Fuzzy arithmetic on LR fuzzy numbers with applications to fuzzy programming

Abstract

In practice, some special LR fuzzy numbers, like the triangular fuzzy number, the Gaussian fuzzy number and the Cauchy fuzzy number, are widely used in many areas to deal with various vague information. With regard to these special LR fuzzy numbers, called regular LR fuzzy numbers in this paper, an operational law is proposed for fuzzy arithmetic, providing a novel approach to analytically and exactly calculating the inverse credibility distribution of some specific arithmetical operations based on the credibility measure. As an application of the operational law, an equivalent form of the expected value operator as well as a theorem for computing the expected value of strictly monotone functions is suggested. Finally, we utilize the operational law to construct a solution framework of fuzzy programming with parameters of regular LR fuzzy numbers, and such type of fuzzy programming problems can be handled by the operational law as the classic deterministic programming without any particular solving techniques.

Keywords

LR fuzzy number regular LR fuzzy number fuzzy arithmetic operational law expected value 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 $R^{+} \to [0, 1]$ such that

L (0) =1;

L (x) <1, ∀ x > 0;

L (x) >0, ∀ x < 1;

L (1) =0 [or L (x) >0, ∀ x and L (+ ∞) =0].

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 $μ_{ξ} (x) = {\begin{matrix} L (\frac{m - x}{α}), & if x \leq m \\ R (\frac{x - m}{β}), & if x \geq m \end{matrix}$ 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 $[\underline{m}, \bar{m}]$ , 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 $\underline{m} = \bar{m}$ .

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) $μ (x) = {\begin{matrix} 0, & if x \leq 2 \\ x / 2 - 1, & if 2 < x \leq 4 \\ e^{(4 - x) / 3}, & if x > 4 . \end{matrix}$ (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) $μ (x) = {\begin{matrix} \frac{x - a}{b - a}, & if a \leq x \leq b \\ \frac{c - x}{c - b}, & if b < x \leq c \\ 0, & otherwise, \end{matrix}$ (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) $μ (x) = e^{- (\frac{x - a}{b})^{2}}, x \in R, b > 0,$ (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) $μ (x) = \frac{1}{1 + {(\frac{x - p}{q})}^{2}}, x \in R, q > 0,$ (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, $L (x) = {\begin{matrix} 1, & if x = 0 \\ 0.8, & if 0 < x \leq 1 / 3 \\ 0.5, & if 1 / 3 < x \leq 2 / 3 \\ 0.2, & if 2 / 3 < x < 1 \\ 0, & otherwise \end{matrix}$ and $R (x) = {\begin{matrix} - 3 x / 2 + 1, & if 0 \leq x < 1 / 3 \\ 0.5, & if 1 / 3 \leq x < 2 / 3 \\ - 3 x / 2 + 3 / 2, & if 2 / 3 \leq x < 1 \\ 0, & otherwise, \end{matrix}$ 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) $μ (x) = {\begin{matrix} 0.2, & if 0 < x < 1 \\ 0.5, & if 1 \leq x < 2 \\ 0.8, & if 2 \leq x < 3 \\ - x / 2 + 5 / 2, & if 3 \leq x < 4 \\ 0.5, & if 4 \leq x < 5 \\ - x / 2 + 3, & if 5 \leq x < 6 \\ 0, & otherwise . \end{matrix}$ (5)

3 Regular fuzzy numbers

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.

3.1 Credibility measure

Suppose that ξ is a fuzzy number with the membership function μ, and r is a real number. The possibility [33] and the necessity [34] of a fuzzy event ξ ≤ r are expressed as follows, $Pos {ξ \leq r} = sup_{x \leq r} μ (x), Nec {ξ \leq r} = 1 - sup_{x > r} μ (x) .$

However, it is not suitable to use the possibility measure or necessity measure merely to measure a fuzzy event in a decision-making system because of the absence of the self-duality. To overcome this deficiency, the credibility measure was proposed in [20] as follows, $Cr {ξ \leq r} = \frac{1}{2} (Pos {ξ \leq r} + Nec {ξ \leq r}) .$ Liu and Liu [20] have also proved that the credibility measure is increasing and self-dual, and $Cr {ξ \leq r} = \frac{1}{2} (sup_{x \leq r} μ (x) + 1 - sup_{x > r} μ (x)) .$ (6)A self-dual measure is absolutely needed in both theory and practice. Hence the credibility measure will be considered in this paper.

3.2 Regular credibility distribution

Let Θ be a nonempty set representing the sample space, $P$ the power set of Θ, and Cr a credibility measure on $P$ . Then $(Θ, P (Θ), Cr)$ is called a credibility space. A fuzzy variable is defined as a function from a credibility space $(Θ, P (Θ), Cr)$ to the set of real numbers. Furthermore, Liu [17] defined the credibility distribution for fuzzy variables as follows.

Definition 3. (Liu [17], Credibility Distribution) The credibility distribution $Φ : R \to [0, 1]$ of a fuzzy variable ξ is defined by $Φ (x) = Cr {θ \in Θ | ξ (θ) \leq x} .$ (7)

Here, Φ (x) is the credibility that the fuzzy variable ξ takes a value less than or equal to x. Liu [18] proved that the credibility distribution Φ is a nondecreasing function on $R$ with Φ (- ∞) =0 and Φ (+ ∞) =1. The credibility distribution plays a key role when studying fuzzy variables just as the probability distribution for random variables.

Example 3.1. On the basis of (1), (6) and (7), the credibility distribution of the LR fuzzy number in Example 2.2 can be figured out as $Φ (x) = {\begin{matrix} 0, & if x \leq 2 \\ (x - 2) / 4, & if 2 < x \leq 4 \\ 1 - \frac{1}{2} e^{(4 - x) / 3}, & if x > 4, \end{matrix}$ which has been depicted in Fig. 6.

Example 3.2. A triangular fuzzy number with the membership function in (2) has the credibility distribution (see Fig. 7) $Φ (x) = {\begin{matrix} 0, & if x \leq a \\ (x - a) / 2 (b - a), & if a < x \leq b \\ (x + c - 2 b) / 2 (c - b), & if b < x \leq c \\ 1, & if x > c \end{matrix}$ (8)denoted by $T (a, b, c)$ , where a < b < c are real numbers.

Example 3.3. A Gaussian fuzzy number with the membership function in (3) has the credibility distribution (see Fig. 8) $Φ (x) = {\begin{matrix} \frac{1}{2} e^{- (\frac{x - a}{b})^{2}}, & if x \leq a \\ 1 - \frac{1}{2} e^{- (\frac{x - a}{b})^{2}}, & if x > a \end{matrix}$ (9)denoted by $N (a, b)$ , where a and b > 0 are realnumbers.

