Fuzzy arithmetic on LR fuzzy numbers with applications to fuzzy programming

1 Introduction

In many research fields, such as optimal control and operations research, some problems are usually described as a programming model or other mathematical relationships with vague information (e.g., demand, time, distance, etc.). To model the imprecision in these problems, the fuzzy set theory introduced by Zadeh [31] can be employed, in which different types of fuzzy numbers and the corresponding fuzzy arithmetic on them are necessary in the modeling process.

In 1975, Zadeh [32] presented the extension principle for fuzzy basic operations involving addition, subtraction, multiplication, division and so on. Since implementing the Zadeh’s extension principle is equal to addressing a nonlinear programming, Kaufmann and Gupta [14] alternatively initialized an interval method for triangular and trapezoidal fuzzy numbers based on α-cut, which is easy to perform with low complexity for simple operations. However, their method may lead to higher powers of α when there are more terms being multiplied. For example, if there are n terms, the result of multiplication would be an nth-order polynomial in α. Further, all fuzzy numbers in the arithmetic procedure are treated as independent fuzzy numbers although most of them are not when based on the extension principle or the interval method. To tackle this problem, some researchers defined requisite constraints or proposed some novel approaches (see, e.g., [16, 25]).

Since the arithmetic based on above approaches are difficult to evaluate and computationally expensive, some approximation methods were introduced. For instance, Dubois and Prade [4] extended usual algebraic operations on real numbers to fuzzy numbers, and suggested a standard approximation to fuzzy arithmetic with efficient computation. Nevertheless, they reminded that frequent uses of the standard approximation for multiplication may lead to wrong results. Consequently Giachetti and Young [8] discussed the error of the standard approximation, and developed a new approximation for triangular and trapezoidal fuzzy numbers to reduce the error. In [9], they next proposed a form using six parameters to define fuzzy numbers, and provided the method for performing fuzzy arithmetic with better accuracy and similar computational speed with the standard approximation. Guerra and Stefanini [10] used piecewise monotonic interpolations to approximate and represent a fuzzy number, and derived a procedure to control the absolute error associated with the arithmetic operations on fuzzy numbers. Besides, some other methods and applications were also investigated recently. For instance, Chutia et al. [3] developed a generalised method to find the membership function for functions of triangular fuzzy numbers. Eslamipoor et al. [7] and Haji et al. [11] proposed different methods to rank fuzzy numbers by using distance method. Furthermore, based on the fuzzy arithmetic and a kind of ranking method called Mehar’s method [15], Hatami and Kazemipoor [12] solved the fully fuzzy linear programming in which all the parameters as well as variables were represented by fuzzy numbers. Madhuri et al. [23] defined a new arithmetic operations of linguistic trapezoidal fuzzy numbers for risk analysis.

The LR fuzzy number initialized by Dubois and Prade [4], especially the triangular fuzzy numbers as its special case, is a commonly used type of fuzzy numbers in various problems over the past few decades. An LR fuzzy number can be represented by its mean value (most likely value), left and right spreads (lower and upper bounds), and shape functions. As to the fuzzy arithmetic on LR fuzzy numbers, Wang and Kuo [28] proposed an alternative operation of fuzzy arithmetic on LR fuzzy numbers by three parameters, and developed a new learning algorithm of a fully fuzzified neural network based on the proposed approximation method. Sorini and Stefanini [27] suggested a parametrization for LR fuzzy numbers, and the parametric representations could be used to model the shapes of the membership functions and obtain operators for the fuzzy arithmetic operations. Chou [2] presented an inverse function arithmetic principle on triangular fuzzy numbers, which could easily interpret the multiplication operation with the membership functions of fuzzy numbers.

