Choquet integrals with respect to the credibility measure

Abstract

The Choquet integral is proven quite reasonable as an integral form with respect to monotone measures, where the credibility measure is a specific case with self-duality. The main objective of this paper is to propose the Choquet integral of measurable functions on the credibility space, which bridges the gap between the Choquet integral and credibility theory. First, the Choquet integrals for nonnegative functions with respect to the credibility measure are introduced, and their properties are investigated such as the monotonicity and translatability. Then, the symmetric Choquet integrals and translatable Choquet integrals of any measurable functions are developed through the use of the Choquet integrals of nonnegative functions. Finally, Choquet integrals on finite sets based on the credibility measure are presented to simplify the calculation procedures.

Keywords

Choquet integral credibility measure symmetric choquet integral translatable choquet integral

1 Introduction

Integrals play an important role in many theoretical and applied fields. Classical integrals include Riemann integral, Lebesgue integral, Riemann-Stieltjes integral, and so on. They are usually used to measuring actual descriptions of the physical world, such as the length, area, and volume, which are σ-additive. Various approaches were developed for general integration in the middle of last century. For example, the Sugeno integral was built based on a normalized monotone measure [10 , 28]. Pan-integral was introduced and its properties discussed by Yang [31], and further investigated by Wang et al., Yang and Song [26 , 32]. It deserves mentioning that the Sugeno integral involves two binary operations, maximum and minimum of real numbers, while pan-integral involves two pan-operations, pan-addition and pan-multiplication. The Choquet integral, however, is a direct generalization of the Lebesgue integral [3]. It merges the information contained in a capacity, which is also called cooperative game, monotone measure or probability measure when σ-additive in other disciplines. In particular, when the set function is a probability measure, the Choquet integral coincides with the expectation in probability theory. Since its inception, the Choquet integral has been attracted by worldwide scholars in many categories such as multi-criteria decision making, classification and recognition [1 , 33].

It is well-known that the development of measure and integral theory goes with the demand of mathematics and its applications. In order to measure a fuzzy event, Zadeh, the founder of fuzzy logic, introduced the concepts of possibility measure and necessary measure, which are proved to be normal, nonnegative and monotone [34, 35]. However, both the possibility measure and necessary measure do not obey the law of truth conservation and are inconsistent with laws of exclude middle and contradiction. This is because they do not satisfy the self-duality property which is intuitive and important in both theory and practice. To address this issue, Liu and Liu presented the concept of credibility measure [13], a self-dual measure. Credibility theory, founded by Liu in 2004 [14] and refined by Liu in 2007 [16], is a new branch of mathematics for studying the behavior of fuzzy phenomena [11 , 20]. Since then, the credibility theory has been developed rapidly and applied widely. For example, Garg employed the credibility theory to construct the membership and nonmembership functions of intuitionistic fuzzy numbers [6]. Jalota et al. proposed some portfolio selection models to fit the uncertain portfolio parameters within the credibilistic framework [9]. Wu et al. provided a method to deal with principal-agent problems under incomplete information based on credibility measure [30].

Taking into consideration that minimum is a special t-norm (in fact, the largest t-norm), ⊤-fuzzy integral, a natural extension of the Sugeno integral, was introduced by scholars in fuzzy set community [29, 36]. Subsequently, the research was further developed by employing many other operations [18 , 24]. Moreover, Hu et al. brought out the concept of H-fuzzy integral by replacing the monotone measure in the ⊤-fuzzy integral with a credibility measure. Du proposed the pan-integral on a credibility space by employing the pan-operations other than t-norms [5]. Murofushi and Sugeno had shown that the Choquet integral is reasonable as an integral with respect to a monotone measure [22]. Inspired by this viewpoint, in the present paper, we present the Choquet integrals with respect to the credibility measure which have not been considered yet. The main contribution of our work is the introduction of the Choquet integrals for nonnegative functions and the symmetric/translatable Choquet integrals of measurable functions based on the credibility measure.

The rest parts of this paper are organized as follows. In Section 2, we recall briefly the basic concepts in the Choquet integral and credibility theory. In Section 3, we introduce the Choquet integrals for nonnegative functions with respect to the credibility measure, and investigate some properties of this type of integral construction such as the monotonicity and translatability. In Section 4, the symmetric Choquet integrals and translatable Choquet integrals of any measurable functions are developed. Choquet integrals on finite sets based on the credibility measure are proposed in Section 5. Section 6 concludes this paper with a summary and suggestions for further research.

2 Preliminaries

In this section, the traditional Choquet integral and credibility theory are reviewed.

Throughout the present paper, let Θ be a nonempty set and $P$ a σ-algebra over Θ. Each element in $P$ is called an event, and the two-tuple $(Θ, P)$ is referred to as a measurable space.

Definition 2.1. [28] Let $(Θ, P)$ be a measurable space. A monotone measure μ on $P$ is a set function μ : A → [0, ∞) satisfying

μ {∅} =0,

$A, B \in P$ , and A ⊆ B imply μ {A} ≤ μ {B}.

Definition 2.2. [15, 16] The set function $Cr : P \to [0, 1]$ is called a credibility measure if it satisfies the following four axioms:

Axiom 1. (Normality) Cr {Θ} =1.

Axiom 2. (Monotonicity) Cr {A} ≤ Cr {B} whenever A ⊆ B.

Axiom 3. (Self-Duality) Cr {A} + Cr {A^c} =1 for any event A.

