A 2-order additive fuzzy measure identification method based on intuitionistic fuzzy sets and its application in credit evaluation

1Introduction

Due to the interaction among attributes, such as complementarity and repeatability, the classical weighted arithmetic mean method is often invalid in the process of multi-attribute decision making (MADM) [1]. In order to overcome this limitation, and effectively describe and deal with any interaction among attributes, Sugeno [2] proposed the concept of fuzzy measure and fuzzy integral. Since then, after nearly 20 years of development, the theory of fuzzy measures had been formed [3, 4]. However, in the practical application process, when there are n attributes, the general fuzzy measure generally needs to determine 2ⁿ - 2 parameters [4, 5]. This complexity greatly limits its practical application ability.

In order to reduce the complexity, Sugeno [2] proposed the λ fuzzy measure, that only needs to determine n parameters, however, the λ fuzzy measure can only represent a certain kind of interaction among attributes, which weakens its performance ability. Thereafter, in order to face with the complexity of discrete fuzzy measures, Grabisch [6] proposed the k-order additive fuzzy measures, including usual additive measures and fuzzy measures. Every discrete fuzzy measure is a k-order additive fuzzy measure for a unique k. The k-order additive fuzzy measures cover all fuzzy measures with any complexity from classical additive measure (k = 1) to general fuzzy measure (k = n). Among them, the 2-order additive fuzzy measure not only merely needs to determine n (n + 1)/2 parameters, but also merely involves the relative importance of attributes and the interaction between attributes, which well solves the contradiction between complexity and performance ability, so it has been widely used [7].

However, because the interaction between attributes is difficult to explain and understand [8], combined with the uncertainty of decision-makers’ cognition, the decision-makers often cannot give accurate quantitative assessment on the interactivity between attributes, but with a certain fuzziness and hesitation, which brings difficulties to decision-making. The existing 2-order additive fuzzy measure identification methods use the subjective methods [9–15] and the objective methods [7, 16, 17] to describe and deal with the interactivity between attributes. The subjective methods are more explanatory than the objective methods, so they have been applied more widely. However, the above methods cannot reflect the fuzziness and hesitation of decision-makers in the assessment of interactivity between attributes.

In [18], Atanassov proposed the concept of intuitionistic fuzzy set (IFS). IFS is an extension of traditional fuzzy set. It considers the information of membership degree, non-membership degree and hesitation degree, so it is more flexible and practical than traditional fuzzy set in dealing with fuzziness and uncertainty [19]. IFS can better and more completely reflect the fuzziness and hesitation of the decision-maker’s judgment of the objective things, so it has been successfully applied in the field of uncertain MADM [20–22].

Based on this observation, in view of the problem that it is difficult to quantitatively assess the interactivity between attributes, the present work uses the IFSs to describe and deal with the interactivity between attributes. As a result, a 2-order additive fuzzy measure identification method based on IFSs is proposed. Based on the proposed method, using the Choquet fuzzy integral as nonlinear integration operator [23], a MADM process is further presented.

This paper is structured as follows: Section 2 introduces the preparatory knowledge employed in this work. Section 3 proposes the 2-order additive fuzzy measure identification method based on IFSs. Section 4 presents the MADM process based on the proposed method. Section 5 describes the analysis results of application example. Section 6 discusses the results obtained and Section 7 concludes this paper.

2Preparatory knowledge

This section introduces the related definitions of IFSs, 2-order additive fuzzy measure and Choquet fuzzy integral reported in the literatures. This is the basis of Section 3 and Section 4.

Definition 1 [18]. Let X be a non-empty set, then A ={ 〈 x, μ_A (x) , ν_A (x) 〉 |x ∈ X  } is called the IFS, where μ_A (x) and ν_A (x) are the membership degree and non-membership degree of element x in X belonging to A respectively, then μ_A : X → [0, 1], ν_A : X → [0, 1] holds, and satisfies 0 ⩽ μ_A (x) + ν_A (x) ⩽1, ∀x ∈ X. In addition, π_A (x) =1 - μ_A (x) - ν_A (x) is called the hesitation degree of element x in X belonging to A, and satisfies 0 ⩽ π_A (x) ⩽1, ∀x ∈ X. For convenience, let α = (μ_α, ν_α) be an IFN, where μ_α ∈ [0, 1], ν_α ∈ [0, 1], 0 ⩽ μ_α + ν_α ⩽ 1, and let Θ be the set of all IFNs.