In this paper, we focus on a specific type of LR fuzzy numbers involving the triangular fuzzy number, the Gaussian fuzzy number, and the Cauchy fuzzy number as special cases, called regular LR fuzzy numbers. First, we discuss the credibility distributions of regular LR fuzzy numbers and give the equivalent conditions for a regular LR fuzzy number. Subsequently, the operational law is presented based on the inverse credibility distribution, which allows fuzzy arithmetic to be calculated exactly instead of approximation or simulation. Based on the proposed operational law, we give an equivalent definition and a simpler proof of linearity for the expected value operator of regular LR fuzzy numbers. Finally, a fuzzy programming model with an expected objective and chance constraints is formulated. We show that this model can be transferred to an equivalent crisp programming model by the operational law, and then solved with the aid of some well-developed optimization software packages.

The rest of this paper is organized as follows. InSection 2, the concept of LR fuzzy number is reviewed. In Section 3, we discuss the credibility distribution of LR fuzzy numbers based on the credibility measure, define the regular fuzzy number as well as the regular LR fuzzy number, and prove the equivalent relationship between them. In Section 4, a novel operational law for regular LR fuzzy numbers is put forward. In Section 5, an equivalent form of the expected value operator for LR fuzzy numbers is presented. In Section 6, the operational law is used to construct a solution framework of fuzzy programming, and then we illustrate it by an example about the purchasing planning problem.

2 LR fuzzy numbers

Dubois and Prade [4] initialized the well-known LR type of representation for fuzzy numbers, where L and R respectively denote the left and right shape functions which can be defined as follows.

Definition 1. (Dubois and Prade [4]) A shape function L (or R) is a decreasing function from R + → [ 0 , 1 ] such that

L (0) =1;

Example 2.1. Different functions can be chosen for L (x) (or R (x)). For instance, as mentioned by Dubois and Prade [6], L (x) = max {0, 1 - x ^p } with p > 0; L (x) = e^-x; and L (x) =1/(1 + x²).

Definition 2. (Dubois and Prade [6]) A fuzzy number ξ is of LR-type if there exist shape functions L (for left) and R (for right), and scalers α > 0, β > 0 with membership function μ ξ ( x ) = { L ( m - x α ) , if x ≤ m R ( x - m β ) , if x ≥ m where the real number m is called the mean value or peak of ξ, and α and β are called the left and right spreads, respectively. Symbolically, ξ is denoted by (m, α, β)  _LR .

Remark 1: If the mean value m is not a real number but an interval [ m _ , m ¯ ], then ξ is called a fuzzy interval or a generalized LR fuzzy number (see, e.g., [26, 35]). In the present paper, we only consider the situation that m _ = m ¯.

Example 2.2: Let L (x) = max {0, 1 - x}, R (x) = e^-x, α = 2, β = 3, and m = 4. Then (4, 2, 3)  _LR denotes an LR fuzzy number with membership function (see Fig. 1)μ ( x ) = { 0 , if x ≤ 2 x / 2 - 1 , if 2 < x ≤ 4 e ( 4 - x ) / 3 , if x > 4 .(1)

Example 2.3. If L (x) and R (x) are both linear functions on the domains {x|0 < L (x) <1} and {x|0 < R (x) <1}, the corresponding LR fuzzy number is a triangular fuzzy number. A triangular fuzzy number ξ determined by the triplet (a, b, c) of real numbers with a < b < c has a membership function (see Fig. 2)μ ( x ) = { x - a b - a , if a ≤ x ≤ b c - x c - b , if b < x ≤ c 0 , otherwise ,(2)which can be denoted as (b, b - a, c - b)  _LR , where the shape functions L and R are L ( x ) = R ( x ) = max { 0 , 1 - x } .

Example 2.4. A fuzzy number ξ is called a Gaussian fuzzy number if it has a membership function (see Fig. 3)μ ( x ) = e - ( x - a b ) 2 , x ∈ R , b > 0 ,(3)which can be denoted as (a, b, b)  _LR , where the shape functions L and R are L ( x ) = R ( x ) = e - x 2 .

Example 2.5. A fuzzy number ξ is called a Cauchy fuzzy number if it has a membership function (see Fig. 4)μ ( x ) = 1 1 + ( x - p q ) 2 , x ∈ R , q > 0 ,(4)which can be denoted as (p, q, q)  _LR , where the shape functions L and R are L ( x ) = R ( x ) = 1 / ( 1 + x 2 ) .