Axiom 4. (Maximality) $Cr {⋃_{i} A_{i}} = sup_{i} Cr {A_{i}}$ for any events {A_i} with $sup_{i} Cr {A_{i}} < 0.5$ .

From this definition, one can see that Cr {∅} =0 and 0 ≤ Cr {A} ≤1 for all $A \in P$ . Thus the credibility measure Cr is a monotone measure on $P$ .

Definition 2.3. [15, 16] Let Θ be a nonempty set, $P$ a σ-algebra over Θ and Cr a credibility measure. Then the triplet $(Θ, P, Cr)$ is called a credibility space.

For the sake of convenience, in the sequel, we denote $R_{+} = [0, \infty)$ and B₊ the Borel field on $R_{+}$ .

A real-valued function $f : Θ \to R_{+}$ is said to be measurable iff $f^{- 1} (B) = {x \in Θ : f (x) \in B} \in P$ for any Borel set B ∈ B₊. The set of all finite nonnegative measurable function is denoted by G. Given f, g ∈ G, we write f ≤ g if f (x) ≤ g (x) for all x ∈ Θ.

For a given f ∈ G, denote f_α = {x ∈ Θ : f (x) ≥ α}, f_{α
⁺} = {x ∈ Θ : f (x) > α}, where $α \in R_{+}$ , and f_α and f_{α
⁺} are referred to as the α-level set and strict α-level set of f, respectively.

Definition 2.4. [22, 28] The Choquet integral of a nonnegative real-valued measurable function f ∈ G with respect to a monotone measure μ over $A \in P$ is defined by $(C) \int_{A} f d μ = \int_{0}^{\infty} μ {f_{α} \cap A} d α,$ where the right-hand side integral is the (improper) Riemann integral.

For a classical measure, the Choquet integral is equal to the usual Lebesgue integral.

Example 2.5. Let Θ = [0, 1], f (x) = x², ∀ x ∈ Θ, and μ (A) = m (A), where m is the Lebesgue measure and A ⊆ Θ. Then $\begin{matrix} (C) \int_{0}^{1} f d μ & = \int_{0}^{\infty} μ {f_{α}} d α \\ = \int_{0}^{1} μ {[\sqrt{α}, 1]} d α \\ = \int_{0}^{1} (1 - \sqrt{α}) d α \\ = \frac{1}{3} = \int_{0}^{1} f d x . \end{matrix}$

3 Choquet integral for nonnegative functions

In this section, the Choquet integral on the credibility space is introduced, and some of its elementary properties are examined.

Definition 3.1. Let f ∈ G and $A \in P$ . The Choquet integral of f on A with respect to Cr, which is denoted by (C) ∫_Af d Cr, is given by $(C) \int_{A} f d Cr = \int_{0}^{\infty} Cr {f_{α} \cap A} d α,$ where the right-hand side integral is the (improper) Riemann integral. When A = Θ, we simply write (C) ∫f d Cr instead of (C) ∫_Θf d Cr.

The following examples are employed to substantiate the conceptual argument.

Example 3.2. Let Θ = [0, 1], f (x) = x², ∀ x ∈ Θ, and $Cr {A} = \frac{1}{2} (sup_{x \in A} f (x) + 1 - sup_{x \in A^{c}} f (x)), \forall A \subseteq Θ$ . Then $\begin{matrix} (C) \int f d Cr & = \int_{0}^{\infty} Cr {f_{α}} d α \\ = \int_{0}^{1} Cr {[\sqrt{α}, 1]} d α \\ = \int_{0}^{1} \frac{2 - α}{2} d α \\ = \frac{3}{4} . \end{matrix}$ Moreover, for 0 ≤ t ≤ 1, $\begin{matrix} (C) \int_{t}^{1} f d Cr & = \int_{0}^{\infty} Cr {f_{α} \cap [t, 1]} d α \\ = \int_{t^{2}}^{1} Cr {[\sqrt{α}, 1]} d α \\ = \int_{t^{2}}^{1} \frac{2 - α}{2} d α \\ = \frac{t^{4} - 4 t^{2} + 3}{4} . \end{matrix}$ For 0 ≤ t < 1, $\begin{matrix} (C) \int_{0}^{t} f d Cr & = \int_{0}^{\infty} Cr {f_{α} \cap [0, t]} d α \\ = \int_{0}^{t^{2}} Cr {[\sqrt{α}, t]} d α \\ = \int_{0}^{t^{2}} \frac{t^{2}}{2} d α \\ = \frac{t^{4}}{2} . \end{matrix}$

Example 3.3. Let Θ = {a, b, c}, Cr be a credibility measure on $(Θ, P)$ with Cr {a} =0.35, Cr {b} =0.15, Cr {c} =0.65.

For this case, there are eight possible events: ∅, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, Θ with their credibilities being 0, 0.35, 0.15, 0.65, 0.35, 0.85, 0.65, 1, respectively. The credibility of A = {a, b}, for example, is obtained by the self-duality axiom (Cr {A} =1 - Cr {c} =0.35).

