The Scalar Mean Chance and Expected Value of Regular Bifuzzy Variables

As a natural extension of the fuzzy variable, a bifuzzy variable is defined as a mapping from a credibility space to the collection of fuzzy variables, which is an appropriate tool to model the two-fold fuzzy phenomena. In order to enrich its theoretical foundation, this paper explores some important measures for regular bifuzzy variables, the most commonly used type of bifuzzy variables. Firstly, we introduce the regular bifuzzy variables’ mean chance measure and some properties, including self-duality and its calculation formulas. Furthermore, we also investigate the mean chance distribution for strictly monotone functions of regular bifuzzy variables based on the proposed operational law. Finally, we present the expected value operator as well as equivalent analytical formulas of the expected value of regular bifuzzy variables and their strictly monotone functions.


Introduction
Since the fuzzy set theory was proposed by Zadeh [1], it has been widely adopted in a variety of academic research and real-world applications to deal with the imprecision implied in judgments, expertise, and linguistic descriptions. As this kind of research was explored in depth, scholars realized that the information of the fuzzy set cannot be accurately obtained in many cases. As a result, they paid close attention to the extensions on the fuzzy set. On the one hand, the incomplete information on the degree of membership attracted attention from scholars, and some well-accepted tools were developed to handle this type of fuzziness, such as the type 2 fuzzy set by Zadeh [2] in 1975, the twofold fuzzy set by Dubois and Prade [3] in 1983, and the intuitionistic fuzzy set by Atanassov [4] in 1986. On the other hand, the scenario with imprecise information on objects of the universal has also received attention. In 2002, Liu [5] initially put forward a new concept, the bifuzzy variable, as a function from a possibility space to the fuzzy variables set; moreover, some important attributes were also defined. As a continuation of [5], Zhou and Liu [6,7] investigated some mathematical properties of this kind of variable, such as chance distribution and expected value operator. Afterwards, Qin and Li [8] gave and proved a sufficient and necessary condition for the chance distribution of the bifuzzy variable.
In addition to theoretical studies, scholars have also considered their research under the bifuzzy environment because such complex uncertainty has extensively existed in practical problems. Xu and Yan [9] developed a multi-objective integer model under the bifuzzy environment to handle a vendor selection problem for material supply in large-scale water conservancy and hydro-power construction projects. Both Chakraborty et al. [10] and Bera et al. [11] investigated the multi-product production-inventory models with bifuzzy parameters. Deng and Qiu [12,13] proposed the bifuzzy discrete event systems and established the supervisory control theories on this type of system under different observation levels successively. Taib et al. [14] formulated a group decision-making model on the basis of conflicting bifuzzy sets and applied it in flood control projects. Shi et al. [15] introduced a novel bifuzzy scheduling model, in order to deal with two-fold uncertainties (ii) L(x) < 1, ∀x > 0; (iii) L(x) > 0, ∀x < 1; (iiii) L(1) = 0 or [L(x) > 0, ∀x and L(+∞) = 0]; (iiiii) L(x) is nonincreasing on [0, +∞), and the fuzzy numberτ has the following membership function where b is called the peak ofτ with µτ (b) = 1, and ι and υ are called the left and right spreads with ι > 0 and υ > 0, respectively, thenτ is said to be an LR-type fuzzy number. Symbolically,τ is written as (b, ι, υ) LR .
Example 1. If a triangular fuzzy numberτ is described as T (a 1 , a 2 , a 3 ) with real numbers a 1 < a 2 < a 3 , and the shape functions satisfy that L(x) = R(x) = max{0, 1 − x}, then the membership function ofτ is presented as Example 2. If an exponential fuzzy numberτ is described as E (m, p, p) with real number p > 0, and the shape functions satisfy L(x) = R(x) = e −x , then the membership function ofτ is presented as Theorem 1. (Zhou et al. [22]) Suppose thatτ is an LR-type fuzzy variable. If the credibility distribution ofτ is not only strictly increasing but continuous on its support, thenτ is called a regular LR fuzzy variable.
In the rest of this paper, for convenience, the triplet (b, ι, υ) LR , whose elements are defined in Definition 5, refers to a regular LR fuzzy variableτ. Generally, ifτ has the same shape functions (i.e., L(·) = R(·)) and same left and right spreads (i.e., ι = υ), we may call it symmetric, which is a well-used type applied in mathematical formulations of fuzzy programmings.