Example 3.4. A Cauchy fuzzy number with the membership function in (4) has the credibility distribution (see Fig. 9) $Φ (x) = {\begin{matrix} \frac{1}{2 + 2 {(\frac{x - p}{q})}^{2}}, & if x \leq q \\ 1 - \frac{1}{2 + 2 {(\frac{x - p}{q})}^{2}}, & if x > q \end{matrix}$ (10)denoted by $C (p, q)$ , where p and q > 0 are real numbers.

Example 3.5. The LR fuzzy number in Example 2.6 with the membership function in (5) has the credibility distribution (see Fig. 10) $Φ (x) = {\begin{matrix} 0, & if x \leq 0 \\ 0.1, & if 0 < x < 1 \\ 0.25, & if 1 \leq x < 2 \\ 0.4, & if 2 \leq x < 3 \\ x / 4 - 1 / 4, & if 3 \leq x < 4 \\ 0.75, & if 4 \leq x < 5 \\ x / 4 - 1 / 2, & if 5 \leq x < 6 \\ 1, & if x \geq 6 . \end{matrix}$

From Examples 3.1∼3.4, it can be seen that the credibility distributions of the LR fuzzy number in Example 2.2, the triangular fuzzy number, the Gaussian fuzzy number, and the Cauchy fuzzy number are all continuous and strictly increasing functions. In order to study this special type of LR fuzzy numbers conveniently, some new concepts are suggested as follows.

Definition 4. (Regular Credibility Distribution) A credibility distribution Φ is said to be regular if it is a continuous and strictly increasing function with respect to x at which 0 < Φ (x) <1, and $lim_{x \to - \infty} Φ (x) = 0, lim_{x \to + \infty} Φ (x) = 1 .$

Definition 5. (Regular Fuzzy Number) A fuzzy number is said to be regular if its credibility distribution is regular.

It is obvious that the LR fuzzy number in Example 2.2, the triangular fuzzy number, the Gaussian fuzzy number, and the Cauchy fuzzy number are all regular, and yet the LR fuzzy number in Example 2.6 is not regular due to its noncontinuous and non-strictly increasing credibility distribution.

As to the regular fuzzy numbers, the concept of the inverse credibility distribution is developed, which will play a significant part in the subsequent content.

Definition 6. (Inverse Credibility Distribution) Let ξ be a fuzzy number with a regular credibility distribution Φ. Then the inverse function Φ^-1 is called the inverse credibility distribution of ξ.

Note that the inverse credibility distribution Φ^-1 is well defined on the open interval (0, 1). If required, we may extend the domain via $Φ^{- 1} (0) = lim_{α ↓ 0} Φ^{- 1} (α), Φ^{- 1} (1) = lim_{α ↑ 1} Φ^{- 1} (α) .$

Example 3.6. The inverse credibility distribution of a triangular fuzzy number $ξ \sim T (a, b, c)$ is (see Fig. 11) $Φ^{- 1} (α) = {\begin{matrix} (2 b - 2 a) α + a, & if α < 0.5 \\ (2 c - 2 b) α + 2 b - c, & if α \geq 0.5 . \end{matrix}$ (11)

Example 3.7. The inverse credibility distribution of a Gaussian fuzzy number $ξ \sim N (a, b)$ is (see Fig. 12) $Φ^{- 1} (α) = {\begin{matrix} a - b \sqrt{- ln (2 α)}, & if α \leq 0.5 \\ a + b \sqrt{- ln (2 - 2 α)}, & if α > 0.5 . \end{matrix}$ (12)

Example 3.8. The inverse credibility distribution of a Cauchy fuzzy number $ξ \sim C (p, q)$ is (see Fig. 13) $Φ^{- 1} (α) = {\begin{matrix} p - q \sqrt{(1 - 2 α) / 2 α}, & if α \leq 0.5 \\ p + q \sqrt{(2 α - 1) / (2 - 2 α)}, & if α > 0.5 . \end{matrix}$

3.3 Equivalent conditions of regular fuzzy numbers

For our purpose, we first propose the followingdefinition.

Definition 7. (Regular LR Fuzzy Number) A fuzzy number is said to be a regular LR fuzzy number if it is an LR fuzzy number with a regular credibility distribution.

Next, equivalent conditions for regular fuzzy numbers as well as regular LR fuzzy numbers can be given via the following theorems.

Theorem 1. (Equivalent Condition I) A fuzzy number is a regular fuzzy number if and only if any one of the following assertions holds,

It is a regular LR fuzzy number;

The credibility distribution Φ is continuous and strictly increasing on {x|0 < Φ (x) <1};

The inverse function of Φ exists and is continuous and strictly increasing on (0, 1).

Proof. It is easy to verify that parts (ii) and (iii) are true according to the definition of regular fuzzy number. Hence we only prove part (i).

On the one hand, if a fuzzy number ξ is a regular LR fuzzy number, then its credibility distribution is regular according to the definition of regular LR fuzzy number, and consequently, it is also a regular fuzzy number.

On the other hand, if we assume that a fuzzy number ξ is a regular fuzzy number, then it can be verified that ξ is an LR fuzzy number with regular credibility distribution as follows.

Since ξ is a regular fuzzy number, its credibility distribution Φ (x) is continuous and strictly increasing on {x|0 < Φ (x) <1}, and $lim_{x \to + \infty} Φ (x) = 1$ . Consequently, concerning the membership function μ (x) of ξ, we can prove that there is only one point taking the membership value of 1 in (- ∞ , + ∞).

Firstly, it is clear that there exists at least one point with membership 1 in (- ∞ , + ∞). Otherwise, if μ (x) <1 for any x ∈ (- ∞ , + ∞), we have $\begin{matrix} Φ (x_{0}) & = \frac{1}{2} (sup_{x \leq x_{0}} μ (x) + 1 - sup_{x > x_{0}} μ (x)) \\ < \frac{1}{2} (1 + 1 - sup_{x > x_{0}} μ (x)) \\ \leq \frac{1}{2} (1 + 1 - 0) = 1 \end{matrix}$ holds for any x₀ ∈ (- ∞ , + ∞), which is contrary to that $lim_{x_{0} \to + \infty} Φ (x_{0}) = 1$ .