For the credibility space $(Θ, P, Cr)$ , take $f (x) = {\begin{matrix} 2, & x = a, \\ 3, & x = b, \\ 1, & x = c . \end{matrix}$

By Definition 3, we have $\begin{matrix} (C) \int_{\emptyset} f d Cr = 0, \\ (C) \int_{{a}} f d Cr = \int_{0}^{2} Cr {a} d α = 2 \cdot Cr {a} = 0.7, \\ (C) \int_{{b}} f d Cr = \int_{0}^{3} Cr {b} d α = 3 \cdot Cr {b} = 0.45, \\ (C) \int_{{c}} f d Cr = \int_{0}^{1} Cr {c} d α = Cr {c} = 0.65, \end{matrix}$ $\begin{matrix} (C) \int_{{a, b}} f d Cr & = \int_{0}^{2} Cr {a, b} d α + \int_{2}^{3} Cr {b} d α \\ = 2 \times 0.35 + 0.15 \\ = 0.85, \\ (C) \int_{{a, c}} f d Cr & = \int_{0}^{1} Cr {a, c} d α + \int_{1}^{2} Cr {a} d α \\ = 0.85 + 0.35 \\ = 1.2, \\ (C) \int_{{b, c}} f d Cr & = \int_{0}^{1} Cr {b, c} d α + \int_{1}^{3} Cr {b} d α \\ = 0.65 + 2 \times 0.15 \\ = 0.95, \\ (C) \int_{Θ} f d Cr & = \int_{0}^{1} Cr {Θ} d α + \int_{1}^{2} Cr {a, b} d α \\ + \int_{2}^{3} Cr {b} d α \\ = 1 + 0.35 + 0.15 \\ = 1.5 . \end{matrix}$

From the above two examples, we can see that for the Choquet integral on credibility space, the additivity with respect to the integration region may not hold, that is, $(C) \int_{A} f d Cr + (C) \int_{B} f d Cr \neq (C) \int_{A \cup B} f d Cr,$ even if A∩ B = ∅. For instance, in Example 3, $(C) \int_{0}^{1} f d Cr \neq (C) \int_{0}^{t} f d Cr + (C) \int_{t}^{1} f d Cr$ for 0 < t < 1; in Example 3, (C) ∫_{a,b}f d Cr ≠ (C) ∫_{a}f d Cr + (C) ∫_{b}f d Cr. Furthermore, the linearity with respect to the integrand does not hold either, namely, $(C) \int_{A} f d Cr + (C) \int_{A} g d Cr \neq (C) \int_{A} (f + g) d Cr .$

Theorem 3.4. Let f ∈ G and $A \in P$ . Then $(C) \int_{A} f d Cr = \int_{0}^{\infty} Cr {f_{α^{+}} \cap A} d α .$

Proof. For any given ɛ > 0, we have $\begin{matrix} (C) \int_{A} f d Cr & = \int_{0}^{\infty} Cr {f_{α} \cap A} d α \\ \geq \int_{0}^{\infty} Cr {f_{α^{+}} \cap A} d α \\ \geq \int_{0}^{\infty} Cr {f_{α + ɛ} \cap A} d (α + ɛ) \\ = \int_{ɛ}^{\infty} Cr {f_{α} \cap A} d α \\ \geq \int_{0}^{\infty} Cr {f_{α} \cap A} d α - ɛ \cdot Cr {A} \\ = (C) \int_{A} f d Cr - ɛ \cdot Cr {A} . \end{matrix}$ Letting ɛ → 0, we get the conclusion.

Proposition 3.5. Assume that f, g ∈ G and $A, B \in P$ , then we have the following properties.

If f ≤ g on A, then (C) ∫_Af d Cr ≤ (C) ∫_Ag d Cr.

If A ⊆ B, then (C) ∫_Af d Cr ≤ (C) ∫_Bf d Cr.

Proof. (1) $(C) \int_{A} 1 d Cr = \int_{0}^{1} Cr {A} d α = Cr {A} .$ $\begin{matrix} (2) (C) \int f \cdot χ_{A} d Cr & = \int_{0}^{\infty} Cr {(f \cdot χ_{A})_{α}} d α \\ = \int_{0}^{\infty} Cr {f_{α} \cap A} d α \\ = (C) \int_{A} f d Cr . \end{matrix}$

(3) From f ≤ g on A, it follows that f_α ∩ A ⊆ g_α ∩ A. Then $\begin{matrix} (C) \int_{A} f d Cr & = \int_{0}^{\infty} Cr {f_{α} \cap A} d α \\ \leq \int_{0}^{\infty} Cr {g_{α} \cap A} d α \\ = (C) \int_{A} g d Cr . \end{matrix}$

(4) By the fact that A ⊆ B, we have f · χ_A ≤ f · χ_B on Θ. Then from (3), the conclusion holds. $\begin{matrix} (5) (C) \int_{A} a \cdot f d Cr & = \int_{0}^{\infty} Cr {(a \cdot f)_{α} \cap A} d α \\ = a \cdot \int_{0}^{\infty} Cr {f_{α / a} \cap A} d (α / a) \\ = a \cdot \int_{0}^{\infty} Cr {f_{α} \cap A} d α \\ = a \cdot (C) \int_{A} f d Cr . \end{matrix}$

□

Corollary 3.6. Let f, g ∈ G and $A, B \in P$ . Then we have the following inequalities.