Definition 2 [18]. Let a₁ = (μ₁, ν₁) and a₂ = (μ₂, ν₂) be two IFNs on a given domain; let λ be a real number, and λ > 0, then the operational rules of IFNs are as follows:

1) a₁ ⊕ a₂ = (μ₁ + μ₂ - μ₁μ₂, ν₁ν₂); 2) a₁ ⊗ a₂ = (μ₁μ₂, ν₁ + ν₂ - ν₁ν₂); 3) λa1=(1-(1-μ1)λ,ν1λ) ; 4) a1λ=(μ1λ,1-(1-ν1)λ) .

Definition 3 [24]. Let α_j = (μ_αj, ν_αj) (j = 1, 2,  ⋯ , n) be a set of IFNs, w_j is the weight of α_j, where w_j ∈ [0, 1], and ∑j=1nwj=1 . Let IFWA : Θⁿ → Θ, if

then IFWA is called the intuitionistic fuzzy weighted average operator.

Definition 4 [25]. Let U ={ x₁, x₂, ⋯ , x_n } be a given domain, A=∑i=1n〈μA(xi),νA(xi)〉/xi is an IFS, then the intuitionistic fuzzy entropy of A is

Definition 5 [26, 27]. Let α = (μ_α, ν_α) be an IFN, where μ_α ∈ [0, 1], ν_α ∈ [0, 1], 0 ⩽ μ_α + ν_α ⩽ 1, when the hesitation degree is not considered, the score function of IFN α can be expressed as [26]

When considering the hesitation degree, the score function of IFN α can be expressed as [27]

Definition 6 [2, 6]. Let X ={ x₁, x₂, ⋯ , x_n } be a set of attributes, let X^* ={ 1, 2, ⋯ , n } be a set of subscripts of attributes. P (X) is the power set of X. If the set function g : P (X) → [0, 1] satisfies the following two conditions:

1) g (∅) =0, g (X) =1; 2) If K ∈ P (X), T ∈ P (X), K ⊆ T, then g (K) ⩽ g (T).

then g is called a fuzzy measure on P (X) [2].

Grabisch [6] proposed the k-order additive fuzzy measures based on pseudo-Boolean function and Möbius transformation. On this basis, the 2-order additive fuzzy measure is then defined as

Definition 7 [13]. Let X ={ x₁, x₂, ⋯ , x_n } be a set of attributes, W ={ w₁, w₂, ⋯ , w_n } is the weight set of X, the Möbius transformation coefficients of x_i and {x_i, x_j} are respectively

Definition 8 [23]. Let f be a nonnegative function defined on X, F is a σ- algebra composed of subsets of X (when X is finite, F is the power set P (X) of X), g is a fuzzy measure defined on F, then the Choquet fuzzy integral of function f on set X for fuzzy measure g is defined as (c)∫fdg=∫0∞g(Fα)dα , where F_α ={ x|f (x) ⩾ α, x ∈ X  }, α ∈ [0, ∞]; ∫0∞g(Fα)dα is the Riemann integral.

When X is a finite set, the elements in X are rearranged as {x₍₁₎, x₍₂₎, ⋯ , x_(n) }, which makes f (x₍₁₎) ⩽ f (x₍₂₎) ⩽ ⋯ ⩽ f (x_(n)). Let H = (c) ∫fdg, then the Choquet fuzzy integral has the following simplified formula:

3A 2-order additive fuzzy measure identification method based on IFSs

Based on IFSs, this section proposes a 2-order additive fuzzy measure identification method. In addition, the correctness of the proposed method is proved theoretically.

Let A ={ A₁, A₂, ⋯ , A_m } be a finite set of alternatives, and C ={ C₁, C₂, ⋯ , C_n } be a set of attributes to compare the alternatives, let C^* ={ 1, 2, ⋯ , n } be a set of subscripts of attributes. The weight vector of attributes is W_C = (w₁, w₂, ⋯ , w_n), where w_i ∈ [0, 1], and ∑i=1nwi=1 . Let D ={ D₁, D₂, ⋯ , D_p } be a set of experts, the weight vector of experts is W_D = (w₁, w₂, ⋯ , w_p), where w_l ∈ [0, 1], and ∑l=1pwl=1 . Based on IFSs, the identification process of 2-order additive fuzzy measure is as follows:

Step 1: Defines the interactivity between attributes.