Next, we can verify that there is at most one point with membership 1 in (- ∞ , + ∞). Otherwise, if there exist two points x₁ < x₂ with μ (x₁) = μ (x₂) =1, then for any x₀ ∈ (x₁, x₂), we have $\begin{matrix} Φ (x_{0}) & = \frac{1}{2} (sup_{x \leq x_{0}} μ (x) + 1 - sup_{x > x_{0}} μ (x)) \\ = \frac{1}{2} (1 + 1 - 1) = \frac{1}{2}, \end{matrix}$ which is contrary to the assumption that Φ (x) is strictly increasing on {x|0 < Φ (x) <1}.

Therefore, there is only one point with membership 1. Denote it by m. That is, μ (m) =1.

For x₀ < m, according to (6) and (7), we have $\begin{matrix} Φ (x_{0}) & = \frac{1}{2} (sup_{x \leq x_{0}} μ (x) + 1 - sup_{x > x_{0}} μ (x)) \\ = \frac{1}{2} (sup_{x \leq x_{0}} μ (x) + 1 - μ (m)) \\ = \frac{1}{2} sup_{x \leq x_{0}} μ (x) . \end{matrix}$ Since Φ (x) is continuous and strictly increasing on {x|0 < Φ (x) <1}, thus μ (x) is continuous and strictly increasing on {x|g (0) < x < m}, where $g (0) = lim_{α ↓ 0} Φ^{- 1} (α)$ .

Similarly, for x₀ > m, we have $\begin{matrix} Φ (x_{0}) & = \frac{1}{2} (sup_{x \leq x_{0}} μ (x) + 1 - sup_{x > x_{0}} μ (x)) \\ = \frac{1}{2} (μ (m) + 1 - sup_{x > x_{0}} μ (x)) \\ = 1 - \frac{1}{2} sup_{x > x_{0}} μ (x) . \end{matrix}$ Since Φ (x) is continuous and strictly increasing on {x|0 < Φ (x) <1}, we have μ (x) is continuous and strictly decreasing on {x|m < x < g (1)}, where $g (1) = lim_{α ↑ 1} Φ^{- 1} (α)$ .

In summary, for the membership function μ (x) of ξ, we have $μ (x) {\begin{matrix} is continuous and stictly incresing, \\ if g (0) < x < m \\ = 1, if x = m \\ is continuous and stictly decresing, \\ if m < x < g (1) \\ = 0, otherwise . \end{matrix}$ It is clear that μ (x) can be represented in the LR-type with shape functions L (x) and R (x), where L (x) and R (x) are continuous and strictly decreasing on {x|0 < L (x) <1} and {x|0 < R (x) <1}. Therefore, ξ is an LR fuzzy number.

Furthermore, since the credibility distribution of ξ is regular, ξ is a regular LR fuzzy number. □

According to Theorem 1, the concepts of regular fuzzy number and regular LR fuzzy number are completely identical. In other words, any regular fuzzy numbers can be represented as an LR fuzzy number with a regular credibility distribution. Concerning the LR-type of representation for the regular fuzzy number, the following theorem can be further obtained.

Theorem 2. (Equivalent Condition II) A fuzzy number is a regular LR fuzzy number if and only if it is an LR fuzzy number and its shape functions L and R are continuous and strictly decreasing on the open intervals {x|0 < L (x) <1} and {x|0 < R (x) <1}, respectively.

Proof. On the one hand, if a fuzzy number ξ is a regular LR fuzzy number, following from the proof of Theorem 1, it is known that the membership function of ξ can be represented in the LR-type with shape functions L (x) and R (x), where L (x) and R (x) are continuous and strictly decreasing on {x|0 < L (x) <1} and {x|0 < R (x) <1}, respectively.

On the other hand, if a fuzzy number ξ is an LR fuzzy number with continuous and strictly decreasing shape functions L and R on {x|0 < L (x) <1} and {x|0 < R (x) <1}, then its credibility distribution Φ is regular, which can be verified in three cases as follows.

Assume that the membership function of ξ is $μ (x) = {\begin{matrix} L (\frac{m - x}{a}), & if x \leq m, a > 0 \\ R (\frac{x - m}{b}), & if x \geq m, b > 0 \end{matrix}$ where L (x) and R (x) are both continuous and decreasing functions on $R^{+}$ and strictly decreasing on the open intervals {x|0 < L (x) <1} and {x|0 < R (x) <1}. Denote $L^{- 1} (0) = lim_{α ↓ 0} L^{- 1} (α)$ and $R^{- 1} (0) = lim_{α ↓ 0} R^{- 1} (α)$ .

Firstly, if x₁ < x₂ ≤ m, according to (6) and (7), we have $\begin{matrix} Φ (x_{1}) & = \frac{1}{2} (sup_{x \leq x_{1}} μ (x) + 1 - sup_{x > x_{1}} μ (x)) \\ = \frac{1}{2} μ (x_{1}) \\ = \frac{1}{2} L (\frac{m - x_{1}}{a}) \end{matrix}$ and $\begin{matrix} Φ (x_{2}) & = \frac{1}{2} (sup_{x \leq x_{2}} μ (x) + 1 - sup_{x > x_{2}} μ (x)) \\ = \frac{1}{2} μ (x_{2}) \\ = \frac{1}{2} L (\frac{m - x_{2}}{a}) . \end{matrix}$ Since L (x) is monotone decreasing on $R^{+}$ and strictly decreasing on {x|0 < L (x) <1}, we have $Φ (x_{1}) \leq Φ (x_{2}), \forall x_{1} < x_{2} \leq m$ and $Φ (x_{1}) < Φ (x_{2}), \forall m - a L^{- 1} (0) < x_{1} < x_{2} \leq m .$

Secondly, if m ≤ x₁ < x₂, similarly, we have $Φ (x_{1}) \leq Φ (x_{2}), \forall m \leq x_{1} < x_{2}$ and $Φ (x_{1}) < Φ (x_{2}), \forall m \leq x_{1} < x_{2} < m + b R^{- 1} (0) .$

Finally, if x₁ < m < x₂, we have $Φ (x_{1}) = \frac{1}{2} L (\frac{m - x_{1}}{a}) < \frac{1}{2} L (0) = \frac{1}{2}$ and $Φ (x_{2}) = 1 - \frac{1}{2} R (\frac{x_{2} - m}{b}) > 1 - \frac{1}{2} R (0) = \frac{1}{2} .$ Then Φ (x₁) < Φ (x₂).