Example 2.6. Generally speaking, the shape function L (or R) defined in Definition 1 may be not continuous or strictly decreasing on the open interval {x|0 < L (x) <1}. For instance, suppose that the shape functions L (x) and R (x) are as follows, L ( x ) = { 1 , if x = 0 0.8 , if 0 < x ≤ 1 / 3 0.5 , if 1 / 3 < x ≤ 2 / 3 0.2 , if 2 / 3 < x < 1 0 , otherwise and R ( x ) = { - 3 x / 2 + 1 , if 0 ≤ x < 1 / 3 0.5 , if 1 / 3 ≤ x < 2 / 3 - 3 x / 2 + 3 / 2 , if 2 / 3 ≤ x < 1 0 , otherwise , which are noncontinuous and non-strictly decreasing, respectively. Letting the mean value, the left and right spreads m = α = β = 3, then we obtain an LR fuzzy number (3, 3, 3)  _LR with the following membership function (see Fig. 5)μ ( x ) = { 0.2 , if 0 < x < 1 0.5 , if 1 ≤ x < 2 0.8 , if 2 ≤ x < 3 - x / 2 + 5 / 2 , if 3 ≤ x < 4 0.5 , if 4 ≤ x < 5 - x / 2 + 3 , if 5 ≤ x < 6 0 , otherwise .(5)

This section first compares the credibility measure with the possibility measure and the necessity measure. Subsequently, we define the regular credibility distribution, the regular fuzzy number, and the inverse credibility distribution, respectively, on the basis of the credibility distribution. Finally, equivalent conditions for regular fuzzy numbers are proved.

4 Operational law

This section gives the operational law for calculating the credibility distribution of strictly monotone function of independent regular LR fuzzy numbers. The notions of independence of fuzzy numbers and strictly monotone function are recalled first.

The independence of fuzzy numbers has been studied by many researchers from different angles, such as Zadeh [33], Nahmias [24], Yager [30], Liu [18], and Liu and Gao [22]. Here we introduce the definition and its equivalent theorem given by Liu and Gao [22].

Definition 8. (Liu and Gao [22]) The fuzzy variables ξ₁, ξ₂, ⋯ , ξ _n are said to be independent if Cr { ξ i ∈ B i , i = 1 , 2 , ⋯ , n } = min 1 ≤ i ≤ n Cr { ξ i ∈ B i } for any Borel sets B₁, B₂, ⋯ , B _n of real numbers.

Theorem 3. (Liu and Gao [22]) The fuzzy variables ξ₁, ξ₂, ⋯ , ξ _n are independent if and only if Cr { ⋃ i = 1 n { ξ i ∈ B i } } = max 1 ≤ i ≤ n Cr { ξ i ∈ B i } for any Borel sets B₁, B₂, ⋯ , B _n of real numbers.

In this paper, we adopt the following definition of strictly monotone functions for our purpose.

Definition 9. A real-valued function f (x₁, x₂, ⋯ , x _n ) is said to be strictly monotone if it is strictly increasing with respect to x₁, x₂, ⋯ , x _m and strictly decreasing with respect to x_m+1, x_m+2, ⋯, x _n , that is, f ( x 1 , ⋯ , x m , x m + 1 , ⋯ , x n ) ≤ f ( y 1 , ⋯ , y m , y m + 1 , ⋯ , y n ) whenever x _i  ≤ y _i for i = 1, 2, ⋯ , m and x _i  ≥ y _i for i = m + 1, ⋯ , n, and f ( x 1 , ⋯ , x m , x m + 1 , ⋯ , x n ) < f ( y 1 , ⋯ , y m , y m + 1 , ⋯ , y n ) whenever x _i  < y _i for i = 1, 2, ⋯ , m and x _i  > y _i for i = m + 1, ⋯ , n.

Example 4.1. The following are strictly monotone functions, f ( x 1 , x 2 ) = x 1 - x 2 , f ( x 1 , x 2 ) = x 1 / x 2 , x 1 , x 2 > 0 .