Theorem 3.7. (Translatability of the Choquet integral). For any constant c satisfying f + c ≥ 0, we have $(C) \int_{A} (f + c) d Cr = (C) \int_{A} f d Cr + c \cdot Cr {A} .$

Proof. From the definition of the Choquet integral directly, noticing that f (x) + c ≥ α for every x ∈ A if α is between 0 and c, we have $\begin{matrix} (C) \int_{A} (f + c) d Cr \\ = \int_{0}^{\infty} Cr {x \in A : f (x) + c \geq α} d α \\ = \int_{0}^{c} Cr {x \in A : f (x) + c \geq α} d α \\ + \int_{c}^{\infty} Cr {x \in A : f (x) + c \geq α} d α \\ = \int_{0}^{c} Cr {A \cap Θ} d α \\ + \int_{c}^{\infty} Cr {x \in A : f (x) \geq α - c} d (α - c) \\ = \int_{0}^{c} Cr {A} d α + \int_{0}^{\infty} Cr {x \in A : f (x) \geq α} d α \\ = c \cdot Cr {A} + (C) \int_{A} f d Cr . \end{matrix}$ Hence, indeed, the conclusion is valid.□

Example 3.8. (Continued from Example 3). For the credibility space $(Θ, P, Cr)$ , take c = -1, then $\begin{matrix} (C) \int (f + c) d Cr & = \int_{0}^{1} Cr {a, b} d α + \int_{1}^{2} Cr {b} d α \\ = 0.35 + 0.15 = 0.5 \\ = (C) \int f d Cr - Cr {Θ} . \end{matrix}$

4 Symmetric and translatable Choquet integrals

In this section, the symmetric and translatable Choquet integrals of measurable functions (not necessarily nonnegative) are presented.

Definition 4.1. The symmetric Choquet integral of real-valued measurable function f on A with respect to Cr, denoted by (C_s) ∫_Af d Cr, is defined by the difference $(C_{s}) \int_{A} f d Cr = (C) \int_{A} f^{+} d Cr - (C) \int_{A} f^{-} d Cr,$ where f⁺ and f^- are the positive and negative parts of f, i.e., $\begin{matrix} f^{+} (x) & = {\begin{matrix} f (x), & f (x) \geq 0, \\ 0, & otherwise . \end{matrix} \\ f^{-} (x) & = {\begin{matrix} - f (x), & f (x) \leq 0, \\ 0, & otherwise . \end{matrix} \end{matrix}$

Example 4.2. (Continued from Example 3). For the credibility space $(Θ, P, Cr)$ , if we take $g (x) = {\begin{matrix} 2, & x = a, \\ - 3, & x = b, \\ 1, & x = c, \end{matrix}$ then $\begin{matrix} (C_{s}) \int g d Cr & = (C) \int_{{a, c}} f d Cr - (C) \int_{{b}} f d Cr \\ = 1.2 - 0.45 \\ = 0.75 . \end{matrix}$

Clearly, the symmetric Choquet integral is symmetric, that is, $(C_{s}) \int_{A} - f d Cr = - (C_{s}) \int_{A} f d Cr$ holds for any real-valued measurable function f. Unfortunately, such an integral loses the translatability, which the Choquet integral with nonnegative integrand has. It means that the equality $(C_{s}) \int_{A} (f + c) d Cr = (C_{s}) \int_{A} f d Cr + c \cdot Cr {A}$ may not be always true for any real number c. For instance, in this example, take A = {b, c} and c = 2, then $\begin{matrix} (C_{s}) \int_{A} (g + 2) d Cr & = \int_{0}^{3} Cr {c} d α - \int_{0}^{1} Cr {b} d α \\ = 3 \times 0.65 - 0.15 = 1.8 \\ \neq (C_{s}) \int_{A} g d Cr + 2 \cdot Cr {A} \\ = 0.2 + 2 \times 0.65 = 1.5 . \end{matrix}$

Theorem 4.3. Let f and g be two real-valued measurable functions. If f ≤ g on A, then $(C_{s}) \int_{A} f d Cr \leq (C_{s}) \int_{A} g d Cr .$

Proof. It follows from the assumption f ≤ g that f⁺ ≤ g⁺ and f^- ≥ g^-. Thus, we have $\begin{matrix} (C_{s}) \int_{A} f d Cr & = (C) \int_{A} f^{+} d Cr - (C) \int_{A} f^{-} d Cr \\ \leq (C) \int_{A} g^{+} d Cr - (C) \int_{A} g^{-} d Cr \\ = (C_{s}) \int_{A} g d Cr . \end{matrix}$

□

The translatability is an important property of the Choquet integral, but the symmetric Choquet integral does not hold. To alleviate this problem, the translatable Choquet integral is introduced as follows.

Definition 4.4. The translatable Choquet integral of f with a lower bound b on A (namely, f (x) ≥ b, ∀ x ∈ A) with respect to Cr, denoted by (C_t) ∫_Af d Cr, is $(C_{t}) \int_{A} f d Cr = (C) \int_{A} (f - b) d Cr + b \cdot Cr {A} .$

This definition is well-defined. That is, if b₁ and b₂ are both lower bounds of function f, then $\begin{matrix} (C) \int_{A} (f - b_{1}) d Cr + b_{1} \cdot Cr {A} \\ = (C) \int_{A} (f - b_{2}) d Cr + b_{2} \cdot Cr {A} . \end{matrix}$

Example 4.5. (Continued from Example 3 and Example 4.) For the credibility space $(Θ, P, Cr)$ , if we take $g (x) = {\begin{matrix} 2, & x = a, \\ - 3, & x = b, \\ 1, & x = c, \end{matrix}$ obviously, -3 is a lower bound of function g. Thus, $\begin{matrix} (C_{t}) \int g d Cr = (C) \int (g + 3) d Cr - 3 \cdot Cr {Θ} \\ = \int_{0}^{4} Cr {a, c} d α + \int_{4}^{5} Cr {a} d α - 3 \cdot Cr {Θ} \\ = 4 \times 0.85 + 0.35 - 3 \\ = 0.75 . \end{matrix}$

