Multi-criteria decision making (MCDM) is regarded as a complex decision tool involving quantitative and qualitative factors, and it refers to the process of systematic evaluation, ranking or selection of limited alternatives under multiple conflicting objectives or criteria. The core challenge lies in the trade-off between decision criteria and uncertainty handling. The trade-off between criteria requires aggregation operators to be de-aggregated, such as weighting operators and geometric operators, but these operators are not applicable when decision criteria influence and interact with each other. The Choquet integral is an aggregate function that can be used to measure the level of uncertainty in a fuzzy set to connect the interactions between decision criteria through the determination of fuzzy measures.

In 1954, the French mathematician Choquet proposed the capacity theory for the first time and proposed the Choquet integral based on the capacity theory [1]. Since then, the study of the Choquet integral from the aspects of properties and measures began. A generalization of the Choquet integral started to surface in [2] in 1987. The Choquet integral was used by Barreneche et al. in 2013 as an aggregation function for the fuzzy rule-based classification system in [3]. A novel aggregation function enhanced by the Choquet integral was introduced in [4] in 2016. The aggregation function, the preaggregation function, and the ordered directional monotone function-referred to as the CC-integral [5], C_F-integral [6], and C_T-integral [7] replace the product operator in the Choquet integral in these generalizations. For the CC-integral, an aggregate class function is obtained by substituting a copula for the product operator in the Choquet integral.

In 1986, Schmeidler first applied Choquet integral to decision-making problems [8]. In 1991, Tanaka and Sugeno applied it to evaluate color printed images [9]. In 1995, Grabisch pointed out the advantages and characteristics of Choquet integral in decision-making. The decision techniques are also presented. So far, Choquet integral has been widely used in the field of decision making [10]-[13]. As an extension of Choquet integral, CC-integral has also been applied to solve MCDM problems [14][15]. The EOL strategy and how to handle the product after use are the most prevalent MCDM issues. Marco et al. first presented the EOL method in 1994, and it has since undergone constant development and improvement [16][17]. The MCDM problem of identifying the EOL strategy of refrigerator composition is a crucial study area to assess the elements of EOL strategy from a holistic perspective. EOL decision-making depends on numerous factors.

The evaluation value is typically fuzzy linguistic evaluation, and the criteria frequently interact with one another when solving the EOL strategy problem of refrigerator building. These language computations are typically transformed into triangle norms, which are difficult to compute. Instead, they are typically transformed into interval values or exact numbers. Consequently, the universal CC-integral must be extended to interval values. This study provides the CC-integral on interval-valued Sugeno probability measures. Before this, Chen proposed the Choquet integral on interval-valued Sugeno probability measures [18], Shang proposed the C_T-integral on interval-valued Sugeno probability measures in 2022 [19]. And taking Copula as the overlap function, it is applied to solve the MCDM problem in the context of EOL policy determination. In contrast to the former, the method presented in this study uses mathematical software to simplify the calculation process, and the findings are more commonly utilized and accurate.

The paper is organized as follows. In Section 2, we introduce some basic concepts, including the definition and theorem of aggregation functions and Choquet integral. In Section 3, we present the expression of the CC-integral on the interval-valued Sugeno probability measure, and provides the discrete expression when the Copula is taken as an overlap function. In Section 4, we apply the CC-integral on the interval-valued Sugeno probability measure to MCDM, and provides a detailed calculation process, demonstrating the effectiveness of the proposed method. Our conclusions are presented in Section 5.

First of all, in this section we use the definition of the theorem for the necessary description and introduction. In the following, let ( X , R θ , μ ) is a measure space, where X be the universal set, R θ be a σ − ring of subsets of X , using a σ -algebra F as R θ is convenient for defining measures, μ be a measure on R θ , n > 0 .

Definition 2.1 [20] A function μ : 2 N → [ 0 , 1 ] is said to be a fuzzy measure if, for all A , B ⊂ N , it satisfies the following properties:

(1) Increasing: if A ⊂ B , then μ ( A ) ⊂ μ ( B ) ;

(2) Boundary conditions: μ ( ∅ ) = 0 and μ ( N ) = 1 .

Theorem 2.1 [20] Let ( X , R θ , μ ) be a measure space. Then Measure μ has the following properties.