Example 4.2. If f (x₁, x₂, ⋯, x _n ) ≤ f (y₁, y₂, ⋯ , y _n ) whenever x _i  ≤ y _i for all i, and f (x₁, x₂, ⋯ , x _n ) < f (y₁, y₂, ⋯ , y _n ) whenever x _i  < y _i for all i, the function is said to be strictly increasing functions. The following are strictly increasing functions, f ( x 1 , x 2 , ⋯ , x n ) = x 1 + x 2 + ⋯ + x n , f ( x 1 , x 2 , ⋯ , x n ) = x 1 x 2 ⋯ x n , x 1 , x 2 , ⋯ , x n ≥ 0 .

Example 4.3. If f (x₁, x₂, ⋯, x _n ) ≥ f (y₁, y₂, ⋯ , y _n ) whenever x _i  ≤ y _i for all i, and f (x₁, x₂, ⋯ , x _n ) > f (y₁, y₂, ⋯ , y _n ) whenever x _i  < y _i for all i, the function is said to be strictly decreasing functions. The following are strictly decreasing functions, f ( x 1 , x 2 , ⋯ , x n ) = - x 1 - x 2 ⋯ - x n , f ( x ) = exp ( - x ) , f ( x ) = 1 / x , x > 0 .

Based on the notions of independence and strictly monotone functions, we present the operational law for regular LR fuzzy numbers as follows.

Theorem 4. (Operational Law) Let ξ₁, ξ₂, ⋯ , ξ _n be independent regular LR fuzzy numbers with credibility distributions Φ₁, Φ₂, ⋯ , Φ _n , respectively. If the function f (x₁, x₂, ⋯ , x _n ) is strictly increasing with respect to x₁, x₂, ⋯ , x _m and strictly decreasing with respect to x_m+1, x_m+2, ⋯ , x _n , then ξ = f ( ξ 1 , ⋯ , ξ m , ξ m + 1 , ⋯ , ξ n ) is a regular LR fuzzy number with inverse credibility distribution Ψ - 1 ( α ) = f ( Φ 1 - 1 ( α ) , ⋯ , Φ m - 1 ( α ) , Φ m + 1 - 1 ( 1 - α ) , ⋯ , Φ n - 1 ( 1 - α ) ) .

Proof. For simplicity, we only prove the case of m = 1 and n = 2. That is, suppose thatξ = f ( ξ 1 , ξ 2 ) ,(13)and f is strictly increasing with respect to ξ₁ and strictly decreasing with respect to ξ₂. Besides, denoteG ( α ) = f ( Φ 1 - 1 ( α ) , Φ 2 - 1 ( 1 - α ) )(14)where Φ 1 - 1 and Φ 2 - 1 are the inverse credibility distributions of ξ₁ and ξ₂, respectively. Based on (13) and (14), we always have{ ξ ≤ G ( α ) } ≡ { f ( ξ 1 , ξ 2 ) ≤ f ( Φ 1 - 1 ( α ) , Φ 2 - 1 ( 1 - α ) ) } .(15)

On the one hand, since f is a strictly monotone function, from Definition 9, we can deduce that ξ 1 ≤ Φ 1 - 1 ( α ) and ξ 2 ≥ Φ 2 - 1 ( 1 - α ) ⇒ f ( ξ 1 , ξ 2 ) ≤ f ( Φ 1 - 1 ( α ) , Φ 2 - 1 ( 1 - α ) ) which implies that{ ξ 1 ≤ Φ 1 - 1 ( α ) } ∩ { ξ 2 ≥ Φ 2 - 1 ( 1 - α ) } ⊆ { f ( ξ 1 , ξ 2 ) ≤ f ( Φ 1 - 1 ( α ) , Φ 2 - 1 ( 1 - α ) ) } .(16)Then it follows from (15) and (16) that { ξ ≤ G ( α ) } ⊇ { ξ 1 ≤ Φ 1 - 1 ( α ) } ∩ { ξ 2 ≥ Φ 2 - 1 ( 1 - α ) } . Since the credibility measure Cr is an increasing set function Liu [17], we haveCr { ξ ≤ G ( α ) } ≥ Cr { { ξ 1 ≤ Φ 1 - 1 ( α ) } ∩ { ξ 2 ≥ Φ 2 - 1 ( 1 - α ) } } .(17)By using the definition of independence (see Definition 8) and the independence of ξ₁ and ξ₂, we getCr { { ξ 1 ≤ Φ 1 - 1 ( α ) } ∩ { ξ 2 ≥ Φ 2 - 1 ( 1 - α ) } } = Cr { ξ 1 ≤ Φ 1 - 1 ( α ) } ∧ Cr { ξ 2 ≥ Φ 2 - 1 ( 1 - α ) } = α ∧ α = α .(18)According to (17) and (18), we obtainCr { ξ ≤ G ( α ) } ≥ α .(19)

