On computing generalized Hukuhara differences of Z-numbers

Abstract

Based on the fact that the Hukuhara difference exists only under very restrictive conditions, in this paper, we present the process of computing the generalized Hukuhara difference of discrete Z-numbers and the generalized difference of continuous Z-numbers respectively. Some examples are given to illustrate the effectiveness of the proposed computing methods.

Keywords

generalized Hukuhara difference generalized difference

1. Introduction

The occurrence of randomness and fuzziness in the real world is inevitable owing to some unexpected situations. Much of the information on which decision are based is uncertain. It is difficult to formalize capability which make rational decisions based on uncertain information. In order to take into account this fact, The notion Z-number introduced by L. A. Zadeh [22] in 2011 has more capability to describe the uncertain information. A Z-number, Z, has two components, Z = (A, B). The first component, A, is a restriction (constraint) on the values which a real-valued uncertain variable, X, is allowed to take. The second component, B, is a measure of reliability (certainty) of the first component. Typically, A and B are described in a natural language. The concept of Z-number is more suitable to describe the knowledge of human being and will be widely used in the uncertain information process. Therefore, the operation of Z-numbers has received more and more attention from researchers.

According to the components of Z-number, Z-number is divided into two categories, i.e., discrete Z-number and continuous Z-number. W. Voxman [16] obtains two canonical representations of discrete fuzzy numbers. In particular, it is shown that any discrete fuzzy number can be represented by a triangular or a block discrete fuzzy number having the same value, ambiguity and fuzziness as the original number. By using α-level sets, G. Wang [17] gives a kind of representation of a discrete fuzzy number and defines a new kind of addition for two discrete fuzzy numbers which preserves the closeness of the operation. He also points out that when the usual addition of two discrete fuzzy numbers is still a discrete fuzzy number. R. Banerjee and S. K. Pal [3] extend the interpretation of the set-theoretic intersection operator to evaluate the intersection of perceptions and discover some of the challenges underlying the implementation of the Z-number in the area of computing with words (CWW). Several approaches of approximate evaluation of a Z-number in order to reduce computational complexity are suggested in [15]. One of the suggested approaches is based on approximation of a fuzzy set of probability densities by means of fuzzy If-THEN rules. The work [1] is devoted to suggest theoretical aspects of such arithmetic operations over discrete Z-numbers as addition, subtraction, multiplication, division, square root of a Z-number and other operations. Let us mention that the Hukuhara difference exists only under very restrictive conditions [6 , 11], then, the author introduce a concept of generalized Hukuhara difference to overcome this shortcoming in [14]. Thus, we can study computing the generalized Hukuhara difference of discrete Z-numbers. However, the generalized Hukuhara difference of discrete fuzzy numbers does not always exist [12, 13], and a new concept of generalized difference of fuzzy numbers is proposed in [14]. Similarly, we can research computing the generalized difference of continuous Z-numbers.

The outline of the rest of the paper is the following: Some related definitions, concepts and lemmas will be recalled in Section 2. In Section 3, we will show the computing process of generalized Hukuhara difference of discrete Z-numbers and give an example to demonstrate the validity of the suggested computing approach. In Section 4, we will present the computing process of the generalized difference of two continuous Z-numbers and more computing process details will be given in a related example. We will conclude with a summary in Section 5.

2. Preliminaries

A fuzzy set $\tilde{a}$ in $ℝ$ is characterized by a membership function $μ_{\tilde{a}} : ℝ \to [0, 1]$ . The α-level set of $\tilde{a}$ is denoted by $[\tilde{a}]^{α} = {x \in ℝ : μ_{\tilde{a}} (x) \geq α}$ for each α ∈ (0, 1]. The 0-level set $[\tilde{a}]^{0}$ is defined as the closure of the set ${x \in ℝ : μ_{\tilde{a}} (x) > 0}$ i.e., $[\tilde{a}]^{0} = cl ({x \in ℝ : μ_{\tilde{a}} (x) > 0})$ . A fuzzy set $\tilde{a}$ is said to be a fuzzy number if it satisfies the following conditions:

$\tilde{a}$ is normal, i.e., there exists an $x_{0} \in ℝ$ such that $μ_{\tilde{a}} (x_{0}) = 1$ ;

$\tilde{a}$ is convex, i.e., $μ_{\tilde{a}} (λ x_{1} + (1 - λ) x_{2}) \geq \min {μ_{\tilde{a}} (x_{1}), μ_{\tilde{a}} (x_{2})}$ for all $x_{1}, x_{2} \in ℝ$ and λ ∈ (0, 1);

$\tilde{a}$ is upper semicontinuous;

The 0-level set $[\tilde{a}]^{0}$ is a compact subset of $ℝ$ .

Let $F$ be the set of all fuzzy numbers on $ℝ$ . Then for any $\tilde{a} \in F$ , it is well known that the α-level set $[\tilde{a}]^{α} = [{\tilde{a}}_{L} (α), {\tilde{a}}_{R} (α)]$ is a non-empty bounded closed interval in $ℝ$ for all α ∈ [0, 1], where ${\tilde{a}}_{L} (α)$ denotes the left-hand end point of $[\tilde{a}]^{α}$ and the ${\tilde{a}}_{R} (α)$ denotes the right one.

A trapezoidal fuzzy number, denoted by $\tilde{a} = 〈 a, b, c, d 〉$ , where a ≤ b ≤ c ≤ d, has α-level ${[\tilde{a}]}^{α} = [a + α (b - a), d - α (d - c)]$ , α ∈ [0, 1]. And if b = c, we say that $\tilde{a}$ is a triangular fuzzy number.

For any $\tilde{a}, \tilde{b} \in F$ and $λ \in ℝ$ , owing to Zadeh’s extension principle [19 –21], the addition and scalar multiplication can be respectively defined for any $x \in ℝ$ by $μ_{\tilde{a} + \tilde{b}} (x) = sup_{x_{1}, x_{2} : x_{1} + x_{2} = x} min {μ_{\tilde{a}} (x_{1}), μ_{\tilde{b}} (x_{2})}$ and $μ_{λ \times \tilde{a}} (x) = μ_{λ \tilde{a}} (x) = {\begin{matrix} μ (\frac{x}{λ}), & if λ \neq 0, \\ \tilde{0}, & if λ = 0 . \end{matrix}$ For any $\tilde{a} \in F$ , we define the fuzzy number $- \tilde{a} \in F$ by $- \tilde{a} = (- 1) \times \tilde{a}$ . By the level set representations of the fuzzy numbers $\tilde{a} \in F$ , we can get that ${[λ \tilde{a}]}^{α} = λ {[\tilde{a}]}^{α} = {\begin{matrix} [λ {\tilde{a}}_{L} (α), λ {\tilde{a}}_{R} (α)], & if λ > 0, \\ {0}, & if λ = 0, \\ [λ {\tilde{a}}_{R} (α), λ {\tilde{a}}_{L} (α)], & if λ < 0, \end{matrix}$ for all $\tilde{a} \in F$ and $λ \in ℝ$ .

Definition 2.1. [4] A random variable, X, is a variable whose possible values x are numerical outcomes of a random phenomenon. Random variables are of two types: discrete and continuous.

To determine a probability that a continuous random variable X takes any value in a closed interval [a, b], denoted P (a ≤ X ≤ b), a concept of probability distribution is used. A probability distribution is a function p (x) such that for that for any two numbers a and b with a ≤ b: $P (a \leq x \leq b) = \int_{a}^{b} p (x) d x,$ where $p (x) \geq 0, \int_{- \infty}^{\infty} p (x) dx = 1 .$

Consider a discrete random variable X with outcomes space {x₁, x₂, … , x_n }. A probability of an outcome X = x_i, denoted P (X = x_i) is defined in terms of a probability distribution. A function p is called a discrete probability distribution or a probability mass function if $P (X = x_{i}) = p (x_{i}),$ where p (x_i) ∈ [0, 1] and $\sum_{i = 1}^{n} p (x_{i}) = 1.$

Consider two independent random variable X₁ and X₂ with probability distributions p₁ and p₂ respectively. Let X₁₂ = X₁ * X₂, * ∈ { + , - } . Below a definition of a probability distribution p₁₂ is given.