In summary, we obtain Φ (x₁) < Φ (x₂) for m - aL^-1 (0) < x₁ < x₂ < m + bR^-1 (0), which means Φ (x) is strictly increasing on (m - aL^-1 (0) , m + bR^-1 (0)), i.e., {x|0 < Φ (x) <1}. Therefore, the credibility distribution of ξ is regular, and consequently, ξ is a regular LR fuzzy number. □

So far, equivalent conditions for regular fuzzy numbers as well as regular LR fuzzy numbers have been proposed. It is shown that all regular fuzzy numbers can be represented as regular LR fuzzy numbers with continuous and strictly decreasing shape functions. In the following three sections of this paper (i.e., Sections 4∼6), we will focus on the fuzzy arithmetic on regular LR fuzzy numbers (i.e., regular fuzzy numbers) and its applications. For the sake of clearness and easy understanding, we prefer to adopt the former phrase, i.e, regular LR fuzzy number.

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 $Cr {ξ_{i} \in B_{i}, i = 1, 2, \dots, n} = min_{1 \leq i \leq n} Cr {ξ_{i} \in B_{i}}$ 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 $Cr {⋃_{i = 1}^{n} {ξ_{i} \in B_{i}}} = max_{1 \leq i \leq n} Cr {ξ_{i} \in B_{i}}$ 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, $\begin{matrix} f (x_{1}, \dots, x_{m}, x_{m + 1}, \dots, x_{n}) \\ \leq & f (y_{1}, \dots, y_{m}, y_{m + 1}, \dots, y_{n}) \end{matrix}$ whenever x_i ≤ y_i for i = 1, 2, ⋯ , m and x_i ≥ y_i for i = m + 1, ⋯ , n, and $\begin{matrix} f (x_{1}, \dots, x_{m}, x_{m + 1}, \dots, x_{n}) \\ < & f (y_{1}, \dots, y_{m}, y_{m + 1}, \dots, y_{n}) \end{matrix}$ 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, $\begin{matrix} f (x_{1}, x_{2}) = x_{1} - x_{2}, \\ f (x_{1}, x_{2}) = x_{1} / x_{2}, x_{1}, x_{2} > 0 . \end{matrix}$

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, $\begin{matrix} f (x_{1}, x_{2}, \dots, x_{n}) = x_{1} + x_{2} + \dots + x_{n}, \\ f (x_{1}, x_{2}, \dots, x_{n}) = x_{1} x_{2} \dots x_{n}, x_{1}, x_{2}, \dots, x_{n} \geq 0 . \end{matrix}$

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, $\begin{matrix} f (x_{1}, x_{2}, \dots, x_{n}) = - x_{1} - x_{2} \dots - x_{n}, \\ f (x) = exp (- x), \\ f (x) = 1 / x, x > 0 . \end{matrix}$

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 $ξ = f (ξ_{1}, \dots, ξ_{m}, ξ_{m + 1}, \dots, ξ_{n})$ is a regular LR fuzzy number with inverse credibility distribution $\begin{matrix} Ψ^{- 1} (α) = & f (Φ_{1}^{- 1} (α), \dots, Φ_{m}^{- 1} (α), \\ Φ_{m + 1}^{- 1} (1 - α), \dots, Φ_{n}^{- 1} (1 - α)) . \end{matrix}$

Proof. For simplicity, we only prove the case of m = 1 and n = 2. That is, suppose that $ξ = f (ξ_{1}, ξ_{2}),$ (13)and f is strictly increasing with respect to ξ₁ and strictly decreasing with respect to ξ₂. Besides, denote $G (α) = f (Φ_{1}^{- 1} (α), Φ_{2}^{- 1} (1 - α))$ (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 ${ξ \leq G (α)} \equiv {f (ξ_{1}, ξ_{2}) \leq f (Φ_{1}^{- 1} (α), Φ_{2}^{- 1} (1 - α))} .$ (15)

On the one hand, since f is a strictly monotone function, from Definition 9, we can deduce that $\begin{matrix} ξ_{1} \leq Φ_{1}^{- 1} (α) and ξ_{2} \geq Φ_{2}^{- 1} (1 - α) \\ \Rightarrow & f (ξ_{1}, ξ_{2}) \leq f (Φ_{1}^{- 1} (α), Φ_{2}^{- 1} (1 - α)) \end{matrix}$ which implies that $\begin{matrix} {ξ_{1} \leq Φ_{1}^{- 1} (α)} \cap {ξ_{2} \geq Φ_{2}^{- 1} (1 - α)} \\ \subseteq & {f (ξ_{1}, ξ_{2}) \leq f (Φ_{1}^{- 1} (α), Φ_{2}^{- 1} (1 - α))} . \end{matrix}$ (16)Then it follows from (15) and (16) that ${ξ \leq G (α)} \supseteq {ξ_{1} \leq Φ_{1}^{- 1} (α)} \cap {ξ_{2} \geq Φ_{2}^{- 1} (1 - α)} .$ Since the credibility measure Cr is an increasing set function Liu [17], we have $\begin{matrix} Cr {ξ \leq G (α)} \\ \geq & Cr {{ξ_{1} \leq Φ_{1}^{- 1} (α)} \cap {ξ_{2} \geq Φ_{2}^{- 1} (1 - α)}} . \end{matrix}$ (17)By using the definition of independence (see Definition 8) and the independence of ξ₁ and ξ₂, we get $\begin{matrix} Cr {{ξ_{1} \leq Φ_{1}^{- 1} (α)} \cap {ξ_{2} \geq Φ_{2}^{- 1} (1 - α)}} \\ = Cr {ξ_{1} \leq Φ_{1}^{- 1} (α)} \land Cr {ξ_{2} \geq Φ_{2}^{- 1} (1 - α)} \\ = α \land α = α . \end{matrix}$ (18)According to (17) and (18), we obtain $Cr {ξ \leq G (α)} \geq α .$ (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 $\begin{matrix} f (ξ_{1}, ξ_{2}) \leq f (Φ_{1}^{- 1} (α), Φ_{2}^{- 1} (1 - α)) \\ \Rightarrow & ξ_{1} \leq Φ_{1}^{- 1} (α) or ξ_{2} \geq Φ_{2}^{- 1} (1 - α) \end{matrix}$ which implies that $\begin{matrix} {f (ξ_{1}, ξ_{2}) \leq f (Φ_{1}^{- 1} (α), Φ_{2}^{- 1} (1 - α))} \\ \subseteq & {ξ_{1} \leq Φ_{1}^{- 1} (α)} \cup {ξ_{2} \geq Φ_{2}^{- 1} (1 - α)} . \end{matrix}$ (20)Then it follows from (15) and (131) that ${ξ \leq G (α)} \subseteq {ξ_{1} \leq Φ_{1}^{- 1} (α)} \cup {ξ_{2} \geq Φ_{2}^{- 1} (1 - α)} .$ (21)Since the credibility measure Cr is an increasing set function, we have $\begin{matrix} Cr {ξ \leq G (α)} \\ \leq & Cr {{ξ_{1} \leq Φ_{1}^{- 1} (α)} \cup {ξ_{2} \geq Φ_{2}^{- 1} (1 - α)}} . \end{matrix}$ (22)By using Theorem 3 and the independence of ξ₁ and ξ₂, we get $\begin{matrix} Cr {{ξ_{1} \leq Φ_{1}^{- 1} (α)} \cup {ξ_{2} \geq Φ_{2}^{- 1} (1 - α)}} \\ = Cr {ξ_{1} \leq Φ_{1}^{- 1} (α)} \lor Cr {ξ_{2} \geq Φ_{2}^{- 1} (1 - α)} \\ = α \lor α = α . \end{matrix}$ (23)According to (22) and (23), we obtain $Cr {ξ \leq G (α)} \leq α .$ (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 $Ψ^{- 1} (α) = f (Φ_{1}^{- 1} (α), Φ_{2}^{- 1} (α), \dots, Φ_{n}^{- 1} (α))$ .