On the other hand, since f is strictly increasing with respect to ξ₁ and strictly decreasing with respect to ξ₂, we can deduce that f ( ξ 1 , ξ 2 ) ≤ f ( Φ 1 - 1 ( α ) , Φ 2 - 1 ( 1 - α ) ) ⇒ ξ 1 ≤ Φ 1 - 1 ( α ) or ξ 2 ≥ Φ 2 - 1 ( 1 - α ) which implies that{ f ( ξ 1 , ξ 2 ) ≤ f ( Φ 1 - 1 ( α ) , Φ 2 - 1 ( 1 - α ) ) } ⊆ { ξ 1 ≤ Φ 1 - 1 ( α ) } ∪ { ξ 2 ≥ Φ 2 - 1 ( 1 - α ) } .(20)Then it follows from (15) and (131) that{ ξ ≤ G ( α ) } ⊆ { ξ 1 ≤ Φ 1 - 1 ( α ) } ∪ { ξ 2 ≥ Φ 2 - 1 ( 1 - α ) } .(21)Since the credibility measure Cr is an increasing set function, we haveCr { ξ ≤ G ( α ) } ≤ Cr { { ξ 1 ≤ Φ 1 - 1 ( α ) } ∪ { ξ 2 ≥ Φ 2 - 1 ( 1 - α ) } } .(22)By using Theorem 3 and the independence of ξ₁ and ξ₂, we getCr { { ξ 1 ≤ Φ 1 - 1 ( α ) } ∪ { ξ 2 ≥ Φ 2 - 1 ( 1 - α ) } } = Cr { ξ 1 ≤ Φ 1 - 1 ( α ) } ∨ Cr { ξ 2 ≥ Φ 2 - 1 ( 1 - α ) } = α ∨ α = α .(23)According to (22) and (23), we obtainCr { ξ ≤ G ( α ) } ≤ α .(24)

Finally, it follows from (19) and (24) that Cr {ξ ≤ G (α)} = α. From the definition of inverse credibility distribution in Definition 6, we know that G (α) is just the inverse credibility distribution of ξ. That is, the inverse credibility distribution of ξ is Ψ - 1 ( α ) = G ( α ) = f ( Φ 1 - 1 ( α ) , Φ 2 - 1 ( 1 - α ) ) .

Furthermore, let us verify that ξ is a regular LR fuzzy number. Since ξ₁ and ξ₂ are regular LR fuzzy numbers, it follows from Theorem 14 that Φ 1 - 1 and Φ 2 - 1 are strictly increasing functions on (0, 1). Considering that f (x₁, x₂) is strictly increasing with respect to x₁ and strictly decreasing with respect to x₂, it is easy to deduce that Ψ - 1 ( α ) = f ( Φ 1 - 1 ( α ) , Φ 2 - 1 ( 1 - α ) ) is a strictly increasing function with respect to α. In other words, the inverse credibility distribution of ξ exists and strictly increasing on (0, 1). According to the assertion in Theorem 1, we obtain that ξ is a regular LR fuzzy number.