Definition 2.2. [4, 5] Let X₁ and X₂ be two independent continuous random variable with probability distributions p₁ and p₂. A probability distribution p₁₂ of X₁₂ = X₁* X₂, * ∈ { + , - } is referred to as a convolution of p₁ and p₂ and is defined as follows:

$\begin{matrix} X_{12} & = & X_{1} + X_{2} : p_{12} (x) = \int_{- \infty}^{\infty} p_{1} (x_{1}) p_{2} (x - x_{1}) d x_{1}, \\ X_{12} & = & X_{1} - X_{2} : p_{12} (x) = \int_{- \infty}^{\infty} p_{1} (x_{1}) p_{2} (x_{1} - x) d x_{1} . \end{matrix}$

Let X₁ and X₂ be two independent discrete random variable with the corresponding outcome spaces X₁ ={ x₁₁, …, x_1n₁ } and X₂ ={ x₂₁, …, x_2n₁ } and the corresponding discrete probability distributions p₁ and p₂. The probability distribution of X₁ * X₂ is the convolution p₁₂ = p₁ ∘ p₂ of p₁ and p₂ which is determined as follows: $p_{12} (x) = \sum_{x = x_{1} * x_{2}} p_{1} (x_{1}) p_{2} (x_{2}),$ for any x ∈ { x₁ * x₂|x₁ ∈ X₁, x₂ ∈ X₂ } , x₁ ∈ X₁, x₂ ∈ X₂ .

Definition 2.3. [22] A Z-number is an ordered pair of fuzzy numbers denoted as $Z = (\tilde{a}, \tilde{b})$ . The first component, $\tilde{a}$ , is a restriction (constraint) on the values which a real-valued uncertain variable, X, is allowed to take. The second component, $\tilde{b}$ , is a measure of reliability (uncertainty) of the first component.

A concept of a Z⁺-number [22] is closely related to the concept of a Z-number. Given a Z-number $Z = (\tilde{a}, \tilde{b})$ , Z⁺-number Z⁺ is a pair consisting of a fuzzy number, $\tilde{a}$ , and a random number R: $Z^{+} = (\tilde{a}, R),$ where $\tilde{a}$ plays the same role as it does in a Z-number $Z = (\tilde{a}, \tilde{b})$ and R plays the role of the probability distribution p, such that $P (\tilde{a}) = \int_{R} μ_{\tilde{a}} (u) p_{R} (u), P (\tilde{a}) \in supp (R) .$

Definition 2.4. If the fuzzy numbers $\tilde{a}$ and $\tilde{b}$ of a Z-number are discrete fuzzy numbers, we say that Z-number is a discrete Z-number. Similarly, if $\tilde{a}$ and $\tilde{b}$ are continuous fuzzy numbers, we say that Z-number is a continuous Z-number.

Definition 2.5. [18] Let $\tilde{a}$ be a discrete fuzzy number. A probability measure of $\tilde{a}$ denoted $P (\tilde{a})$ is defined as $P (\tilde{a}) = \sum_{i = 1}^{n} μ_{\tilde{a}} (x_{i}) p (x_{i}) .$

Definition 2.6. [18] Let $\tilde{a}$ be a continuous fuzzy number. A probability measure of $\tilde{a}$ denoted $P (\tilde{a})$ is defined as $P (\tilde{a}) = \int μ_{\tilde{a}} (x) p (x) dx .$

3. gH-difference of discrete Z-numbers

Hukuhara difference of discrete Z-numbers is introduced in [1]. It is well known that the Hukuhara difference exists only under very restrictive conditions [6 , 10]. In this section, we will solve the problem of difference of two discrete Z-numbers by invoking generalized Hukuhara difference [14]. First, we introduce the definition of generalized Hukuhara difference.

