ⋯ , ω _m )  ^T is a weight vector corresponding to the linguistic type data set {l _{α 1} , l _{α 2} , ⋯ , l _{α m}  }, ω _j  ⩾ 0 (j = 1, 2, ⋯ , m), and ∑ j = 1 m ω j = 1 .

Definition 4 [6]. Let B ={ x, μ _B  (x) , ν _B  (x) |x ∈ X } be an IVIFS, where μ _B  (x) and ν _B  (x) are, respectively, the degree of membership μ _B  : X → [0, 1] and non-membership v _B  : X → [0, 1] of element x in X, and μ _B  (x) ⊂ [0, 1], ν _B  (x) ⊂ [0, 1], μ _B  (x) + ν _B  (x) ⩽ 1, ∀ x ∈ X.

On the basis of definition 4, combining intuitionistic linguistic numbers and pure linguistic variables, Peng and Ye [46] proposed the concept and arithmetic of interval-valued intuitionistic pure linguistic sets (IVIPLSs).

Definition 5 [46]. Let the linguistic scale variable h α ( x ) ∈ H ¯ , and X be a nonempty set. Then, the IVIPLS can be expressed as follows. B = { x [ h α ( x ) , ( μ B ( x ) , ν B ( x ) ) ] | x ∈ X } .(7)

Here, μ _B  (x) and ν _B  (x) represent the degree of membership and non-membership of the linguistic variable value h_α(x), respectively. B can also be written as B ={ h_α(x), (μ _B  (x) , ν _B  (x)) |x ∈ X }. The general form of an interval-valued intuitionistic pure linguistic number (IVIPLN) is abbreviated as h_α(x), ([a, b] , [c, d]), where [a, b] ⊂ [0, 1], [c, d] ⊂ [0, 1], b + d ⩽ 1. Let θ₁ = h_α(x1), ([a₁, b₁] , [c₁, d₁]) and θ₂ = h_α(x2), ([a₂, b₂] , [c₂, d₂]) be any two interval-valued intuitionistic pure linguistic variables. Then, the following properties can be obtained. θ 1 ⊕ θ 2 = 〈 h α ( x 1 ) , ( [ a 1 , b 1 ] , [ c 1 , d 1 ] ) 〉 + 〈 h α ( x 2 ) , ( [ a 2 , b 2 ] , [ c 2 , d 2 ] ) 〉 = h α ( x 1 ) + α ( x 2 ) , ( [ a 1 + a 2 - a 1 a 2 , b 1 + b 2 - b 1 b 2 ] , [ c 1 c 2 , d 1 d 2 ] ) ,(8) λ ( θ 1 ⊕ θ 2 + ⋯ + θ j ) = h ∑ j = 1 n λ α ( x j ) , ( [ ( 1 - ∏ j = 1 n ( 1 - a j ) λ , 1 - ∏ j = 1 n ( 1 - b j ) λ ] , [ ∏ j = 1 n c j λ , ∏ j = 1 n d j λ ] ) .(9)

Definition 6 [46]. Let θ = h_α(x), ([a, b] , [c, d]) be an IVIPLN, and S (θ) be the score function of θ. S ( θ ) = h θ ( x ) × ( a - c + b - d ) 2 = h θ ( x ) ( a - c + b - d ) / 2 .(10)

Here, h_-θ(x) ⩽ S (θ) ⩽ h_θ(x), and obviously, a greater value of S (θ), implies a greater value of θ.

Entropy was first proposed by Shannon [47], and has been widely utilized in social economy, engineering technology and other fields [48–52]. The basic idea of the entropy weight method (EWM) is to determine the objective weight according to the variability of certain criteria. In general, if the information entropy of the criteria is lower, these criteria play a greater role in the comprehensive evaluation, and therefore have higher weighs. The calculation procedure of the EWM is as follows.