Let us return to the above counterexample, it follows that, for A = {b, c}, $\begin{matrix} (C_{t}) \int_{A} (g + 2) d Cr \\ = (C) \int_{A} (g + 2 + 1) d Cr - Cr {A} \\ = \int_{0}^{4} Cr {c} d α - Cr {b, c} \\ = 4 \times 0.65 - 0.65 = 1.95 \\ = (C_{t}) \int_{A} g d Cr + 2 \cdot Cr {b, c} \\ = 0.65 + 2 \times 0.65 = 1.95 . \end{matrix}$

Furthermore, we have $\begin{matrix} (C_{s}) \int_{\emptyset} g d Cr = (C_{t}) \int_{\emptyset} g d Cr = 0, \\ (C_{s}) \int_{{a}} g d Cr = (C_{t}) \int_{{a}} g d Cr = 0.7, \\ (C_{s}) \int_{{b}} g d Cr = (C_{t}) \int_{{b}} g d Cr = - 0.45, \\ (C_{s}) \int_{{c}} g d Cr = (C_{t}) \int_{{c}} g d Cr = 0.65, \\ (C_{s}) \int_{{a, b}} g d Cr = 0.25 \neq (C_{t}) \int_{{a, b}} g d Cr = 0.7, \\ (C_{s}) \int_{{a, c}} g d Cr = (C_{t}) \int_{{a, c}} g d Cr = 1.2, \\ (C_{s}) \int_{{b, c}} g d Cr = 0.2 \neq (C_{t}) \int_{{b, c}} g d Cr = 0.65, \\ (C_{s}) \int_{Θ} g d Cr = (C_{t}) \int_{Θ} g d Cr = 0.75 . \end{matrix}$

From this example, we can see that the symmetric and translatable Choquet integrals are indeed two different concepts.

However, the translatable Choquet integral is not symmetric. In the aforementioned example, $\begin{matrix} (C_{t}) \int_{A} - g d Cr & = (C) \int_{A} (- g + 1) d Cr - Cr {A} \\ = \int_{0}^{4} Cr {b} d α - Cr {A} \\ = 4 \times 0.15 - 0.65 = - 0.05 \\ \neq - (C_{t}) \int_{A} g d Cr = - 0.65 . \end{matrix}$

The following theorem shows that the translatable Choquet integral for real-valued measurable functions presented in Definition 4 keeps the translatability.

Theorem 4.6. Let f be a measurable function on credibility space $(Θ, P, Cr)$ . Then, $(C_{t}) \int_{A} (f + c) d Cr = (C_{t}) \int_{A} f d Cr + c \cdot Cr {A}$ for any real number c.

Proof. Let b be a lower bound of function f, then b + c is a lower bound of function f + c, thus,

$\begin{matrix} (C_{t}) \int_{A} (f + c) d Cr \\ = (C) \int_{A} ((f + c) - (b - c)) d Cr + (b + c) \cdot Cr {A} \\ = (C) \int_{A} (f - b) d Cr + (b + c) \cdot Cr {A} \\ = (C_{t}) \int_{A} f d Cr + c \cdot Cr {A} . \end{matrix}$

□

Theorem 4.7. Let f be a measurable function with a lower bound b (b < 0) on A. Then, $\begin{matrix} (C_{t}) \int_{A} f d Cr & = \int_{b}^{0} (Cr {f_{α} \cap A} - Cr {A}) d α \\ + \int_{0}^{\infty} Cr {f_{α} \cap A} d α . \end{matrix}$ Specially, when A = Θ, $(C_{t}) \int f d Cr = \int_{0}^{\infty} Cr {f_{α}} d α - \int_{b}^{0} Cr {f_{α}^{c}} d α .$

Proof. By Definition 4, we have $\begin{matrix} (C_{t}) \int_{A} f d Cr = (C) \int_{A} (f - b) d Cr + b \cdot Cr {A} \\ = \int_{0}^{\infty} Cr {(f - b)_{α} \cap A} d α + b \cdot Cr {A} \\ = \int_{0}^{\infty} Cr {f_{α + b} \cap A} d α + b \cdot Cr {A} \\ = \int_{b}^{\infty} Cr {f_{α} \cap A} d α + b \cdot Cr {A} \\ = \int_{b}^{0} (Cr {f_{α} \cap A} - Cr {A}) d α \\ + \int_{0}^{\infty} Cr {f_{α} \cap A} d α . \end{matrix}$

□

Example 4.8. (Continued from Example 4). Compute the translatable Choquet integral of g by Theorem 4. $\begin{matrix} (C_{t}) \int g d Cr \\ = \int_{0}^{\infty} Cr {g_{α}} d α - \int_{- 3}^{0} (1 - Cr {g_{α}}) d α \\ = \int_{0}^{1} Cr {a, c} d α + \int_{1}^{2} Cr {a} d α - \int_{- 3}^{0} Cr {b} d α \\ = 0.85 + 0.35 - 3 \times 0.15 \\ = 0.75, \end{matrix}$ which accords with the result of Example 4.