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 $Ψ^{- 1} (α) = f (Φ_{1}^{- 1} (1 - α), Φ_{2}^{- 1} (1 - α), \dots, Φ_{n}^{- 1} (1 - α))$ .

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 $ξ_{1} \sim N (3, 2)$ and $ξ_{2} \sim N (5, 3)$ are two Gaussian fuzzy numbers, and the parameters a = 2 and b = 1, the inverse credibility distribution of 2ξ₁ - ξ₂ is $\begin{matrix} Ψ^{- 1} (α) = 2 Φ_{1}^{- 1} (α) - Φ_{2}^{- 1} (1 - α) \\ = {\begin{matrix} 2 (3 - 2 \sqrt{- ln (2 α)}) \\ - (5 + 3 \sqrt{- ln (2 - 2 (1 - α))}), & if α \leq 0.5 \\ 2 (3 + 2 \sqrt{- ln (2 - 2 α)}) \\ - (5 - 3 \sqrt{- ln (2 (1 - α))}), & if α > 0.5 \end{matrix} \\ = {\begin{matrix} 1 - 7 \sqrt{- ln (2 α)}, & if α \leq 0.5 \\ 1 + 7 \sqrt{- ln (2 - 2 α)}, & if α > 0.5 \end{matrix} \end{matrix}$ according to (12) for α ∈ (0, 1). It is obvious that 2ξ₁ - ξ₂ is also a Gaussian fuzzy number, more clearly, $2 ξ_{1} - ξ_{2} \sim N (1, 7)$ .

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 $ξ_{1} \sim T (1, 2, 3)$ and $ξ_{2} \sim T (3, 4, 5)$ 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

$\begin{matrix} Ψ^{- 1} (α) & = & (2 α + 1) / (2 (1 - α) + 3) \\ = & (2 α + 1) / (5 - 2 α) . \end{matrix}$ (25)

5 Expected value

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 $E [ξ] = \int_{0}^{+ \infty} Cr {ξ \geq r} d r - \int_{- \infty}^{0} Cr {ξ \leq r} d r$ 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, then $E [ξ] = \int_{0}^{1} Φ^{- 1} (α) d α$ (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 $\begin{matrix} E [ξ] & = \int_{0}^{+ \infty} Cr {ξ \geq x} d x - \int_{- \infty}^{0} Cr {ξ \leq x} d x \\ = \int_{0}^{+ \infty} (1 - Φ (x)) d x - \int_{- \infty}^{0} Φ (x) d x \\ = \int_{Φ (0)}^{1} Φ^{- 1} (α) d α + \int_{0}^{Φ (0)} Φ^{- 1} (α) d α \\ = \int_{0}^{1} Φ^{- 1} (α) d α . \end{matrix}$ □

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 $ξ \sim T (a, b, c)$ can be calculated according to (11) and (132) as $\begin{matrix} E [ξ] & = \int_{0}^{0.5} ((1 - 2 α) a + 2 α b) d α \\ + \int_{0.5}^{1} ((2 - 2 α) b + (2 α - 1) c) d α \\ = \frac{a + 2 b + c}{4} . \end{matrix}$

Example 5.2. The expected value of a Gaussian fuzzy number $ξ \sim N (a, b)$ can be calculated according to (12) and (132) as $\begin{matrix} E [ξ] = & \int_{0}^{0.5} (a - b \sqrt{- ln (2 α)}) d α \\ + \int_{0.5}^{1} (a + b \sqrt{- ln (2 - 2 α)}) d α \end{matrix}$ $\begin{matrix} = & 0.5 a - b \int_{0}^{0.5} \sqrt{- ln (2 α)} d α \\ + 0.5 a + b \int_{0.5}^{1} \sqrt{- ln (2 - 2 α)} d α \\ = & a - 0.5 b \int_{0}^{1} \sqrt{- ln t} d t - 0.5 b \int_{1}^{0} \sqrt{- ln t} d t \\ = & a . \end{matrix}$

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 $\begin{matrix} E [ξ] = & \int_{0}^{1} f (Φ_{1}^{- 1} (α), \dots, Φ_{m}^{- 1} (α), \\ Φ_{m + 1}^{- 1} (1 - α), \dots, Φ_{n}^{- 1} (1 - α)) d α . \end{matrix}$

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 $E [ξ^{2}] = \int_{0}^{1} (Φ^{- 1} (α))^{2} d α .$ Providing that $ξ \sim T (1, 2, 3)$ 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 $E [ξ^{2}] = \int_{0}^{1} (1 + 2 α)^{2} d α = 13 / 3 .$

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 $E [ξ_{1} / ξ_{2}] = \int_{0}^{1} Φ_{1}^{- 1} (α) / Φ_{2}^{- 1} (1 - α) d α .$ Providing that $ξ_{1} \sim T (1, 2, 3)$ and $ξ_{2} \sim T (3, 4, 5)$ are two triangular fuzzy numbers, it is easy to have that the expected value of ξ₁/ξ₂ is $E [ξ_{1} / ξ_{2}] = \int_{0}^{1} \frac{2 α + 1}{5 - 2 α} d α = - 3 ln 0.6 - 1 \approx 0.5325$ 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 $\begin{matrix} E [a ξ + b η] & = \int_{0}^{1} (a Φ^{- 1} (α) + b Ψ^{- 1} (α)) d α \\ = a \int_{0}^{1} Φ^{- 1} (α) d α + b \int_{0}^{1} Ψ^{- 1} (α) d α \\ = aE [ξ] + bE [η] . \end{matrix}$ Similarly, if a ≤ 0 and b ≥ 0, we have $\begin{matrix} E [a ξ + b η] & = \int_{0}^{1} (a Φ^{- 1} (1 - α) + b Ψ^{- 1} (α)) d α \\ = a \int_{0}^{1} Φ^{- 1} (α) d α + b \int_{0}^{1} Ψ^{- 1} (α) d α \\ = aE [ξ] + bE [η] . \end{matrix}$ 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. □

6 Fuzzy programming

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.