The above proof is also applicable the general cases. Here we only consider the case of m = 1, n = 2 for simplicity. □

Remark 2. If the function f (x₁, x₂, ⋯ , x _n ) is strictly increasing with respect to x₁, x₂, ⋯ , x _n , then ξ = f (ξ₁, ξ₂, ⋯ , ξ _n ) is a regular LR fuzzy number with inverse credibility distribution Ψ - 1 ( α ) = f ( Φ 1 - 1 ( α ) , Φ 2 - 1 ( α ) , ⋯ , Φ n - 1 ( α ) ).

Remark 3. If the function f (x₁, x₂, ⋯ , x _n ) is strictly decreasing with respect to x₁, x₂, ⋯ , x _n , then ξ = f (ξ₁, ξ₂, ⋯ , ξ _n ) is a regular LR fuzzy number with inverse credibility distribution Ψ - 1 ( α ) = f ( Φ 1 - 1 ( 1 - α ) , Φ 2 - 1 ( 1 - α ) , ⋯ , Φ n - 1 ( 1 - α ) ).

Example 4.4. Let ξ₁ and ξ₂ be independent regular LR fuzzy numbers with credibility distributions Φ₁ and Φ₂, respectively. Since the function f (x₁, x₂) = ax₁ - bx₂ is strictly increasing with respect to x₁ and strictly decreasing with respect to x₂ for any constants a > 0 and b > 0, it follows from Theorem 1 that aξ₁ - bξ₂ is a regular LR fuzzy number, and its inverse credibility distribution is Ψ - 1 ( α ) = a Φ 1 - 1 ( α ) - b Φ 2 - 1 ( 1 - α ) . Providing that ξ 1 ∼ N ( 3 , 2 ) and ξ 2 ∼ N ( 5 , 3 ) are two Gaussian fuzzy numbers, and the parameters a = 2 and b = 1, the inverse credibility distribution of 2ξ₁ - ξ₂ is Ψ - 1 ( α ) = 2 Φ 1 - 1 ( α ) - Φ 2 - 1 ( 1 - α ) = { 2 ( 3 - 2 - ln ( 2 α ) ) - ( 5 + 3 - ln ( 2 - 2 ( 1 - α ) ) ) , if α ≤ 0.5 2 ( 3 + 2 - ln ( 2 - 2 α ) ) - ( 5 - 3 - ln ( 2 ( 1 - α ) ) ) , if α > 0.5 = { 1 - 7 - ln ( 2 α ) , if α ≤ 0.5 1 + 7 - ln ( 2 - 2 α ) , if α > 0.5 according to (12) for α ∈ (0, 1). It is obvious that 2ξ₁ - ξ₂ is also a Gaussian fuzzy number, more clearly, 2 ξ 1 - ξ 2 ∼ N ( 1 , 7 ).

Example 4.5. Let ξ₁ and ξ₂ be independent positive regular LR fuzzy numbers with credibility distributions Φ₁ and Φ₂, respectively. Since f (x₁, x₂) = x₁/x₂ is a strictly monotone function for x₁, x₂ > 0, the quotient ξ₁/ξ₂ is a regular LR fuzzy number with the inverse credibility distribution Ψ - 1 ( α ) = Φ 1 - 1 ( α ) / Φ 2 - 1 ( 1 - α ) . Providing that ξ 1 ∼ T ( 1 , 2 , 3 ) and ξ 2 ∼ T ( 3 , 4 , 5 ) are two triangular fuzzy numbers, it follows from (11) that Φ 1 - 1 ( α ) = 2 α + 1 and Φ 2 - 1 ( α ) = 2 α + 3. Thus the inverse credibility distribution of ξ₁/ξ₂ is

Ψ - 1 ( α ) = ( 2 α + 1 ) / ( 2 ( 1 - α ) + 3 ) = ( 2 α + 1 ) / ( 5 - 2 α ) .(25)

In this section, we first give an equivalent form of the expected value for regular LR fuzzy numbers using the inverse credibility distribution, and then present a theorem for calculating the expected value of strictly monotone functions based on the equivalent form and the proposed operational law.