Theorem 4.9. Let f and g be two lower-bounded measurable functions. If f ≤ g on A, then $(C_{t}) \int_{A} f d Cr \leq (C_{t}) \int_{A} g d Cr .$

Proof. Suppose that b is a lower bound of f, thus we have 0 ≤ f - b ≤ g - b. Therefore, $\begin{matrix} (C_{t}) \int_{A} f d Cr & = (C) \int_{A} (f - b) d Cr + b \cdot Cr {A} \\ \leq (C) \int_{A} (g - b) d Cr + b \cdot Cr {A} \\ = (C_{t}) \int_{A} g d Cr . \end{matrix}$

□

Obviously, if f is a nonnegative measurable function, then both the symmetric and translatable Choquet integrals of f coincide with its classical Choquet integral with respect to the credibility measure. By the symmetric and translatable Choquet integrals, the class of integrable functions is considerably enlarged. For some typical properties, eg, monotonicity (with respect to the integrand), translatability and symmetry, comparisons of these three types of Choquet integrals are summarized in Table 1, where the notation √ denotes the current integral fulfills the corresponding property, while × does not fulfill.

Table 1
Properties of different types of Choquet integrals based on the credibility measure

Property Monotonicity Translatability Symmetry

Classical type √ √ meaningless

Symmetric type √ × √

Translatable type √ √ ×

Property	Monotonicity	Translatability	Symmetry
Classical type	√	√	meaningless
Symmetric type	√	×	√
Translatable type	√	√	×

5 Choquet integrals on finite sets

Let Θ = {x₁, x₂, …, x_n} denote a finite set in this section. If the values of nonnegative function f, f (x₁) , f (x₂) , …, f (x_n), are rearranged into a nondecreasing order as $f (x_{1}^{*}) \leq f (x_{2}^{*}) \leq \dots \leq f (x_{n}^{*}),$ where $(x_{1}^{*}, x_{2}^{*}, \dots, x_{n}^{*})$ is a permutation of (x₁, x₂, …, x_n). Thus, we have $\begin{matrix} (C) \int f d Cr \\ = \sum_{i = 1}^{n} (f (x_{i}^{*}) - f (x_{i - 1}^{*})) Cr {x_{i}^{*}, x_{i + 1}^{*}, \dots, x_{n}^{*}}, \end{matrix}$ with a convention $f (x_{0}^{*}) = 0$ .

In fact, $\begin{matrix} (C) \int f d Cr = \int_{0}^{\infty} Cr {f_{α}} d α \\ = \sum_{i = 1}^{n} \int_{f (x_{i - 1}^{*})}^{f (x_{i}^{*})} Cr {f_{α}} d α \\ = \sum_{i = 1}^{n} (f (x_{i}^{*}) - f (x_{i - 1}^{*})) Cr {x_{i}^{*}, x_{i + 1}^{*}, \dots, x_{n}^{*}} . \end{matrix}$

Example 5.1. (Continued from Example 3). For the nonnegative function f on credibility space $(Θ, P, Cr)$ , it can be represented simply by (1, c), (2, a), (3, b). Then, $\begin{matrix} (C) \int f d Cr & = Cr {Θ} + (2 - 1) \cdot Cr {a, b} \\ + (3 - 2) \cdot Cr {b} \\ = 1 + 0.35 + 0.15 \\ = 1.5 . \end{matrix}$ If we take A = {b, c}, then $\begin{matrix} (C) \int_{A} f d Cr & = Cr {b, c} + (3 - 1) \cdot Cr {b} \\ = 0.65 + 2 \times 0.15 \\ = 0.95 . \end{matrix}$

Furthermore, we can reformulate the Choquet integral on a finite set as follows: $\begin{matrix} (C) \int f d Cr \\ = \sum_{i = 1}^{n} f (x_{i}^{*}) (Cr {x_{i}^{*}, \dots, x_{n}^{*}} - Cr {x_{i + 1}^{*}, \dots, x_{n}^{*}}), \end{matrix}$ with a convention ${x_{n + 1}^{*}, x_{n}^{*}} = \emptyset$ .

Example 5.2. (Continued from Example 3). For the function f on credibility space $(Θ, P, Cr)$ , $\begin{matrix} (C) \int f d Cr = 1 \cdot (Cr {Θ} - Cr {a, b}) \\ + 2 \cdot (Cr {a, b} - Cr {b}) + 3 \cdot Cr {b} \\ = 1 \times (1 - 0.35) + 2 \times (0.35 - 0.15) + 3 \times 0.15 \\ = 1.5 . \end{matrix}$ If we take A = {b, c}, then $\begin{matrix} (C) \int_{A} f d Cr & = 1 \cdot (Cr {b, c} - Cr {b}) + 3 \cdot Cr {b} \\ = 1 \times (0.65 - 0.15) + 3 \times 0.15 \\ = 0.95 . \end{matrix}$

Denote a set function $π : P \to [0, \infty)$ given as $\begin{matrix} π (A) & = {\begin{matrix} f (x_{i}^{*}) - f (x_{i - 1}^{*}), & A = {x_{i}^{*}, \dots, x_{n}^{*}}, \\ 0, & otherwise . \end{matrix} \end{matrix}$ Then we can verify that $f (x) = \sum_{x \in A \in P} π (A), \forall x \in Θ .$ Namely, π is a partition of f. Furthermore, if π and Cr are taken as $| P |$ -dimensional vectors, then we have $(C) \int f d Cr = 〈 π, Cr 〉,$ where 〈 · , · 〉 the inner product of vectors π and Cr.