According to the supermodular game theory, the interactivity between attributes is defined as follows:

Definition 9. Let any two attributes C_i and C_j (i≠j) in attribute set C have partial order relation, the supremum C_i ∨ C_j and the infimum C_i ∧ C_j are in C, then C is called a sub-lattice [28]. Let f be a real-valued function defined on the sub-lattice C, C ⊆ Rⁿ. For ∀C_i, C_j ∈ C, when f (C_i ∨ C_j) + f (C_i ∧ C_j) > f (C_i) + f (C_j), f is a supermodular function [28], then it is said that there is complementarity between C_i and C_j (i≠j); when f (C_i ∨ C_j) + f (C_i ∧ C_j) < f (C_i) + f (C_j), f is a submodular function [28], then it is said that there is repeatability between C_i and C_j (i≠j); particularly, when f (C_i ∨ C_j) + f (C_i ∧ C_j) = f (C_i) + f (C_j), then it is said that there is independence between C_i and C_j (i≠j).

Step 2: Calculate the individual assessment result of each expert on the interactivity between C_i and C_j (i≠j).

According to Definition 9, the expert D_l (l = 1, 2, ⋯ , p) employs the IFN α = (μ_α, ν_α) to assess the interactivity between C_i and C_j (i≠j) ( Cn2 pairs in total). When the expert thinks that there is complementarity between C_i and C_j (i≠j), μ_α > 0.5, and the larger μ_α is, the stronger complementarity is. When the expert thinks that there is repeatability between C_i and C_j (i≠j), ν_α > 0.5, and the larger ν_α is, the stronger repeatability is. In addition, when the expert thinks that C_i and C_j (i≠j) are independent of each other, μ_α = ν_α = 0.5. Thus, the individual assessment result ξ˜ijl=(μξ˜ijl,νξ˜ijl) (l = 1, 2, ⋯ , p) of the interactivity between C_i and C_j (i≠j) is then given.

Step 3: Calculate the group assessment result of p experts on the interactivity between C_i and C_j (i≠j).

Given the weight vector W_D of experts, using Equation (1) to aggregate the opinions of p experts, the group assessment result ξ˜ij of the interactivity between C_i and C_j (i≠j) is further calculated as

Step 4: Determine the intuitionistic fuzzy interaction degree S(ξ˜ij) between C_i and C_j (i≠j).

With Equation (3), the intuitionistic fuzzy interaction degree S(ξ˜ij) between C_i and C_j (i≠j) is defined as follows:

Definition 10. Let ξ˜ij=(μξ˜ij,νξ˜ij) be the group assessment result of the interactivity between C_i and C_j (i≠j), according to Equation (3), if S(ξ˜ij) satisfies

Thus, according to Definition 10, the intuitionistic fuzzy interaction degree S(ξ˜ij) between C_i and C_j (i≠j) can be determined.

Step 5: Calculate the Möbius transformation coefficients m_i and m_ij of attributes.

According to the weight w_i of the i-th attribute and the intuitionistic fuzzy interaction degree S(ξ˜ij) between C_i and C_j (i≠j), using Equation (6), the Möbius transformation coefficients m_i and m_ij of attributes can be calculated as

Step 6: Identify the 2-order additive fuzzy measure g_K.

According to the Möbius transformation coefficients m_i and m_ij of attributes, using Equation (5), the 2-order additive fuzzy measure g_K can be identified as

Theorem 1. The fuzzy measure identified by steps 1 to 6 is a 2-order additive fuzzy measure.

To prove that the fuzzy measure identified by steps 1 to 6 is a 2-order additive fuzzy measure, it is only necessary to prove that the determined Möbius transformation coefficients satisfy the following constrained conditions [6]:

1) m (∅) =0; 2) m_i ⩾ 0, ∀i ∈ C^*; 3) ∑_i∈C^*m_i + ∑_{i,j}⊂C^*m_ij = 1; 4) m_i + ∑_{j∈K^*∖{i}}m_ij ⩾ 0, ∀K ⊂ C.