Expected value is the average value of a fuzzy variable in the sense of fuzzy measure. It has been defined in several ways. For instance, Dubois and Prade [5], Heilpern [13], Yager [29] gave the different definitions, respectively. In 2002, Liu and Liu [20] presented a general definition of expected value for fuzzy variables via the credibility distribution as follows.

Definition 10. (Liu and Liu [20]) Let ξ be a fuzzy variable. Then the expected value of ξ is defined by E [ ξ ] = ∫ 0 + ∞ Cr { ξ ≥ r } d r - ∫ - ∞ 0 Cr { ξ ≤ r } d r provided that at least one of the two integrals is finite.

In the following, we provide an equivalent form of the expected value for regular LR fuzzy numbers by means of the inverse credibility distribution.

Theorem 5. Let ξ be a regular LR fuzzy number. If its expected value exists, thenE [ ξ ] = ∫ 0 1 Φ - 1 ( α ) d α(26)where Φ^-1 is the inverse credibility distribution of ξ.

Proof. It follows from the definitions of expected value operator and credibility distribution that E [ ξ ] = ∫ 0 + ∞ Cr { ξ ≥ x } d x - ∫ - ∞ 0 Cr { ξ ≤ x } d x = ∫ 0 + ∞ ( 1 - Φ ( x ) ) d x - ∫ - ∞ 0 Φ ( x ) d x = ∫ Φ ( 0 ) 1 Φ - 1 ( α ) d α + ∫ 0 Φ ( 0 ) Φ - 1 ( α ) d α = ∫ 0 1 Φ - 1 ( α ) d α . □

Theorem 5 implies that the value of the expected value E [ξ] is just the area surrounded by two axes, α = 1, and the curve of the inverse credibility distribution Φ^-1. Figure 14 shows the geometric interpretation of the expected value of a regular LR fuzzy number.

Example 5.1. The expected value of a triangular fuzzy number ξ ∼ T ( a , b , c ) can be calculated according to (11) and (132) as E [ ξ ] = ∫ 0 0.5 ( ( 1 - 2 α ) a + 2 α b ) d α + ∫ 0.5 1 ( ( 2 - 2 α ) b + ( 2 α - 1 ) c ) d α = a + 2 b + c 4 .

Example 5.2. The expected value of a Gaussian fuzzy number ξ ∼ N ( a , b ) can be calculated according to (12) and (132) as E [ ξ ] = ∫ 0 0.5 ( a - b - ln ( 2 α ) ) d α + ∫ 0.5 1 ( a + b - ln ( 2 - 2 α ) ) d α = 0.5 a - b ∫ 0 0.5 - ln ( 2 α ) d α + 0.5 a + b ∫ 0.5 1 - ln ( 2 - 2 α ) d α = a - 0.5 b ∫ 0 1 - ln t d t - 0.5 b ∫ 1 0 - ln t d t = a .

According to the operational law and the equivalent form of the expected value, a theorem for calculating the expected value of strictly monotone functions is proved as follows.

Theorem 6. (Expected Value of Strictly Monotone Functions) Let ξ₁, ξ₂, ⋯ , ξ _n be independent regular LR fuzzy numbers with credibility distributions Φ₁, Φ₂, ⋯ , Φ _n , respectively. If the function f (x₁, x₂, ⋯, x _n ) is strictly increasing with respect to x₁, x₂, ⋯ , x _m and strictly decreasing with respect to x_m+1, x_m+2, ⋯ , x _n , then the expected value of fuzzy number ξ = f (ξ₁, ⋯ , ξ _m , ξ_m+1, ⋯ , ξ _n ) is E [ ξ ] = ∫ 0 1 f ( Φ 1 - 1 ( α ) , ⋯ , Φ m - 1 ( α ) , Φ m + 1 - 1 ( 1 - α ) , ⋯ , Φ n - 1 ( 1 - α ) ) d α .

Proof. It follows immediately from Theorems 4 and 5.