Remark 1.
Clearly, for Example 1, the credibility distribution ofτ can be deduced via its membership function according to Zhou et al. [22]. Thus, following from Definition 4 and the decreasing shape functions L(·) and R(·), the corresponding continuous and strictly increasing credibility distribution is obtained. Analogously, the same conclusions can be drawn for Example 2. Thus, these two LR-type fuzzy variables are obviously regular.
The credibility distributions and their inverse functions, named inverse credibility distributions, of the triangular fuzzy number and the exponential fuzzy variable are developed in Examples 3 and 4. Example 3. If a triangular fuzzy numberτ is described by T (a 1 , a 2 , a 3 ), then its credibility distribution Υτ and inverse credibility distribution Υ −1 τ are given as and Example 4. If an exponential fuzzy numberτ is described by E (m, p, p), then its credibility distribution Υτ and inverse credibility distribution Υ −1 τ are given as and Apparently, note that for the triangular fuzzy number defined in Example 3, when a 2 = 0.5(a 1 + a 3 ), it is said to be symmetric. Meanwhile, by virtue of the parametric form of an exponential fuzzy number, the variable defined in Example 4 is also absolutely a symmetric regular LR fuzzy variable.

Bifuzzy Variables
In addition, several definitions of extension of the fuzzy variable have been proposed aiming at describing the two-fold uncertain phenomena, such as the bifuzzy variable, type-2 fuzzy variable, intuitionistic fuzzy variable, etc. In this paper, the concept of the bifuzzy variable introduced by Liu [6] is adopted for further study, which is shown as follows.
Definition 6. (Liu [19]) Suppose that the triplet (Θ, P(Θ), Cr) represents a credibility space. Then, a bifuzzy variableκ can be regarded as a function from this space to the collection of fuzzy variables.

Remark 2.
Generally speaking, we may viewκ as a fuzzy variableκ defined on the universal collection of fuzzy variables. For the sake of convenience,κ is called a primary fuzzy number, and for each θ ∈ Θ,κ(θ) is called a secondary fuzzy number. Therefore, the set of all secondary fuzzy numbers is exactly the above universal collection.

Regular Bifuzzy Variables
On the foundation of the above definitions, we restrict our attention to the special type of bifuzzy variable, named the regular bifuzzy variable, and its definition and some elementary properties are discussed here as follows.

Definition 9.
A regular bifuzzy variable,κ, is a regular LR fuzzy variableκ defined on the special set of regular LR fuzzy variables satisfying the following conditions: (i) the left(and right) shape functions are the same; (ii) the left and right spreads are all denoted by two positive numbers; (iii) for θ ∈ Θ, the peak ofκ(θ) is determined byκ(θ).
Obviously, the bifuzzy variable defined in Example 6 is called regular, and yet, in Example 5, it is not this type due to the discontinuity of the membership function of its primary fuzzy number. For simplicity, the general form of a regular bifuzzy variable is given in the following.

Remark 3.
Suppose thatκ is a regular bifuzzy variable. If ι 1 > 0 and υ 1 > 0, the general form ofκ can be written asκ = (κ, ι 1 , υ 1 ) L 1 R 1 , whereκ = (b, ι 2 , υ 2 ) L 2 R 2 with real numbers ι 2 > 0 and υ 2 > 0. That is, the membership function of the primary fuzzy numberκ is and for any θ ∈ Θ, the membership function of the secondary fuzzy numberκ(θ) is Remark 4. Suppose thatκ is a regular bifuzzy variable If one of the following two conditions holds: (i) for each θ ∈ Θ,κ(θ) is a real number instead of a regular LR fuzzy variable; (ii) the nonempty set Θ contains only one singleton; then a regular LR fuzzy variable can be taken as a special type ofκ.

Example 7.
In particular, a regular bifuzzy variableκ is said to be triangular if its primary fuzzy number and secondary fuzzy numbers are both triangular. Roughly speaking,κ is a triangular fuzzy number taking triangular fuzzy values. Therefore, it is not hard to see that the variable defined in Example 6 is clearly a triangular regular bifuzzy variable. Analogously, a regular bifuzzy variable is said to be exponential if it is an exponential fuzzy number taking exponential fuzzy values.
For the purpose of deriving the arithmetic operations for independent regular bifuzzy variables, we recall the definition of the monotonicity of functions in the following.

Mean Chance Measure
In this section, the concept of mean chance measure of regular bifuzzy variables is presented, and some elementary properties are proved. In addition, the analysis formulas are discussed below. After that, the operational law for the measurability of the strictly monotone function of the regular bifuzzy variables is discussed, which will have extensive applications in solving the uncertain programmings, such as dependent-chance programming and chance-constrained programming, without simulation. In other words, using the operational law, the bifuzzy optimization problems, which consist of strictly monotone functions and regular bifuzzy variables, can be converted to their corresponding crisp equivalent programmings and then solved with well-developed software.