6.1 Fuzzy chance-constrained programming

Assume that x is a decision vector, ξ = (ξ₁, ξ₂, ⋯ , ξ_n) is an n-dimensional fuzzy vector, f ( x , ξ) is the objective function, and g_j ( x , ξ) is the constraint function for j = 1, 2, ⋯ , p. Since the objective function f ( x , ξ) is also a fuzzy variable, it cannot be minimized directly. Instead, we may minimize its expected value, i.e., E [f ( x , ξ)]. Besides, considering that the fuzzy constraints g_j ( x , ξ) ≤0, j = 1, 2, ⋯ , p, do not define a crisp feasible set, it is naturally desired that the fuzzy constraints hold with confidence levels α₁, α₂, ⋯ , α_p. Then we have a set of chance constraints as follows, $Cr {g_{j} (x, ξ) \leq 0} \geq α_{j}, j = 1, 2, \dots, p .$

Therefore, in order to formulate decision systems with fuzzy parameters, we present the following fuzzy chance-constrained programming model, ${\begin{matrix} min_{x} E [f (x, ξ)] \\ subject to : \\ \begin{matrix} Cr {g_{j} (x, ξ) \leq 0} \geq α_{j}, j = 1, 2, \dots, p . \end{matrix} \end{matrix}$ (27)The target of model (27) is to obtain a decision with the minimum expected objective value E [f (x, ξ)] subject to a series of chance constraints.

Definition 11.A vector x is called a feasible solution to the fuzzy programming model (27) if $Cr {g_{j} (x, ξ) \leq 0} \geq α_{j}$ holds for j = 1, 2, ⋯ , p.

Definition 12. A feasible soulution x ^* is called an optimal solution to the fuzzy programming model (27) if $E [f (x^{*}, ξ)] \leq E [f (x, ξ)]$ holds for any feasible solution x .

6.2 Crisp equivalent model

For the fuzzy programming model (27), if the fuzzy vector ξ consists of regular LR fuzzy numbers, a crisp equivalent form can be obtained by using the following theorems.

Theorem 8. Assume that the objective function f ( x , ξ₁, ξ₂, ⋯ , ξ_n) is strictly increasing with respect to ξ₁, ξ₂, ⋯, ξ_m and strictly decreasing with respect to ξ_m+1, ξ_m+2, ⋯ , ξ_n. If ξ₁, ξ₂, ⋯ , ξ_n are independent regular LR fuzzy numbers, then the expected objective function E [f (x, ξ₁, ξ₂, ⋯ , ξ_n)] in model (27) equals to $\begin{matrix} \int_{0}^{1} & f (x, Φ_{1}^{- 1} (α), \dots, Φ_{m}^{- 1} (α), \\ Φ_{m + 1}^{- 1} (1 - α), \dots, Φ_{n}^{- 1} (1 - α)) d α \end{matrix}$ where $Φ_{i}^{- 1}$ is the inverse credibility distribution of ξ_i for i = 1, 2, ⋯ , n.

Proof. It follows from Theorem 4 that the inverse credibility distribution of f (x, ξ₁, ξ₂, ⋯ , ξ_n) is $\begin{matrix} Ψ^{- 1} (x, α) = & f (x, Φ_{1}^{- 1} (α), \dots, Φ_{m}^{- 1} (α), \\ Φ_{m + 1}^{- 1} (1 - α), \dots, Φ_{n}^{- 1} (1 - α)) . \end{matrix}$ Using Theorem 5, we have $E [f (x, ξ_{1}, \dots, ξ_{n})] = \int_{0}^{1} Ψ^{- 1} (x, α) d α$ . □

Theorem 9. Assume that the constraint function g_j ( x , ξ₁, ξ₂, ⋯ , ξ_n) is strictly increasing with respect to ξ₁, ξ₂, ⋯ , ξ_{k
_j} and strictly decreasing with respect to ξ_{k_j+1}, ξ_{k_j+2}, ⋯ , ξ_n. If ξ₁, ξ₂, ⋯ , ξ_n are independent regular LR fuzzy numbers, then the chance constraint $Cr {g_{j} (x, ξ_{1}, ξ_{2}, \dots, ξ_{n}) \leq 0} \geq α$ (28)holds if and only if $\begin{matrix} g_{j} (x, Φ_{1}^{- 1} (α), \dots, Φ_{k_{j}}^{- 1} (α), \\ Φ_{k_{j} + 1}^{- 1} (1 - α), \dots, Φ_{n}^{- 1} (1 - α)) \leq 0 \end{matrix}$ where $Φ_{i}^{- 1}$ is the inverse credibility distribution of ξ_i for i = 1, 2, ⋯ , n.

Proof. It follows from the operational law in Theorem 4 that the inverse credibility distribution of g_j (x, ξ₁, ξ₂, ⋯ , ξ_n) is $\begin{matrix} Ψ^{- 1} (x, α) = & g_{j} (x, Φ_{j}^{- 1} (α), \dots, Φ_{k_{j}}^{- 1} (α), \\ Φ_{k_{j} + 1}^{- 1} (1 - α), \dots, Φ_{n}^{- 1} (1 - α)) . \end{matrix}$ On the other hand, it is obvious that (28) holds if and only if Ψ^-1 ( x , α) ≤0. □

Theorem 10. Assume that f ( x , ξ₁, ξ₂, ⋯ , ξ_n) is strictly increasing with respect to ξ₁, ξ₂, ⋯ , ξ_m and strictly decreasing with respect to ξ_m+1, ξ_m+2, ⋯ , ξ_n, and g_j (x, ξ₁, ξ₂, ⋯ , ξ_n) is strictly increasing with respect to ξ₁, ξ₂, ⋯ , ξ_{k
_j} and strictly decreasing with respect to ξ_{k_j+1}, ξ_{k_j+2}, ⋯ , ξ_n for j = 1, 2, ⋯ , p . If ξ₁, ξ₂, ⋯ , ξ_n are independent regular LR fuzzy numbers, then the fuzzy programming model (27) is equivalent to the crisp mathematical programming ${\begin{matrix} min_{x} \int_{0}^{1} f (x, Φ_{1}^{- 1} (α), \dots, Φ_{m}^{- 1} (α), \\ Φ_{m + 1}^{- 1} (1 - α), \dots, Φ_{n}^{- 1} (1 - α)) d α . \\ subject to : \\ \begin{matrix} g_{j} (x, Φ_{1}^{- 1} (α_{j}), \dots, Φ_{k_{j}}^{- 1} (α_{j}), \\ Φ_{k_{j} + 1}^{- 1} (1 - α_{j}), \dots, Φ_{n}^{- 1} (1 - α_{j})) \leq 0, \\ j = 1, 2, \dots, p \end{matrix} \end{matrix}$ (29)where $Φ_{i}^{- 1}$ is the inverse credibility distribution of ξ_i for i = 1, 2, ⋯ , n.