□

Example 5.3. Let ξ be a nonnegative regular LR fuzzy number with credibility distribution Φ. Since f (x) = x² is a strictly increasing function on [0, + ∞), it follows from Theorem 4 that the square ξ² is a regular LR fuzzy number with the inverse credibility distribution (Φ^-1 (α)) ². Then its expected value is E [ ξ 2 ] = ∫ 0 1 ( Φ - 1 ( α ) ) 2 d α . Providing that ξ ∼ T ( 1 , 2 , 3 ) is a triangular fuzzy number, the inverse credibility distribution of ξ is Φ^-1 (α) =1 + 2α by means of (11). Hence the expected value of ξ² is E [ ξ 2 ] = ∫ 0 1 ( 1 + 2 α ) 2 d α = 13 / 3 .

Example 5.4. Let ξ₁ and ξ₂ be independent positive regular LR fuzzy numbers with credibility distributions Φ₁ and Φ₂, respectively. Since f (x₁, x₂) = x₁/x₂ is a strictly monotone function for x₁, x₂ > 0, the quotient ξ₁/ξ₂ is a regular LR fuzzy number with the inverse credibility distribution ϒ ( α ) = Φ 1 - 1 ( α ) / Φ 2 - 1 ( 1 - α ). Then its expected value is E [ ξ 1 / ξ 2 ] = ∫ 0 1 Φ 1 - 1 ( α ) / Φ 2 - 1 ( 1 - α ) d α . Providing that ξ 1 ∼ T ( 1 , 2 , 3 ) and ξ 2 ∼ T ( 3 , 4 , 5 ) are two triangular fuzzy numbers, it is easy to have that the expected value of ξ₁/ξ₂ is E [ ξ 1 / ξ 2 ] = ∫ 0 1 2 α + 1 5 - 2 α d α = - 3 ln 0.6 - 1 ≈ 0.5325 according to (25) in Example 4.5.

Liu and Liu [21] have shown that the expected value operator defined in (132) has the linearity for independent fuzzy numbers. However, the proof is somewhat complicated. Here, we show that the same conclusion for regular LR fuzzy numbers can be obtained through a relatively simple derivation on the basis of Theorems 5 and 6.

Theorem 7. (Linearity of Expected Value Operator) Let ξ and η be independent regular LR fuzzy numbers with finite expected values. Then for any real numbers a and b, we have E [ a ξ + b η ] = aE [ ξ ] + bE [ η ] .

Proof. Suppose that the credibility distributions of ξ and η are Φ and Ψ, respectively. If a ≥ 0 and b ≥ 0, it follows from Theorems5 and 6 that E [ a ξ + b η ] = ∫ 0 1 ( a Φ - 1 ( α ) + b Ψ - 1 ( α ) ) d α = a ∫ 0 1 Φ - 1 ( α ) d α + b ∫ 0 1 Ψ - 1 ( α ) d α = aE [ ξ ] + bE [ η ] . Similarly, if a ≤ 0 and b ≥ 0, we have E [ a ξ + b η ] = ∫ 0 1 ( a Φ - 1 ( 1 - α ) + b Ψ - 1 ( α ) ) d α = a ∫ 0 1 Φ - 1 ( α ) d α + b ∫ 0 1 Ψ - 1 ( α ) d α = aE [ ξ ] + bE [ η ] . For simplicity, we only prove the above two cases. It is easy to deduce that the other two cases (i.e., a ≥ 0 and b ≤ 0; a ≤ 0 and b ≤ 0) can be also verified similarly. That is, for any real numbers a and b, the equality E [aξ + bη] = aE [ξ] + bE [η] holds. □

To handle the uncertain programming, Charnes and Cooper [1] initialized the chance-constrained programming which offers a powerful approach of modeling stochastic decision systems. Following the idea of stochastic chance-constrained programming, a framework of fuzzy chance-constrained programming was presented by Liu and Iwamura [19]. As a fuzzy version of the chance-constrained programming in [1], in this section, we present a fuzzy chance-constrained programming model integrating the expected objective with some chance constraints, and then show that the model can be converted to a crisp equivalent mathematical model by using the proposed operational law.