Proof.

Since S(ξ˜ij)∈[-1,1] , and w_j ∈ [0, 1], we have

Sum the two sides of the above inequality to j, we obtain

Multiply both sides of the above inequality by w_i, we can get

Sum the two sides of the above inequality to i, the following inequality can be given

Therefore, we have

That is to say

Because P > 0, and w_i ⩾ 0, we can get mi=wiP⩾0 .

Since S(ξ˜ij)∈[-1,1] , and w_j ∈ [0, 1], we have -(1-wi)⩽∑j∈K*∖{i}S(ξ˜ij)wj⩽1-wi . Thus, the following inequality can be given

Hence, we get m_i + ∑_{j∈K^*∖{i}}m_ij ⩾ 0, ∀K ⊂ C. Q.E.D.

4A MADM process based on the proposed method

Using the Choquet fuzzy integral as nonlinear integration operator, this section presents a MADM process based on the proposed method.

Step 1: Calculate the group intuitionistic fuzzy assessment value x˜jik .

Let C_i ={ C_i1, C_i2, ⋯ , C_ik, ⋯ , C_ini } (k = 1, 2, ⋯ , n_i) be a set of sub-attributes under the i-th attribute. According to the type of sub-attribute (such as positive, negative and neutral), the expert D_l (l = 1, 2, ⋯ , p) employs the IFN to assess the performance of the alternative A_j (j = 1, 2, ⋯ , m) under the sub-attribute C_ik (i = 1, 2, ⋯ , n), the individual intuitionistic fuzzy assessment value x˜jikl=(μx˜jikl,νx˜jikl) of the k-th sub-attribute of the alternative A_j under the i-th attribute is then given.

Given the weight vector W_D of experts, using Equation (1) to aggregate the opinions of p experts, the group intuitionistic fuzzy assessment value x˜jik of the k-th sub-attribute of the alternative A_j under the i-th attribute is further calculated as

Step 2: Calculate the overall weight w_i of the i-th attribute and the relative weight wik′ of the k-th sub-attribute under the i-th attribute.

By using Equation (2), the intuitionistic fuzzy entropy E_ik of the k-th sub-attribute under the i-th attribute is calculated as

According to the definition of information entropy, if the information entropy of an attribute is smaller, the information quantity provided by the attribute is larger, and the weight of the attribute is also greater [29]. Therefore, the entropy weight w_ik of the k-th sub-attribute under the i-th attribute can be expressed as [30]

That is to say, the overall weight vector of sub-attributes is W_i = (w_i1, w_i2, ⋯ , w_ik, ⋯ , w_ini), i = 1, 2, ⋯ , n. The overall weight w_i of the i-th attribute is then obtained as

It can be proved that ∑i=1nwi=1 holds. Therefore, the overall weight vector of attributes is W_C = (w₁, w₂, ⋯ , w_i, ⋯ , w_n). The relative weight wik′ of the k-th sub-attribute under the i-th attribute is then obtained as

Step 3: Calculate the intuitionistic fuzzy assessment score SL(x˜ji) of the alternative A_j under the i-th attribute.

According to x˜jik=(μx˜jik,νx˜jik) and wik′ , with Equation (1), the intuitionistic fuzzy assessment value x˜ji of the alternative A_j under the i-th attribute is calculated as

Hence, with Equation (4), the intuitionistic fuzzy assessment score SL(x˜ji) of the alternative A_j under the i-th attribute is further calculated as

Step 4: Identify the 2-order additive fuzzy measure g_K.

Given the weight vector W_D of experts, the intuitionistic fuzzy interaction degree S(ξ˜ij) between C_i and C_j (i≠j) can be determined by steps 1 to 4 in Section 3. Thus, according to w_i and S(ξ˜ij) , the 2-order additive fuzzy measure g_K can be further identified by step 5 and step 6 in Section 3.

Step 5: Calculate the Choquet fuzzy integral values and the ranking of alternatives.