Proof. It follows from Theorems 8 and 9 immediately. □

As a result, based upon Theorem 10, if the objective function f ( x , ξ) and the constraint functions g_j (x, ξ), j = 1, 2, …, p, are strictly monotone and ξ consists of independent regular LR fuzzy numbers, we can convert the fuzzy programming model (27) to the crisp model (273). From the mathematical viewpoint, there is no difference between the crisp mathematical programming and classical programming except for an integral. Thus we may solve such type of fuzzy optimization problems within the framework of classic deterministic optimization requiring no particular solvingtechniques.

6.3 Numerical example

In the following, a purchasing planning problem is given to illustrate the solution framework of fuzzy programming via the proposed fuzzy arithmetic operations.

Consider a company which plans to purchase some machines to build a new plant. This plant is to supply three types of components for its downstream plant in this company. Each type of component is produced by different machines, and thus three types of machines should be purchased. Denote by x_i the number of the i-th type of machine purchased for i = 1, 2, 3,respectively.

The price of the i-th type of machine is a_i, and the total capital available for this procurement plan is a. Then we have a constraint on the capital budgeting as $a_{1} x_{1} + a_{2} x_{2} + a_{3} x_{3} \leq a .$

Another constraint for this purchasing planning problem is the limitation of maximum space available for the machines. Denote by b_i the space occupied by the i-th type of machine for i = 1, 2, 3, respectively, and by b the total available space. Then we have the following constraint, $b_{1} x_{1} + b_{2} x_{2} + b_{3} x_{3} \leq b .$

The production capacity of the i-th type of machine is η_i, and the demand of the i-th type of component produced by the i-th type of machine from the downstream plant is ξ_i, i = 1, 2, 3. Since the demand should be fulfilled, that is, shortage is not allowed, we have η_ix_i ≥ ξ_i, i = 1, 2, 3. In practice, the production capacity η_i and the future demand ξ_i are usually uncertain. Here we suppose that they are fuzzy variables. In this case, η_ix_i ≥ ξ_i does not define a crisp constraint. If the manager sets α_i as the confidence level to be achieved of meeting the demands of the i-th type of component, then we have the following chance constraints, $Cr {η_{i} x_{i} \geq ξ_{i}} \geq α_{i}, i = 1, 2, 3 .$

Assume that the profit produced by per i-th type of machine is τ_i for i = 1, 2, 3. Then the total profit is τ₁x₁ + τ₂x₂ + τ₃x₃. The profits τ_i, i = 1, 2, 3, are allocated by the company according to the sales of its final products, which are usually affected by season, competitors and other factors. Consequently, the profits are assumed as fuzzy variables in this paper, and the objective is to maximize the expected value of the total profit, i.e., $E [τ_{1} x_{1} + τ_{2} x_{2} + τ_{3} x_{3}] .$

In the end, we have the following integer programming model for this purchasing planning problem, ${\begin{matrix} max E [τ_{1} x_{1} + τ_{2} x_{2} + τ_{3} x_{3}] \\ subject to : \\ \begin{matrix} a_{1} x_{1} + a_{2} x_{2} + a_{3} x_{3} \leq a \\ b_{1} x_{1} + b_{2} x_{2} + b_{3} x_{3} \leq b \\ Cr {η_{i} x_{i} \geq ξ_{i}} \geq α_{i}, i = 1, 2, 3 \\ x_{i} are positive integers, i = 1, 2, 3 . \end{matrix} \end{matrix}$ (30)

Providing that ξ_i, η_i and τ_i are independent regular LR fuzzy numbers with credibility distribution ϒ_i, Ψ_i and Φ_i, respectively, i = 1, 2, 3, then we can convert model (30) to the following deterministic form according to Theorem 10, ${\begin{matrix} max x_{1} \int_{0}^{1} Φ_{1}^{- 1} (α) d α + x_{2} \int_{0}^{1} Φ_{2}^{- 1} (α) d α \\ + x_{3} \int_{0}^{1} Φ_{3}^{- 1} (α) d α \\ subject to : \\ \begin{matrix} a_{1} x_{1} + a_{2} x_{2} + a_{3} x_{3} \leq a \\ b_{1} x_{1} + b_{2} x_{2} + b_{3} x_{3} \leq b \\ ϒ_{i}^{- 1} (α_{i}) - x_{i} Ψ_{i}^{- 1} (1 - α_{i}) \leq 0, i = 1, 2, 3 \\ x_{i} are positive integers, i = 1, 2, 3, \end{matrix} \end{matrix}$ (31)which can be easily solved by classical numerical methods or intelligent algorithms.

For example, according to model (31) and the data listed in Table 1, we obtain the following linear integer programming model, ${\begin{matrix} max 4.25 x_{1} + 6.25 x_{2} + 4.5 x_{3} \\ subject to : \\ \begin{matrix} 5 x_{1} + 6 x_{2} + 4 x_{3} - 600 \leq 0 \\ 7 x_{1} + 6 x_{2} + 8 x_{3} - 800 \leq 0 \\ 204.8990 - 17.1283 x_{1} \leq 0 \\ 186.1237 - 25.2138 x_{2} \leq 0 \\ 207.3485 - 21.1711 x_{3} \leq 0 \\ x_{i} are positive integers, i = 1, 2, 3 . \end{matrix} \end{matrix}$ (32)

By using LINGO, the optimal total profit is obtained as 627.5, and the optimal solution is $(x_{1}^{*}, x_{2}^{*}, x_{3}^{*}) = (12, 62, 42) .$

Obviously, model (30) with parameters listed in Table 1 is hard to be solved by traditional approaches but fuzzy simulation (which has been suggested in [17]), since three types of fuzzy numbers, i.e., Gaussian fuzzy numbers, Cauchy fuzzy numbers, and triangular fuzzy numbers, simultaneously appear in the same model, which makes it more complicated. However, as shown in models (31) and (32), it can be transformed to a deterministic form and then be easily solved by classical methods following from the proposed solution framework of fuzzy programming with parameters of regular LR fuzzy numbers.