The standard Hukuhara difference (H-difference) is defined by $\tilde{a} \underset{H}{⊖} \tilde{b} = \tilde{c}$ if there is a fuzzy numbers $\tilde{c}$ satisfying $\tilde{a} = \tilde{b} + \tilde{c}$ [9, 10]. As it is know, in most case, for any two fuzzy numbers $\tilde{a}$ , $\tilde{b}$ the Hukuhara difference do not always exist. A generalization of the H-difference was introduced in [14]. Given two fuzzy numbers $\tilde{a}, \tilde{b} \in F$ , the generalized Hukuhara difference (gH-difference. for short), denoted by $\tilde{a} \underset{gH}{⊖} \tilde{b} = \tilde{c}$ , is the fuzzy number $\tilde{c}$ , if it exists, such that $\tilde{a} \underset{gH}{⊖} \tilde{b} = \tilde{c} \Leftrightarrow {\begin{matrix} (i) & \tilde{a} = \tilde{b} + \tilde{c} or \\ (ii) & \tilde{b} = \tilde{a} - \tilde{c} . \end{matrix}$

Lemma 3.1. [2] In terms of α-level set, we have

$\begin{matrix} {[\tilde{a} \underset{gH}{⊖} \tilde{b}]}^{α} = [min {{\tilde{a}}_{L} (α) - {\tilde{b}}_{L} (α), {\tilde{a}}_{R} (α) - {\tilde{b}}_{R} (α)}, \\ max {{\tilde{a}}_{L} (α) - {\tilde{b}}_{L} (α), {\tilde{a}}_{R} (α) - {\tilde{b}}_{R} (α)}] . \end{matrix}$

Now let us consider the procedures underlying computation of gH-difference $Z_{12} = Z_{1} \underset{gH}{⊖} Z_{2}$ of Z-numbers $Z_{1} = ({\tilde{a}}_{1}, {\tilde{b}}_{1})$ and $Z_{2} = ({\tilde{a}}_{2}, {\tilde{b}}_{2})$ , where $Z_{12} = ({\tilde{a}}_{12}, {\tilde{b}}_{12})$ .

First, it is clearly that gH-difference $Z_{12} = Z_{1} \underset{gH}{⊖} Z_{2}$ exists only if gH-difference of two fuzzy number ${\tilde{a}}_{1}$ and ${\tilde{a}}_{2}$ , ${\tilde{a}}_{12} = {\tilde{a}}_{1} \underset{gH}{⊖} {\tilde{a}}_{2}$ , exists. So we suppose that ${\tilde{a}}_{12} = {\tilde{a}}_{1} \underset{gH}{⊖} {\tilde{a}}_{2}$ exists. Then we can computed ${\tilde{a}}_{12}$ by the definition of gH-difference or Lemma 3.1. We should proceed to compute the corresponding Z⁺-numbers, and the Z⁺-number $Z_{12}^{+} = Z_{1}^{+} \underset{gH}{⊖} Z_{2}^{+}$ is defined as: $Z_{12}^{+} = ({\tilde{a}}_{1} \underset{gH}{⊖} {\tilde{a}}_{2}, R_{1} - R_{2}),$ where R₁ and R₂ are respresented by discrete probability distributions: $\begin{matrix} p_{1} = \frac{p_{1} (x_{11})}{x_{11}} + \frac{p_{1} (x_{12})}{x_{12}} + \dots + \frac{p_{1} (x_{1 n})}{x_{1 n}}, \\ p_{2} = \frac{p_{2} (x_{21})}{x_{21}} + \frac{p_{2} (x_{22})}{x_{22}} + \dots + \frac{p_{2} (x_{2 n})}{x_{2 n}}, \end{matrix}$ for which one necessarily has $\begin{matrix} \sum_{i = 1}^{n} p_{1} (x_{1 i}) = 1, \\ \sum_{i = 1}^{n} p_{2} (x_{2 i}) = 1 . \end{matrix}$

It is important to point out that R₁ - R₂, defined in accord with Definition 2.2, is a convolution p₁₂ = p₁ ∘ p₂ of discrete probability distributions. Next, according to the result of ${\tilde{a}}_{12}$ , we will discuss in two different cases.