Step 1. Consider the decision matrix A = (x _ij ) _n×m obtained for n alternatives and m attribute indicators. A = [ x 11 , x 12 , ⋯ , x 1 m x 21 , x 22 , ⋯ , x 2 m ⋮ ⋮ ⋯ ⋮ x n 1 , x n 2 , ⋯ , x nm ] .(11)

Step 2. Benefit-type criteria and cost-type criteria can be normalized using Equations (12) and (13), respectively. The normalized matrix B = ( x ij ′ ) n × m can then be obtained. x ij ′ = x ij - min i { x ij } max i { x ij } - min i { x ij } ,(12) x ij ′ = max i { x ij } - x ij max i { x ij } - min i { x ij } ,(13) B = [ x 11 ′ , x 12 ′ , ⋯ , x 1 m ′ x 21 ′ , x 22 ′ , ⋯ , x 2 m ′ ⋮ ⋮ ⋯ ⋮ x n 1 ′ , x n 2 ′ , ⋯ , x nm ′ ] .(14)

Step 3. Calculate the entropy of each criterion. H j = - 1 ln n ∑ i = 1 n ( x ij ′ ∑ i = 1 n x ij ′ × ln x ij ′ ∑ i = 1 n x ij ′ ) ,(15) where 0 ⩽ H _j  ⩽ 1. In addition, if x ij ′ = 0 , then x ij ′ ∑ i = 1 n x ij ′ × ln x ij ′ ∑ i = 1 n x ij ′ = 0 .

Step 4. The entropy weight value of the jth criterion ω _j can be obtained as ω j = 1 - H j m - ∑ j = 1 m H j .(16)

Step 1. Obtain a complete linguistic decision matrix L. Each expert DM _i  (1 ⩽ i ⩽ n) gives a judgment for attribute C _j  (1 ⩽ j ⩽ m) using the linguistic terms in Table 2. L = [ θ 11 , θ 12 , ⋯ , θ 1 m θ 21 , θ 22 , ⋯ , θ 2 m ⋮ ⋮ ⋯ ⋮ θ n 1 , θ n 2 , ⋯ , θ nm ] ,(17) where θ _ij  = h_{α(x ij )}, ([a _ij , b _ij ] , [c _ij , d _ij ]).

Step 2. Calculate the score function value S (θ) of θ _ij in matrix L using Equation (10), and then compute the score function matrix L¹ as L 1 = [ S ( θ ) 11 , S ( θ ) 12 , ⋯ , S ( θ ) 1 m S ( θ ) 21 , S ( θ ) 22 , ⋯ , S ( θ ) 2 m ⋮ ⋮ ⋯ ⋮ S ( θ ) n 1 , S ( θ ) n 2 , ⋯ , S ( θ ) nm ] ,(18)

where S ( θ ) ij = h α ( x ij ) × ( a ij - c ij + b ij - d ij ) 2 = h α ( x ij ) × ( a ij - c ij + b ij - d ij ) / 2 = h α ij .

Step 3. Compute the normalized score function matrix L¹ and the normalized matrix L².

By combining linguistic variables using Equations (12), and/or (13), interval-valued pure linguistic normalization equations can be obtained.

The normalization of the benefit-type criteria is as follows: h α ij ′ = h f - 1 .(19)

The normalization of the cost-type criteria is as follows: h α ij ′ = h f - 1 [ ( f ( max i α ij ) - f ( α ij ) ) / ( f ( max i α ij ) - f ( min i α ij ) ) ] .(20)

The normalized matrix L 2 = ( α ij ′ ) n × m is calculated using Equatios (19) and (20).

Step 4. The linguistic entropy H _j can be obtained as H ( h α ij ′ ) = - 1 ln n ∑ i = 1 n g ( α ij ′ ∑ i = 1 n α ij ′ ) × ln g ( α ij ′ ∑ i = 1 n α ij ′ ) .(21)

Specifically, if g ( α ij ′ ∑ i = 1 n α ij ′ ) = 0 , then ∑ i = 1 n g ( α ij ′ ∑ i = 1 n α ij ′ ) × ln g ( α ij ′ ∑ i = 1 n α ij ′ ) = 0 .