Reordering the intuitionistic fuzzy assessment score SL(x˜ji) (i = 1, 2, ⋯ , n) of the alternative A_j (j = 1, 2, ⋯ , m) under the i-th attribute from small to large, the SL(x˜j(i)) can be then obtained. Substituting the SL(x˜j(i)) and the 2-order additive fuzzy measure g_K into Equation (7), the Choquet fuzzy integral value H_j of the alternative A_j can be calculated. Simultaneously, the ranking of alternatives can be given, where the larger H_j is, the better the alternative A_j is.

5Application example

This section takes the credit evaluation of the big data listed companies in China as an application example to illustrate the feasibility and effectiveness of the proposed method (see Section 3).

6Discussion

From the results and analysis of the previous section, we observed that Method 1 was able to obtain the higher discrimination value than Method 2 (Method 1 was 0.7102, and Method 2 was 0.6903), and the slightly lower Choquet fuzzy integral mean value than Method 2 (Method 1 was 0.6912, and Method 2 was 0.6959).

Compared with Method 2, Method 1 can obtain the higher discrimination value. Since the variance of interaction degrees determined by Method 1 (its value was equal to 0.0684) was higher than that of Method 2 (its value was equal to 0.0355). According to Equation (6), the variances of m_i and m_ij increase with the increase of the variance of interaction degrees. From Equations (5) and (7), we can further see that the variance of g (K) and the variance of H also increase correspondingly. Thus, according to the algorithm of discrimination [33], the discrimination value becomes larger.

Compared with Method 2, Method 1 can obtain the slightly lower Choquet fuzzy integral mean value. Since the average value of interaction degrees determined by Method 1 (its value was equal to 0.2718) was higher than that of Method 2 (its value was equal to 0.2087). According to Equation (6), when the average value of interaction degrees increases, the mean value of P increases, meanwhile, the mean value of m_i decreases and the mean value of m_ij increases. However, because |m_i| ⪢ |m_ij|, from Equation (5), we can further see that the mean value of g (K) also decreases correspondingly. Thus, according to Equation (7), the mean value of H becomes smaller.

The variance or average value of interaction degrees determined by Method 1 was higher than that of Method 2, which was closely related to that the experts used the IFN to assess the interactivity between attributes, so it can fully express their opinions.

7Conclusion

In this paper, the proposed method defines the interactivity between attributes by using the supermodular game theory, so that the interaction between attributes is easier to explain and understand, which lays a solid foundation for experts to qualitatively assess the interactivity between attributes. The proposed method allows the experts to assess the interactivity between attributes by using the IFN, which better preserves the assessment information of experts and embodies the fuzziness and hesitation of interactivity assessment. Furthermore, the proposed method uses the IFWA to aggregate the opinions of all experts, which considers the weights of experts, thereby ensuring the rationality of decision-making. In addition, the proposed method uses the score function of IFN to define and calculate the intuitionistic fuzzy interaction degree between attributes, so the transformation from qualitative description to quantitative characterization is finally realized. Therefore, using the IFSs, this work successfully solves the problem that it is difficult to quantitatively assess the interactivity between attributes in the identification process of 2-order additive fuzzy measure.

This work proposed a 2-order additive fuzzy measure identification method based on IFSs. Obviously, on the one hand, compared with the objective method describing and dealing with the interactivity between attributes (as in [7, 16] and [17]), the proposed method has subjectivity. On the other hand, compared with the subjective method describing and dealing with the interactivity between attributes (as in [9–14] and [15]), the proposed method has fuzziness and hesitation.

Future application example analysis will consider increasing the number of samples and experts to improve the persuasiveness of application example analysis results and the stability of interactivity assessment. Furthermore, because the sample time may affect the analysis results of application example, it is necessary to update the sample data from 2016 to 2020 in the future. In addition, Method 1 should be compared with other methods except Method 2, such as the diamond pairwise comparisons method [9, 10, 14], proportional scaling method [11], multicriteria correlation preference information method [12], etc.

According to the supermodular game theory, Definition 9 gives the definition of interactivity between attributes. However, this definition is still relatively general. The detailed theoretical analysis of the connotation of interactivity between attributes needs to be completed in the future. The existing 2-order additive fuzzy measure identification methods use the subjective methods and the objective methods to describe and deal with the interactivity between attributes. Thus, combining the results of subjective methods and objective methods is one of the main research directions in the future. In addition, constructing a Group Decision Support System will reduce the complexity of Method 1 in practical application.