[Case 1:] If ${\tilde{a}}_{1} = {\tilde{a}}_{2} + {\tilde{a}}_{12}$ . That is to say, there exist H-difference $\tilde{a} \underset{H}{⊖} \tilde{b}$ . Then we can compute the gH-difference of Z-numbers according to the process of calculating H-difference of Z-numbers [1]. It requires to determine μ_{p
₁₂} () when compute ${\tilde{b}}_{12}$ . Furthermore, it requires determination of all the distributions p₁₂. We know that $p_{1} (x_{1}) = \sum_{x = x_{2 i} + x_{12, j}} p_{2} (x_{2 i}) p_{12} (x_{12, j}) .$ Thus, given p₂ (x₂) and p₁ (x_1i), then the problem of determination of distribution p₁₂ (x₁₂) can be converted to the following form: ${\begin{array}{l} p_{12} (x_{12, 1}) a_{11} + \dots + p_{12} (x_{12, n}) a_{1 n} = p_{1} (x_{11}) \\ p_{12} (x_{12, 1}) a_{21} + \dots + p_{12} (x_{12, n}) a_{2 n} = p_{1} (x_{12}) \\ ⋮ \\ p_{12} (x_{12, 1}) a_{i 1} + \dots + p_{12} (x_{12, n}) a_{2 n} = p_{1} (x_{1 i}) \\ ⋮ \\ p_{12} (x_{12, 1}) a_{n 1} + \dots + p_{12} (x_{12, n}) a_{n n} = p_{1} (x_{1 i}) p_{12} (x_{12, 1}) + \dots p_{12} (x_{12, n}) = 1 \end{array}$ (1) where $a_{ij} = {\begin{matrix} p_{2} (x_{2 j}), & if x_{1 i} = x_{2 j} + x_{12, j} \\ 0, & otherwise \end{matrix} .$ Let u₁ = p₁₂ (x_12,1) , ⋯ , u_n = p₁₂ (x_12,n) and w₁ = p₁ (x₁₁) , ⋯ , w_n = p₁ (x_1n). (3) can be written as the system of linear algebraic equations: ${\begin{array}{l} u_{1} a_{11} + u_{2} a_{12} + \dots + u_{n} a_{1 n} = w_{1} \\ u_{1} a_{21} + u_{2} a_{22} + \dots + u_{n} a_{2 n} = w_{2} \\ ⋮ \\ u_{1} a_{i 1} + u_{2} a_{i 2} + \dots + u_{n} a_{i n} = w_{i} \\ ⋮ \\ u_{1} a_{n 1} + u_{2} a_{n 2} + \dots + u_{n} a_{n n} = w_{n} u_{1} + u_{2} + \dots + u_{n} = 1 \end{array}$ (2) for u₁, ⋯ , u_n are unknown variables and w₁, ⋯ , w_n are the given values.

When the rank of the coefficient matrix is equal to the rank of augmented matrix, i.e., $rank (A_{1}) = rank (\bar{A_{1}}) = n$ , the system (4) has a unique solution, where $A_{1} = [\begin{matrix} a_{11}, a_{12}, \dots, a_{1 n} \\ a_{21}, a_{22}, \dots, a_{2 n} \\ ⋮ \\ a_{i 1}, a_{i 2}, \dots, a_{2 n} \\ ⋮ \\ a_{n 1}, a_{n 2}, \dots, a_{nn} \\ 1, 1, \dots, 1 \end{matrix}]$ and $\bar{A_{1}} = [\begin{matrix} a_{11}, a_{12}, \dots, a_{1 n}, w_{1} \\ a_{21}, a_{22}, \dots, a_{2 n}, w_{2} \\ ⋮ \\ a_{i 1}, a_{i 2}, \dots, a_{2 n}, w_{i} \\ ⋮ \\ a_{n 1}, a_{n 2}, \dots, a_{nn}, w_{n} \\ 1, 1, \dots, 1, 1 \end{matrix}] .$