(1) Monotonicity: whenever E ∈ R θ , F ∈ R θ , and E ⊂ F implies μ ( E ) ≤ μ ( F ) ;

(2) Continuity from below: whenever E i ∈ R θ , i = 1 , 2 , ⋯ , and { E i } is nondecreasing, implying that μ ( ∪ i = 1 ∞ ( E i ) ) = lim i → ∞ μ ( E i ) ;

(3) Continuity from above: whenever E i ∈ R θ , i = 1 , 2 , ⋯ , and { E i } is nonincreasing, and there exists i 0 such that μ ( E i 0 ) < ∞ implying that μ ( ∩ i = 1 ∞ ( E i ) ) = lim i → ∞ μ ( E i ) ;

Definition 2.2 [21] A bivariate function C : [ 0 , 1 ] 2 → [ 0 , 1 ] is called a Copula, iff

(1) C ( x , y ) + C ( x 1 , y 1 ) ≥ C ( x , y 1 ) + C ( x 1 , y ) ;

(2) C ( x , 0 ) = C ( 0 , x ) = 0 ;

(3) C ( x , 1 ) = C ( 1 , x ) = x ,

For all x , x 1 , y , y 1 ∈ [ 0 , 1 ] with x ≤ x 1 and y ≤ y 1 :

Copula are functions that link (two-dimensional) probability distribution functions to their one-dimensional margins, playing an significant role in the theory of probabilistic metric spaces and statistics [22]. Table 1 presents some examples of Copulas, such as t-norms, overlap functions and other non-associative copulas. These functions play a crucial role in the subsequent research.

Definition 2.3 [18] Nonnegative set function μ is said to satisfy the σ − λ rules iff:

where λ ∈ ( − 1 s u p μ , ∞ ) ∪ 0 , E i ⊂ ℱ , E i ∩ E j = ∅ for all i , j = 1 , 2 , ⋅ ⋅ ⋅ , and i ≠ j .

Table 1. Examples of copulas.

Definition 2.4 [18] A set function ν : A → R + , ν _ : A → R + , ν ¯ : A → R + , ν = [ ν _ , ν ¯ ] is called an interval-valued fuzzy measure iff:

(1) ν _ ( ∅ ) = 0 , ν ¯ ( ∅ ) = 0 ;

(2) if E , F ⊂ X , and E ⊂ F , then ν _ ( E ) ≤ ν _ ( F ) , ν ¯ ( E ) ≤ ν ¯ ( F ) ;

(3) for every E ⊂ X , ν _ ( E ) ≤ ν ¯ ( E ) ;

Definition 2.5 [18] If ν _ and ν ¯ follow the σ − λ rules, and ν _ ( X ) = 1 , ν ¯ ( X ) = 1 , then ν = [ ν _ , ν ¯ ] is referred an interval-valued Sugeno probability measure based on σ − λ rules, denoted by h λ = h _ λ , h ¯ λ .

Definition 2.6 [18] Let X be a finite set with power set 2 X , a set function ν : 2 X → [ ν _ , ν ¯ ] ⊂ L ( 0 , 1 ) is called regular interval fuzzy measure iff:

(1) Boundary Conditions: ν _ ( ∅ ) = ν ¯ ( ∅ ) = 0 , ν _ ( X ) = 1 , ν ¯ ( X ) = 1 ;

(2) Monotonicity: for any M , N ∈ 2 X with M ⊂ N , then ν _ ( M ) ≤ ν _ ( N ) , ν ¯ ( M ) ≤ ν ¯ ( N ) .

Definition 2.7 [18] Suppose X be a finite set with power set 2 X , a set function ν : 2 X → [ ν _ , ν ¯ ] ⊂ L ( 0 , 1 ) is called regular λ -interval fuzzy measure iff:

(1) ν _ ( ∅ ) = ν ¯ ( ∅ ) = 0 , ν _ ( X ) = 1 , ν ¯ ( X ) = 1 ;

(2) if E ⊂ X , F ⊂ X , E ∩ F = ∅ , then ν _ ( E ∪ F ) = ν _ ( E ) + ν _ ( F ) + λ ν _ ( E ) ν _ ( F ) ,

and,

where L ( 0 , 1 ) is the set of all closed subintervals of the unit interval.

Theorem 2.2 [20][27] The following formulas determine the parameters λ = ( λ 1 , λ 2 ) of the regular interval Sugeno probability measure:

Proof. Because λ = ( λ 1 , λ 2 ) and take into account that X = { x 1 , x 2 , ⋯ , x n } is a finitee set, g λ i = g λ ( x i ) ( i = 1 , 2 , ⋯ , n ) is said to be a measure of density. Then,

The definition of the classical Lebesgue integral takes into account additive measures, while the Choquet integral is a generalization of the Lebesgue integral and it considers fuzzy measures. In this section, by introducing the definition of discrete Choquet integral, the generalization of Choquet integral, namely CC-integral, is studied, and the expression of CC-integral on Sugeno probability measure is proposed.