Mean Chance Distribution
In order to construct the chance of the occurrence of bifuzzy events, various definitions of the primitive chance have been suggested in Liu [19], including Pos-Pos, Nec-Nec, and Cr-Cr, etc. However, by referring to Liu and Liu [23], none of them are scalar values, not to mention self-dual, and so they are not suitable measures in terms of some mathematical properties, which may lead to counter-intuitive results in practice. Therefore, in light of the concept of the mean chance in [23], for a regular bifuzzy variable, we initialize its mean chance distribution through the Riemann integral as follows.
Definition 11. Suppose thatκ is a regular bifuzzy variable. Then, it has the mean chance distribution For simplicity, we could denote it as Ψκ : R → [0, 1].
According to Equation (15) and the characteristics of credibility measure, we could easily obtain the conclusion that the mean chance distribution is non-decreasing with respect to x. Specifically, Ψκ(−∞) = 0 and Ψκ(+∞) = 1. Following this, in order to prove the properties of the mean chance measure of bifuzzy variables, its self-duality should be firstly established.
Theorem 2. (Self-duality) Suppose thatκ is a regular bifuzzy variable. Then, for any x ∈ R, we have That is, the mean chance measure is self-dual.
Proof of Theorem 2. It is noteworthy that the self-duality of the credibility measure has been proven by Liu and Liu [18], i.e., Cr{τ ≥ x} = 1 − Cr{τ ≤ x}, whereτ is a fuzzy variable and x is a real number. Based on this, it follows from the definition of the mean chance measure that Thus, this theorem is naturally verified.

Theorem 3.
Suppose thatκ is a regular bifuzzy variable. If the credibility distribution of the primary fuzzy numberκ ofκ is described as Υκ, then the mean chance of {κ ≤ x} is calculated by whereξ is a secondary fuzzy number ofκ, expressed as (y, ι, υ) LR , with respect to y.
Proof of Theorem 3. Following from Definition 9, it is obvious thatκ(θ) is a regular LR fuzzy number for a determined θ. Consequently, if x is a real number, then Cr{κ(·) ≤ x} is a measurable function of θ. In fact, through Equation (15), the mean chance of the occurrence of the bifuzzy event {κ ≤ x} could be regarded as the expectation of the fuzzy variable Cr{κ(·) ≤ x}. In other words, whereξ is called the secondary fuzzy number for any real number y. This is simply Equation (17).
In the subsequent content, by taking advantage of the simple form of the credibility distribution of the symmetric regular LR fuzzy variable, the following two kinds of regular bifuzzy variables are adopted as examples to illustrate how to calculate their corresponding mean chance distributions. For simplicity, we only show the detailed processes of Theorem 4, and similar results can be obtained by Theorem 3.

Operational Law
In particular, to provide arithmetic operations for analytically and explicitly calculating the mean chance distribution of the strictly monotone and continuous function of independent regular bifuzzy variables, we first review the operational law for regular LR fuzzy numbers introduced by Zhou et al. [22], which is documented as follows.
Proof of Theorem 6. At first, it is obvious that the primary fuzzy number ofκ can be written asκ = g(κ 1 ,κ 2 , · · · ,κ n ), and on account of Theorem 5, its inverse credibility distribution could be formulated by Subsequently, following from Theorems 4, this theorem is naturally verified.
Theoretically, it is not easy to obtain the measurable function, i.e., Cr{g(ζ 1 ,ζ 2 , · · · ,ζ n ) ≤ x}, directly in Equation (26). Thus, considering the relationship between the credibility distribution and the inverse credibility distribution of a regular LR fuzzy number, we can further derive the equivalent form of the above theorem via an inverse operation.

Remark 6.
Note that for any β ∈ (0, 1), the inverse credibility distributions of the regular LR fuzzy variablesζ 1 ,ζ 2 , · · · ,ζ n are determined, respectively. Thus, the function g( is strictly monotone and continuous with regard to α, and so we could simulate the value of the root in Equation (29) through the golden section or bisection method.

Expected Value Operator
Usually, as one of the fundamental measures for uncertain events, the expected value has been given great attention by researchers, such as Dubois and Prade [24], Heilpern [25], Liu and Liu [18], etc. It should be mentioned that the first two are both defined on the basis of possibility measure, which has been proven not to satisfy the self-duality. Thus, in light of the work by Liu and Liu [18] and the self-duality of the mean chance measure demonstrated above, the expected value operator of a regular bifuzzy variableκ, defined by Choquet integrals, is given in this paper. Theorem 8. Suppose thatκ is a regular bifuzzy variable, and Ch M is the mean chance measure. Then, the expected value of the variableκ, E[κ], can be stated as where integrals are defined in case either of the two takes a finite value.
Proof of Theorem 8. Following from Definition 8 and Theorem 3, we have That is, this theorem is naturally verified.