p₁₂ is just the solution u₁, ⋯ , u_n of system (2). It is noticeable that not for each pair of p₁ and p₂, there exists p₁₂ with p₁ = p₂ ∘ p₁₂. However, if there exists such p₂ for any p₁, the exact solution p₁₂ exists. Denote N and M are the number of distributions p₁ and p₂, respectively. Then it needs to repeat as the exact solution to (2) M times and any p₁₂ to be found convolves with the all distributions p₂. Thus, μ_{p
₁₂} (p₁₂) can be obtained by $\begin{array}{l} μ_{p_{12}} (p_{12}) \\ = \max_{{p_{1}, p_{2} | p_{1} = p_{2} \circ p_{12}}} \min {μ_{p_{1}} (p_{1}), μ_{p_{2}} (p_{2})} . \end{array}$

That is, max is taken over all the M cases when the same exact solution p₁₂ to (2) repeats.

[Case 2:] If ${\tilde{a}}_{2} = {\tilde{a}}_{1} - {\tilde{a}}_{12}$ . Next it is needed to compute ${\tilde{b}}_{12}$ . This requires to determine μ_{p
₁₂} () and all the distributions p₁₂. Since $p_{2} (x_{2}) = \sum_{x = x_{1 i} - x_{12, j}} p_{1} (x_{1 i}) p_{12} (x_{12, j}),$ the problem of determination of p₁₂ (x₁₂) can be formalized with given p₁ (x₁) and p₂ (x_2i), and the form is as follows: ${\begin{matrix} p_{12} (x_{12, 1}) a_{11} + \dots + p_{12} (x_{12, n}) a_{1 n} = p_{2} (x_{21}) \\ p_{12} (x_{12, 1}) a_{21} + \dots + p_{12} (x_{12, n}) a_{2 n} = p_{2} (x_{22}) \\ ⋮ \\ p_{12} (x_{12, 1}) a_{i 1} + \dots + p_{12} (x_{12, n}) a_{2 n} = p_{2} (x_{2 i}) \\ ⋮ \\ p_{12} (x_{12, 1}) a_{n 1} + \dots + p_{12} (x_{12, n}) a_{nn} = p_{2} (x_{2 n}) \\ p_{12} (x_{12, 1}) + \dots + p_{12} (x_{12, n}) = 1 \end{matrix},$ (3) where $a_{ij} = {\begin{matrix} p_{1} (x_{1 j}), & if x_{2 i} = x_{1 j} + x_{12, j} \\ 0, & otherwise \end{matrix} .$ Let u₁ = p₁₂ (x_12,1) , ⋯ , u_n = p₁₂ (x_12,n) and v₁ = p₁ (x₁₁) , ⋯ , v_n = p₁ (x_1n). (3) can be written as the system of linear algebraic equations: ${\begin{array}{l} u_{1} a_{11} + u_{2} a_{12} + \dots + u_{n} a_{1 n} = v_{1} \\ u_{1} a_{21} + u_{2} a_{22} + \dots + u_{n} a_{2 n} = v_{2} \\ ⋮ \\ u_{1} a_{i 1} + u_{2} a_{i 2} + \dots + u_{n} a_{i n} = v_{i} \\ ⋮ \\ u_{1} a_{n 1} + u_{2} a_{n 2} + \dots + u_{n} a_{n n} = u_{1} + u_{2} + \dots + u_{n} = 1 \end{array}$ (4) for u₁, ⋯ , u_n are unknown variables and v₁, ⋯ , v_n are the given values.