7 Conclusion

In this paper, we mainly concentrated on the fuzzy arithmetic on regular LR fuzzy numbers. The major results of this study include the following aspects: 1) the notion of regular LR fuzzy number was defined, and it is proved that a fuzzy number is a regular LR fuzzy number if and only if it is a regular fuzzy number. In other words, it has a continuous and strictly increasing credibility distribution; 2) an operational law for independent regular LR fuzzy numbers was proposed for fuzzy arithmetic; 3) an equivalent definition of the expected value operator and a theorem for calculating the expected value of strictly monotone functions were represented; 4) a solution framework of fuzzy programming was constructed based on the proposed operational law.

Acknowledgments

This work was supported in part by a grant from the Ministry of Education Funded Project for Humanities and Social Sciences Research (No. 14YJC630124).

References

Charnes

and Cooper

W.W.

, Management Models and Industrial Applications of Linear Programming, Wiley, 1961.

Chou

C.C.

, The canonical representation of multiplication operation on triangular fuzzy numbers, Computers & Mathematics with Applications 45(10) (2003), 1601–1610.

Chutia

, Mahanta

and Datta

, Arithmetic of triangular fuzzy variable from credibility theory, International Journal of Energy, Information and Communications 2(3) (2011), 9–20.

Dubois

and Prade

, Operations on fuzzy numbers, International Journal of Systems Science 9(6) (1978), 613–626.

Dubois

and Prade

, The mean value of a fuzzy number, Fuzzy Sets and Systems 24(3) (1987), 279–300.

Dubois

and Prade

, Possibility Theory, Plenum Press, 1988.

Eslamipoor

and Haji

M.J.

, Sepehriar

, Proposing a revised method for ranking fuzzy numbers, Journal of Intelligent and Fuzzy Systems 25(2) (2013), 373–378.

Giachetti

R.E.

and Young

R.E.

, Analysis of the error in the standard approximation used for multiplication of triangular and trapezoidal fuzzy numbers and the development of a new approximation, Fuzzy Sets and Systems 91(1) (1997), 1–13.

Giachetti

R.E.

and Young

R.E.

, A parametric representation of fuzzy numbers and their arithmetic operators, Fuzzy Sets and Systems 91(2) (1997), 185–202.

10.

Guerra

M.L.

and Stefanini

, Approximate fuzzy arithmetic operations using monotonic interpolations, Fuzzy Sets and Systems 150(1) (2005), 5–33.

11.

Haji

M.J.

, Zare

H.K.

, Eslamipoor

and Sepehriar

, A developed distance method for ranking generalized fuzzy numbers, Neural Computing and Applications 25(3-4) (2014), 727–731.

12.

Hatami

and Kazemipoor

, Solving fully fuzzy linear programming with symmetric trapezoidal fuzzy numbers using Mehar’s method, Journal of Mathematical and Computational Science 4(2) (2014), 463–470.

13.

Heilpern

, The expected value of a fuzzy number, Fuzzy Sets and Systems 47(1) (1992), 81–86.

14.

Kaufmann

and Gupta

M.M.

, Fuzzy Mathematical Models in Engineering and Management Science, Plenum Press, 1988.

15.

Kaur

and Kumar

, Mehar’s method for solving fully fuzzy linear programming problems with LR fuzzy parameters, Applied Mathematical Modelling 37(12) (2013), 7142–7153.

16.

Klir

G.J.

, Fuzzy arithmetic with requisite constraints, Fuzzy Sets and Systems 91(2) (1997), 165–175.

17.

Liu

, Theory and Practice of Uncertain Programming, Springer, 2002.

18.

Liu

, Uncertainty Theory, Springer, 2004.

19.

Liu

and Iwamura

, Chance-constrained programming with fuzzy parameters, Fuzzy Sets and Systems 94(2) (1998), 227–237.

20.

Liu

and Liu

, Expected value of fuzzy variable and fuzzy expected value models, IEEE Transactions on Fuzzy Systems 10(4) (2002), 445–450.

21.

Liu

and Liu

, Expected value operator of random fuzzy variable and random fuzzy expected value models, International Journal of Uncertainty, Fuzziness & Knowledge-Based Systems 11(2) (2003), 195–215.

22.

Liu

and Gao

, The independence of fuzzy variables in credibility theory and its applications, International Journal of Uncertainty, Fuzziness & Knowledge-Based Systems 15(Supp 2) (2007), 1–20.

23.

Madhuri

K.U.

, Babu

S.S.

and Shankar

N.R.

, Fuzzy risk analysis based on the novel fuzzy ranking with new arithmetic operations of linguistic fuzzy numbers, Journal of Intelligent and Fuzzy Systems 26(5) (2014), 2391–2401.

24.

Nahmias

, Fuzzy variables, Fuzzy Sets and Systems 1(2) (1978), 97–110.

25.

Pushpa

and Vasuki

, Estimation of confidence level ‘h’ in fuzzy linear regression analysis using shape preserving operations, International Journal of Computer Applications 68(17) (2013), 19–25.

26.

Saneeifard

, Ranking LR fuzzy numbers with weighted averaging based on levels, International Journal of Industrial Mathematics 1(2) (2009), 163–173.

27.

Sorini

and Stefanini

, Some parametric forms for LR fuzzy numbers and LR fuzzy arithmetic, 9th International Conference on Intelligent Systems Design and Applications, Pisa, Italy, 2009, pp. 312–317.

28.

Wang

H.F.

and Kuo

C.Y.

, Three-parameter fuzzy arithmetic approximation of LR fuzzy numbers for fuzzy neural networks, International Journal of Uncertainty, Fuzziness & Knowledge-Based Systems 14(2) (2006), 211–233.

29.

Yager

, A procedure for ordering fuzzy subsets of the unit interval, Information Sciences 24(2) (1981), 143–161.

30.

Yager

, On the specificity of a possibility distribution, Fuzzy Sets and Systems 50(3) (1992), 279–292.

31.

Zadeh

L.A.

, Fuzzy sets, Information and Control 8(3) (1965), 338–353.

32.

Zadeh

L.A.

, The concept of a linguistic variable and its application to approximate reasoning-I, Information Sciences 8(3) (1975), 199–249.

33.

Zadeh

L.A.

, Fuzzy sets as a basis for a theory of possibility, Fuzzy Sets and Systems 1(1) (1978), 3–28.

34.

Zadeh

L.A.

, Atheory of approximate reasoning, in: Mathematical Frontiers of the Social and Policy Sciences, Westview Press, 1979, pp. 69–129.

35.

Zimmermann

H.J.

, Fuzzy Set Theory and Its Applications, Kluwer, 1991.