Step 5. The interval-valued intuitionistic pure linguistic entropy weights can be obtained as ω j = ( 1 - H ( h α ij ′ ) ) / ( m - ∑ j = 1 m H ( h α ij ′ ) ) .(22)

Step 6. The comprehensive linguistic information value of each alternative is calculated using Equations (5) and (6), and then all alternatives are ranked and the best one is selected.

The algorithm flow of the IVIPLEW method for a complete linguistic decision matrix is shown in Fig. 1. Obviously, the time complexity of the presented algorithm is O(n).

OPEN IN VIEWER

Fig. 1

Algorithm flowchart.

Due to the uncertainties in the subjective judgment of attributes by experts, it is difficult to assign a specific linguistic judgment to an attribute. Therefore, several missing values appear in the linguistic decision information matrix. Failure to handle the missing values properly will affect the final evaluation results. Thus, it is necessary to deal with MAGDM problems in which some values are missing in a scientific and reasonable manner. A practical method is proposed to supplement missing values.

Step 1. An incomplete linguistic decision matrix L is obtained, in which the missing values are indicated by ɛ. The incomplete decision matrix can be represented as L = [ θ 11 , θ 12 , ⋯ , θ 1 m ɛ , θ 22 , ⋯ , θ 2 m ⋮ ⋮ ⋯ ⋮ θ n 1 , ɛ , ⋯ , θ nm ] .(23)

Step 2. According to the incomplete linguistic information matrix, and from Equation (9), the expected value of the IVIPL is calculated using the equation E ( θ ) = λ ( θ 1 ⊕ θ 2 + ⋯ + θ j ) = h ∑ j = 1 n λ α ( x j ) , ( [ ( 1 - ∏ j = 1 n ( 1 - a j ) λ , 1 - ∏ j = 1 n ( 1 - b j ) λ ] , [ ∏ j = 1 n c j λ , ∏ j = 1 n d j λ ] ) ,(24) where λ = 1 n - μ , in which μ indicates the number of missing values in a column. After completing the missing value linguistic decision matrix, the rest of the procedure is consistent with subsection 4.1.

The proposed method takes the weighted arithmetic average value of the index evaluation into account. Compared with other methods, this reduces the volatility of the data and prevents other factors from having an undue influence. Based on the description of the proposed MAGDM method, the procedure of the complete and incomplete IVIPLEW method is as shown in Fig. 2.

OPEN IN VIEWER

Fig. 2

Flow chart of the proposed method.

5.1 Case description

The decision-making background of this case study comes from an information technology company, which we refer to as company C. Company C is a comprehensive high-tech enterprise focusing on computer information system integration, application system development, operation and maintenance services, and providing industry solutions. The company is committed to providing customers with professional services such as information planning and design, management consulting, and industry solutions. In addition, it provides smart government, smart education, smart camp, smart manufacturing, smart medical, and other solutions using cloud computing and big data as the core.

As the development scale of company C has continued to expand in recent years, it has faced unprecedented development opportunities. Unfortunately, the company has management problems that are insufficient to meet the current challenges. First, the company has not established an effective system for evaluating and selecting talented staff. Second, changes in the environment have forced the company to speed up the construction of its management system. Finally, the company’s organizational system is not yet perfect. Therefore, the company intends to form a new department to meet the current management challenges. Each member of the senior management of company C believes that choosing the right department manager will effectively improve the current shortcomings. Therefore, the objective of this case study is to help company C to select a suitable department manager from several candidates.

To perform effective talent selection, the human resources department of company C has developed a series of evaluation processes that include a written test and an interview. During the interview stage, an evaluation team of five experts controls and implements the whole process of talent selection. Three of the five experts are currently in the human resources department, with the chief executive officer and chief operating officer making up the evaluation team. The evaluation criteria for candidates are professional skills (C₁), management skills (C₂), computer skills (C₃), safety consciousness (C₄), and communication skills (C₅). The raw dataset was formed by the five experts based on the candidates’ resumes and interview performance. Officials from the human resources department will make the final decision based on various evaluation results.