When the rank of the coefficient matrix is equal to the rank of augmented matrix, i.e., $rank (A_{2}) = rank (\bar{A_{2}}) = n$ , the system (4) has a unique solution, where $A_{2} = [\begin{matrix} a_{11}, a_{12}, \dots, a_{1 n} \\ a_{21}, a_{22}, \dots, a_{2 n} \\ ⋮ \\ a_{i 1}, a_{i 2}, \dots, a_{2 n} \\ ⋮ \\ a_{n 1}, a_{n 2}, \dots, a_{nn} \\ 1, 1, \dots, 1 \end{matrix}]$ and $\bar{A_{2}} = [\begin{matrix} a_{11}, a_{12}, \dots, a_{1 n}, v_{1} \\ a_{21}, a_{22}, \dots, a_{2 n}, v_{2} \\ ⋮ \\ a_{i 1}, a_{i 2}, \dots, a_{2 n}, v_{i} \\ ⋮ \\ a_{n 1}, a_{n 2}, \dots, a_{nn}, v_{n} \\ 1, 1, \dots, 1, 1 \end{matrix}] .$

p₁₂ is just the solution u₁, ⋯ , u_n of system (2). It is noticeable that not for each pair of p₁ and p₂, there exists p₁₂ with p₂ = p₁ ∘ p₁₂. However, if there exists such p₂ for any p₁, the exact solution p₁₂ exists. Denote N and M are the number of distributions p₁ and p₂, respectively. Then it needs to repeat as the exact solution to (4) N times and any p₁₂ to be found convolves with the all distributions p₁. Thus, μ_{p
₁₂} (p₁₂) can be obtained by $\begin{matrix} μ_{p_{12}} (p_{12}) \\ = max_{{p_{1}, p_{2} | p_{2} = p_{1} \circ p_{12}}} min {μ_{p_{1}} (p_{1}), μ_{p_{2}} (p_{2})} . \end{matrix}$

That is, max is taken over all the N cases when the same exact solution p₁₂ to (4) repeats.

Finally, we construct ${\tilde{b}}_{12}$ . What is only known is a fuzzy restriction on p₁₂ described by the membership function μ_{p
₁₂}. It is well known that $P ({\tilde{a}}_{12})$ is a fuzzy set ${\tilde{b}}_{12}$ with the membership function $μ_{{\tilde{b}}_{12}}$ defined as follows: $μ_{{\tilde{b}}_{12}} (b_{12, s}) = sup (μ_{p_{12, s}} (p_{12, s}))$ (5) subject to $b_{12, s} = \int_{R} μ_{{\tilde{a}}_{12}} (x) p_{12, s} (x_{k}) dx .$ (6)

By solving the problem (5)-(6), $Z_{12} = ({\tilde{a}}_{12}, {\tilde{b}}_{12})$ can be obtained.

Now, let us consider the following example about gH-difference of discrete Z-number.

Example 3.2. Let us consider gH-difference of discrete Z-numbers $Z_{1} = ({\tilde{a}}_{1}, {\tilde{b}}_{1})$ and $Z_{2} = ({\tilde{a}}_{2}, {\tilde{b}}_{2})$ , denoted $Z_{12} = Z_{1} \underset{gH}{⊖} Z_{2}$ . And

$\begin{matrix} {\tilde{a}}_{1} = & \frac{0.2}{10} + \frac{0.6}{11} + \frac{1}{12} + \frac{0.9}{13} + \frac{0.5}{14} + \frac{0.1}{15}, \\ {\tilde{b}}_{1} = & \frac{0.3}{0.57} + \frac{0.5}{0.67} + \frac{1}{0.78} + \frac{0.9}{0.80} + \frac{0.2}{0.92} + \frac{0.1}{0.99}; \\ {\tilde{a}}_{2} = & \frac{0.1}{4} + \frac{0.2}{5} + \frac{0.4}{6} + \frac{0.6}{7} + \frac{0.8}{8} + \frac{1}{9} + \frac{0.9}{10} + \frac{0.7}{11} \\ + \frac{0.5}{12} + \frac{0.3}{13} + \frac{0.1}{14}, \\ {\tilde{b}}_{2} = & \frac{0.2}{0.613} + \frac{0.4}{0.697} + \frac{0.5}{0.776} + \frac{0.7}{0.813} + \frac{0.8}{0.824} + \frac{1}{0.871} \\ + \frac{0.8}{0.925} + \frac{0.6}{0.928} + \frac{0.5}{0.937} + \frac{0.2}{0.952} + \frac{0.1}{0.973}; \end{matrix}$ For better illustration in this figure and the other figures in this paper, we mainly use continuous representation for graphs of membership functions, where discrete points are connected by continuous curves.