Think of N = { 1 , ⋯ , n } as a finite set.

Definition 3.1 [28] Suppose ν : 2 N → [ 0 , 1 ] be a fuzzy measure. The discrete Choquet integral is the function ℭ m : [ 0 , 1 ] n → [ 0 , 1 ] , defined, for all of

observe that the equation can be also written as,

which we call the Choquet integral in its expanded form.

The discrete Choquet integral in its enlarged form can be combined with copulas to create a family of aggregation functions by simply replacing the product operation with a copula, according to Reference [5]. We refer to these functions as CC-integral.

Definition 3.2 [5] Suppose ν : 2 N → [ 0 , 1 ] be a fuzzy measure and C : [ 0 , 1 ] 2 → [ 0 , 1 ] be a bivariate copula. The Choquet-like Copula-based integral with respect to ν is defined as a function ℭ m : [ 0 , 1 ] n → [ 0 , 1 ] , given, for all x ∈ [ 0 , 1 ] n , by,

where

(1) ( f ( x ( 1 ) ) , ⋯ , f ( x ( n ) ) ) is an increasing permutation on the input f ( x ) , that is, 0 ≤ f ( x ( 1 ) ) ≤ ⋯ ≤ f ( x ( n ) ) ;

(2) f ( x ( 0 ) ) = 0 ;

(3) A ( i ) = { ( i ) , ⋯ , ( n ) } is the subset of indices corresponding to the n − i + 1 largest components of x → .

Definition 3.3 For a non-negative measurable interval-valued function F : Ω → I ( R + ) , and any set A ∈ A , the Choquet integral of F is characterized by its measurable selections:

where the set S F consists of all measurable selections f : Ω → R + .

Definition 3.4 [29] If there is a C-integral function h : Ω → P 0 ( R + ) , for any measurable selection f ∈ S F , A ∈ A has,

then F is said to be C-integrally bounded.

Theorem 3.1 [18] If C-be integrally bounded, F be a non-negative measurable, A ∈ A ; then, X is C-integrable on A and,

Remark If ( c ) ∫ A f _   d μ and ( c ) ∫ A f ¯   d μ exists and is bounded, then the interval-valued function f is C-integrable on A .

Theorem 3.2 Suppose f is an interval-valued function on X = { x 1 , x 2 , ⋯ , x n } and C is a Copula. Then, the CC-integral of f as regards the interval-valued Sugeno probability measure g λ on X is proposed by,

where x ′ 1 , x ′ 2 , ⋯ , x ′ n is a permutation of x 1 , x 2 , ⋯ , x n such that f ( x ′ 0 ) ≤ f ( x ′ 1 ) ≤ ⋯ ≤ f ( x ′ n ) , f ( x ′ 0 ) = [ 0 , 0 ] , X ′ i = { x ′ i , x ′ i + 1 , ⋯ , x ′ n } , i = 1 , 2 , ⋯ , n and X ′ n + 1 = ∅ .

Proof. Due to f is an interval-valued function on X , according to Theorem 3.1, we have,

Because the CC-integral is monotonic and continuous, let us assume that f _ and f ¯ are real-valued functions on x , respectively. Meanwhile, noted then nonnegativity and the monotonicity of the fuzzy measure, we can obtain

where C can be T M , T P , T L , T H P , O B , O m M , O α , C F , C L , C D i v , when C is T L ,

T L ( x , y ) = max { 0 , x + y − 1 } , we have an expression of the C_TL-integral on interval-valued Sugeno probability measure:

It is worth noting that when C is O m M , O m M ( x , y ) = min { x , y } max { x 2 , y 2 } , Copula is an overlap function, we have an expression of the C O m M -integral on interval-valued Sugeno probability measure:

MCDM is a decision-making process in which a decision maker needs to evaluate, compare, and eventually select (or rank, classify) one or more alternatives from a limited set of alternatives under multiple, often conflicting criteria. It can systematically deal with complex decision scenarios, especially when the decision involves multiple conflicting objectives, limited alternatives and subjective preferences, MCDM provides a scientific, transparent and defensible framework to transform multi-dimensional conflicts into computable optimization problems.