5.2 Selection process

Five candidates participated in the assessment. As the experts cannot provide accurate evaluation results for each criterion, they used the linguistic terms in Table 2 to measure the degree of matching. These five criteria are expressed as C ={ C₁, C₂, C₃, C₄, C₅ }, the experts are referred to as E ={ E₁, E₂, E₃, E₄, E₅ }, and the five candidates are denoted as A ={ A₁, A₂, A₃, A₄, A₅ }.

Step 1. Obtain the IVIPL variable decision matrices, as shown in Tables 3–7.

Step 2. Supplement any incomplete decision information matrices. Take ɛ₅₂ as an example. The expected IVIPL value is calculated using Equation (24), from which we obtain the missing value ɛ₅₂ = 〈h₂, ([0.527, 0.668] , [0.141, 0.265]) 〉. Similarly, the missing value ɛ₂₄ = 〈h_2.5, ([0.654, 0.755] , [0.119, 0.221]) 〉 is obtained for candidate A₃. Therefore, after supplementing the missing values of ɛ₅₂ and ɛ₂₄, complete decision matrices have been obtained.

Step 3. Use Equation (10) to calculate the score function value for each attribute. The final results are listed in Tables 8–12.

Step 4. As all attributes are benefit-type criteria, Equation (19) is used to normalize the score function values. The results are shown in Tables 13–17. These Tables show the normalized results for the candidates.

Step 5. The linguistic entropy can be calculated using Equation (21), and then the linguistic entropy weight is given by Equation (22). Table 18 shows the results with respect to the linguistic entropy and the linguistic entropy weight.

Step 6. Calculate the final evaluation result using Equation (5). Table 19 shows the results.

As can be seen from Table 19, A₄ ≻ A₅ ≻ A₃ ≻ A₂ ≻ A₁, so the optimal candidate is A₄.

The effectiveness of the IVIPLEW method proposed in this paper is now verified by comparing its performance that of the IVIPL information aggregation method. In addition, a sensitivity analysis is carried out to verify the robustness of the proposed method.

5.3.2 Sensitivity analysis

The weights obtained by the IVIPLEW method are relatively objective, but seldom consider the experience and professional knowledge of decision-makers. Reasonable attribute weights should consider the subjective judgment of experts and objective factors. Therefore, Equation (25) is used to provide a comprehensive consideration of the subjective weighting of experts and the weight obtained from the evaluation. ω j = α ω j 1 + ( 1 - α ) ω j 2 ,(25) where ω j 1 represents the weight calculated by the IVIPLEW method and ω j 2 represents the weight given by the experts’ personal preferences. α is a preference factor that reflects the decision-maker’s preference for the weight obtained by the expert’s experience and the weight obtained by the IVIPLEW method. A larger value of α suggests that the weight obtained by the IVIPLEW method is of more use to the decision-makers; otherwise, the weight obtained by expert experience and knowledge is of more use to the decision-makers.

The weights assigned by the experts to each attribute are ω j 2 = ( 0.3 , 0.25 , 0.2 , 0.15 , 0.1 ) . The weights obtained by IVIPLEW, ω j 1 , are listed in Table 17. This paper now analyzes the ranking for different scenarios of α. Figures 3 and 4 show the evaluation results and the ranking for different combinations of α.

OPEN IN VIEWER

Fig. 3

Evaluation results for different combinations of α.

OPEN IN VIEWER

Fig. 4

Ranking for different combinations of α.

As can be seen from Fig. 3, only the evaluation result of A₄ increases as α increases. This indicates that the attribute weights obtained by IVIPLEW have a positive effect on A₄, whereas the evaluation results of A₁, A₂, A₃, and A₅ may decrease as α increases, which shows that the subjective attribute weights assigned by experts have a positive effect on A₁, A₂, A₃, and A₅. In addition, it can be seen from Fig. 4 that when α is in the interval [0, 0.4), the best candidate is A₅, and when α is in the interval 0.4, 1], the best choice is A₄. Therefore, when choosing a different preference factor α, the best selection may change. Further analysis indicates that because A₄ and A₅ have similar final evaluation results, they are sensitive to the attribute weights, whereas candidates A₁, A₂, and A₃ are some way behind the other two, so changes in the weights have no effect on the ranking of the evaluation results for these three candidates.