Example 5.3. (Continued from Example 3). For the function f on credibility space $(Θ, P, Cr)$ , then $\begin{matrix} π (B) & = {\begin{matrix} 1, & B = Θ, \\ 1, & B = {a, b}, \\ 1, & B = {b}, \\ 0, & otherwise . \end{matrix} \end{matrix}$ Take $P = {\emptyset, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, Θ}$ , then π = (0, 0, 1, 0, 1, 0, 0, 1) ′ and Cr = (0, 0.35, 0.15, 0.65, 0.35, 0.85, 0.65, 1) ′. Thus, $(C) \int f d Cr = 〈 π, Cr 〉 = 0.15 + 0.35 + 1 = 1.5 .$ Moreover, if we take A = {b, c}, then $\begin{matrix} π (B) & = {\begin{matrix} 1, & B = {b, c}, \\ 2, & B = {b}, \\ 0, & otherwise . \end{matrix} \end{matrix}$ Take $P^{*} = {\emptyset, {b}, {c}, {b, c}}$ , then π^* = (0, 2, 0, 1) ′ and Cr^* = (0, 0.15, 0.65, 0.65) ′. Thus, $(C) \int_{A} f d Cr = 〈 π^{*}, {Cr}^{*} 〉 = 2 \times 0.15 + 0.65 = 0.95 .$

In the following numerical example, we show how to generate the credibility measures based on a possibility distribution.

Example 5.4. Let Θ be the universe of positive integers, and $F$ be a normal fuzzy set of “small integers” defined by $F = \frac{1}{1} + \frac{0.8}{2} + \frac{0.5}{3} + \frac{0.2}{4}$ . Then the proposition “X is a small integer” associates X with the possibility distribution $Pos {x} = F (x)$ , where x ∈ Θ and Pos {x} denotes the possibility that X is x given that X is a small integer. Due to space limitations, in Table 2, we only list the possibility measures, necessary measures and credibility measures on some specific sets for later use induced by the proposition. In the table, Pos {A} (Nec {A} and Cr {A}, respectively) signifies the possibility (necessary and credibility, respectively) that A contains a small integer, which are computed by: ∀A ⊆ Θ, $\begin{matrix} Pos {A} & = sup_{x \in A} Pos {x}, \\ Nec {A} & = 1 - Pos {A^{c}}, \\ Cr {A} & = \frac{1}{2} (Pos {A} + Nec {A}) . \end{matrix}$

Table 2
Possibility measures, necessary measures and credibility measures induced by $F$

A Pos {A} Nec {A} Cr {A}

{3} 0.5 0 0.25

{2, 3, 4} 0.8 0 0.4

{1, 2, 3, 4, 5} 1 1 1

A	Pos {A}	Nec {A}	Cr {A}
{3}	0.5	0	0.25
{2, 3, 4}	0.8	0	0.4
{1, 2, 3, 4, 5}	1	1	1

Let $f = \frac{0.3}{1} + \frac{0.7}{2} + \frac{1}{3} + \frac{0.7}{4} + \frac{0.3}{5}$ . Then $\begin{matrix} (C) \int f d Cr = \int_{0}^{1} Cr {f_{α}} d α = (1 - 0.7) \cdot Cr {3} \\ + (0.7 - 0.3) \cdot Cr {2, 3, 4} + 0.3 \cdot Cr {1, 2, 3, 4, 5} \\ = 0.535 . \end{matrix}$ Moreover, it follows that $\begin{matrix} (C) \int f d Pos = \int_{0}^{1} Pos {f_{α}} d α = 0.3 \cdot Pos {3} \\ + 0.4 \cdot Pos {2, 3, 4} + 0.3 \cdot Pos {1, 2, 3, 4, 5} \\ = 0.77, \\ (C) \int f d Nec = \int_{0}^{1} Nec {f_{α}} d α = 0.3 \cdot Nec {3} \\ + 0.4 \cdot Nec {2, 3, 4} + 0.3 \cdot Nec {1, 2, 3, 4, 5} \\ = 0.3 . \end{matrix}$ In fact, we always have $(C) \int_{A} f d Nec \leq (C) \int_{A} f d Cr \leq (C) \int_{A} f d Pos .$ Therefore, we can interpret the Choquet integral with respect to the credibility measure as a compromise between the Choquet integrals with respect to the possibility measure and necessary measure.

The following example is presented to illustrate the application of the credibility measure based Choquet integral to multiple attribute decision making.

Example 5.5. With the development of science and technology, the rapid rise of the emerging technology industry becomes a bright spot in the global economy. Many financial institutions and companies have many venture capital investments in emerging technology enterprises (ETEs). A commercial bank wants to invest in one of four potential ETEs f_i (i = 1, 2, 3, 4). There are four attributes that are used to evaluate the potential ETEs: the technical advancement (c₁), the development of science and technology (c₂), the employment creation (c₃), the industrialization and infrastructure (c₄). Assume that the attribute values are represented by real numbers up to 10. The decision makers evaluate these ETEs in terms of the above four attributes whose credibilities are 0.6, 0.25, 0.4 and 0.1, respectively, and give the decision information as shown in Table 3.

Table 3

Decision information about four emerging technology enterprises

	c ₁	c ₂	c ₃	c ₄
f ₁	8	5	6	9
f ₂	7	8	4	8
f ₃	9	6	8	5
f ₄	8	7	7	10

Note that the Choquet integral is a flexible aggregation tool in information fusion. Then, we use the proposed concept to select the best one out of these four ETEs.