Theorem 9.
Suppose thatκ is a regular bifuzzy variable. Then, it has the expected value where Ψκ represents the mean chance distribution ofκ.
Proof of Theorem 9. According to Equation (15), it is known that the mean chance distribution Ψκ(x) is strictly increasing and continuous, and so denote Ψκ(x) = which is the commonly used form to calculate the expected value in uncertainty theory. Thus, Equation (36) is naturally obtained.
Theorem 10. Suppose thatκ is a regular bifuzzy variable. If the mean chance distribution ofκ is described as Ψκ, and its inverse function is denoted as Ψ −1 κ , thenκ has the expected value Proof of Theorem 10. Following from Theorem 9, through the change of variables of the integral, set β = Ψκ(x), and so x = Ψ −1 κ (β). Then, we have Thus, Equation (37) is naturally obtained.
Thus far, two methodologies have been introduced to mathematically formulate the expected value of the regular bifuzzy variable on the foundation of its mean chance distribution. However, it seems somewhat complicated to calculate the expectation using the above methods. Alternatively, in this regard, the following equivalent analytical formulas are proposed to obtain the expected value in a straightforward manner on the basis of works by Li [26] and Zhou et al. [22]. Theorem 11. Suppose thatκ is a regular bifuzzy variable and the primary fuzzy numberκ ofκ has the credibility distribution Υκ. Then, the expected value ofκ can be calculated by whereξ is a secondary fuzzy number ofκ, expressed as (y, ι, υ) LR , with respect to y, and Υξ represents the credibility distribution ofξ.
By using Theorem 12, for triangular and exponential regular bifuzzy variables, we may obtain the following conclusions for their expected values. Particularly, for Theorems 9-11, the same results will be developed.
Not surprisingly, the expected value of a strictly monotone function with regular bifuzzy variables can be deduced via its mean chance distribution. Furthermore, based on the operational law for regular LR fuzzy variables, a novel formula is constructed for computing the expected value. Consequently, we could explicitly obtain the expectation without having to calculate the mean chance distribution. Theorem 13. Suppose thatκ i is a regular bifuzzy variable and the primary fuzzy numberκ i ofκ i has the inverse credibility distribution Υ −1 κ i , for i = 1, 2, · · · , n. Ifκ 1 ,κ 2 , · · · ,κ n are independent, and the function g(s 1 , s 2 , · · · , s n ) is strictly increasing with reference to s 1 , s 2 , · · · , s t and strictly decreasing with reference to s t+1 , s t+2 , · · · , s n , then denoteκ = g(κ 1 ,κ 2 , · · · ,κ n ) and the expected value ofκ is written as whereζ i is a secondary fuzzy number ofκ i , expressed as with respect to β.
Denote the function g 2 (x 1 , x 2 ) = x 1 /x 2 . For x 1 ≥ 0 and x 2 > 0, g 2 (x 1 , x 2 ) is strictly increasing with regard to x 1 and strictly decreasing with regard to x 2 . Therefore, ifκ 1 is non-negative andκ 2 is positive, then g 2 (κ 1 ,κ 2 ) has the expected value Accordingly, for Example 12, it is obvious that Furthermore, the linearity of the expected value operator of independent bifuzzy variables defined in Liu [6] can be mathematically proved.

Conclusions
In this paper, we restricted our attention on the special case of bifuzzy variables, namely regular bifuzzy variables, especially the most frequently used types, e.g., triangular and exponential regular bifuzzy variables. Two kinds of important measures were investigated in-depth, mean chance and expected value. The major contributions of this research include the following aspects: (1) the notion of mean chance measure of regular bifuzzy variables was presented, and some properties were provided; (2) the operational law for regular bifuzzy variables was proposed, and on this foundation, the mean chance distribu-tion of strictly monotone functions of this type of variable was developed; (3) an equivalent form of the expected value operator was constructed, and some theorems for computing the expected value of regular bifuzzy variables, as well as their strictly monotone functions, were also deduced.
Moreover, we believe that the proposed arithmetic operations and operational law are of great importance for optimization and decision making in the bifuzzy environment. In uncertain fields, dependent-chance programming, chance-constrained programming, and the expected value model are three frequently utilized approaches in handling practical problems, such as the vehicle routing problem, joint replenishment problem, and hub location problem, etc. Our proposed arithmetic operations enable the implementation of these programmings with strictly monotone functions and regular bifuzzy variables and provide a sound theoretical foundation for modeling bifuzzy phenomena, which is also one of the critical future directions of this research.