By Definition 2, we can compute the credibility measures as follows: $\begin{matrix} Cr {c_{1}, c_{2}} = Cr {c_{1}, c_{4}} = Cr {c_{1}, c_{2}, c_{4}} = 0.6, \\ Cr {c_{2}, c_{3}} = Cr {c_{3}, c_{4}} = Cr {c_{2}, c_{3}, c_{4}} = 0.4, \\ Cr {c_{1}, c_{3}} = Cr {c_{1}, c_{3}, c_{4}} = 0.75, \\ Cr {c_{2}, c_{4}} = 0.25, Cr {c_{1}, c_{2}, c_{3}} = 0.9 . \end{matrix}$

Then, the aggregation of f₁ by the Choquet integral with respect to the credibility measure is $\begin{matrix} (C) \int f_{1} d Cr & = 5 \cdot Cr {Θ} + Cr {c_{1}, c_{3}, c_{4}} \\ + 2 \cdot Cr {c_{1}, c_{4}} + Cr {c_{4}} \\ = 7.05 . \end{matrix}$ Similarly, we have $\begin{matrix} (C) \int f_{2} d Cr & = 6.05, \\ (C) \int f_{3} d Cr & = 8, \\ (C) \int f_{4} d Cr & = 7.8 . \end{matrix}$

Hence, we can rank all the ETEs in descending order as $f_{3} ≻ f_{4} ≻ f_{1} ≻ f_{2},$ where ≻ denotes “preferred to”, and the best alternative is f₃ among these ETEs.

The weighted average (WA) operator is the most common and elementary aggregation tool in practice. For argument variables x₁, x₂, …, x_n associated with nonnegative weights w₁, w₂, …, w_n, their weighted arithmetic mean is $WA (x_{1}, x_{2}, \dots, x_{n}) = \frac{1}{W} \sum_{i = 1}^{n} w_{i} x_{i},$ where $W = \sum_{i = 1}^{n} w_{i}$ . If, in this example, the credibilities serve as the weights, then we can obtain that $\begin{matrix} WA (f_{1}) = 6.926, & WA (f_{2}) = 6.370, \\ WA (f_{3}) = 7.852, & WA (f_{4}) = 7.667 . \end{matrix}$ Then, it produces the same rankings and the best alternative is also f₃, which are in accord with our conclusions.

Suppose that there is another ETE $f = \frac{10}{c_{1}} + \frac{0}{c_{2}} + \frac{10}{c_{3}} + \frac{10}{c_{4}}$ , then we have WA (f) =8.148. Thus, the best alternative is f, which is inconsistent with our intuition. However, by the proposed method, we have (C) ∫f d Cr = 7.5. Therefore, the best alternative is still f₃.

6 Conclusions

In this study, we introduce the Choquet integral of measurable functions with respect to the credibility measure. When compared with H-fuzzy integral and Du’s Pan-integral, the Choquet integral, other than ⊤-fuzzy integral and Pan-integral, is adopted as the integral construction on the credibility space. The main contributions of this paper are listed as follows:

We introduce the Choquet integrals for nonnegative measurable functions on the credibility space. And it is proved that this new type of integral satisfies some important properties such as the monotonicity and translatability.

We propose the symmetric Choquet integrals and translatable Choquet integrals for measurable functions that are not necessarily nonnegative with respect to the credibility measure. It is shown that the developed symmetric Choquet integrals are symmetric, while the translatable Choquet integrals remain the translatability.

We present three numerical methods for computing the Choquet integrals on finite sets based on the credibility measure. By each method, one can obtain the same result as that by the conventional definition. An example for the investment selection is presented to illustrate the proposed method for multiple attribute decision making.

Our further work will focus on Choquet integrals of measurable functions with respect to other specific monotone measures. The relationships among them are deserved to be extensively examined.

Footnotes

Acknowledgements

The author is extremely grateful to the anonymous referees for their valuable comments and suggestions which helped to improve the presentation of this paper. This research was supported by the National Natural Science Foundation of China (Grant no. 61806182), the Scientific Research Fund for Young Teachers of Zhengzhou University (Grant no. 32220326), the Research Base Program of New Disciplines in Economics and Management of Zhengzhou University (Grant no. 101/32610168) and the Training Project for Young Backbone Teachers of Colleges and Universities of Henan Province.

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Choquet integrals with respect to the credibility measure

Abstract

Keywords

1 Introduction

2 Preliminaries

3 Choquet integral for nonnegative functions

4 Symmetric and translatable Choquet integrals

Table 1 Properties of different types of Choquet integrals based on the credibility measure Property Monotonicity Translatability Symmetry Classical type √ √ meaningless Symmetric type √ × √ Translatable type √ √ ×

Table 2 Possibility measures, necessary measures and credibility measures induced by F A Pos {A} Nec {A} Cr {A} {3} 0.5 0 0.25 {2, 3, 4} 0.8 0 0.4 {1, 2, 3, 4, 5} 1 1 1

Footnotes

Acknowledgements

References

Table 1
Properties of different types of Choquet integrals based on the credibility measure

Property Monotonicity Translatability Symmetry

Classical type √ √ meaningless

Symmetric type √ × √

Translatable type √ √ ×

Table 2
Possibility measures, necessary measures and credibility measures induced by $F$

A Pos {A} Nec {A} Cr {A}

{3} 0.5 0 0.25

{2, 3, 4} 0.8 0 0.4

{1, 2, 3, 4, 5} 1 1 1