At the first step, we proceed to the corresponding Z⁺-number, $Z_{12}^{+} = ({\tilde{a}}_{12}, R_{12}) = ({\tilde{a}}_{1} \underset{gH}{⊖} {\tilde{a}}_{2}, R_{1} - R_{2}) .$ It is easy to proof there exist a fuzzy number ${\tilde{a}}_{12}$ such that ${\tilde{a}}_{2} = {\tilde{a}}_{1} - {\tilde{a}}_{12}$ , where ${\tilde{a}}_{12} = \frac{0.1}{6} + \frac{0.4}{5} + \frac{0.8}{4} + \frac{1}{3} + \frac{0.7}{2} + \frac{0.3}{1} .$ Thus, ${\tilde{a}}_{12}$ is the gH-difference of fuzzy numbers ${\tilde{a}}_{1}$ and ${\tilde{a}}_{2}$ , i.e., ${\tilde{a}}_{1} \underset{gH}{⊖} {\tilde{a}}_{2} = {\tilde{a}}_{12} .$ Let us mention that H-difference of ${\tilde{a}}_{1}$ and ${\tilde{a}}_{2}$ does not exist.

At the second step, we should determine whether there exist such distributions p₁₂ such that (4). Thus, we should consider all possible pairs of distributions p₁ and p₂ to determine for which pairs there exist distributions p₁₂ such that (4). Let us consider the following distributions: $p_{1} = \frac{0}{10} + \frac{0.2}{11} + \frac{0.3}{12} + \frac{0.2}{13} + \frac{0.1}{14} + \frac{0.2}{15}$ and $\begin{matrix} p_{2} & = & \frac{0}{4} + \frac{0.04}{5} + \frac{0.06}{6} + \frac{0.1}{7} + \frac{0.17}{8} + \frac{0.21}{9} \\ + \frac{0.14}{10} + \frac{0.14}{11} + \frac{0.09}{12} + \frac{0.03}{13} + \frac{0.02}{14} . \end{matrix}$

Denote u₁ = p₁₂ (x_12,1) , u₂ = p₁₂ (x_12,2) , …, u₆ = p₁₂ (x_12,6) , the (4) will be formalized as follows:

${\begin{array}{l} 0 \cdot u_{1} + 0 \cdot u_{2} + 0 \cdot u_{3} + 0 \cdot u_{4} + 0 \cdot u_{5} + 0 \cdot u_{6} = 0 \\ 0.2 \cdot u_{1} + 0 \cdot u_{2} + 0 \cdot u_{3} + 0 \cdot u_{4} + 0 \cdot u_{5} + 0 \cdot u_{6} = 0.04 \\ 0.3 \cdot u_{1} + 0.2 \cdot u_{2} + 0.2 \cdot u_{3} + 0 \cdot u_{4} + 0 \cdot u_{5} + 0 \cdot u_{6} = 0.06 \\ 0.2 \cdot u_{1} + 0.3 \cdot u_{2} + 0.2 \cdot u_{3} + 0 \cdot u_{4} + 0 \cdot u_{5} + 0 \cdot u_{6} = 0.1 \\ 0.1 \cdot u_{1} + 0.2 \cdot u_{2} + 0.3 \cdot u_{3} + 0.2 \cdot u_{4} + 0 \cdot u_{5} + 0 \cdot u_{6} = 0.17 \\ 0.2 \cdot u_{1} + 0.1 \cdot u_{2} + 0.2 \cdot u_{3} + 0.3 \cdot u_{4} + 0.2 \cdot u_{5} + 0 \cdot u_{6} = 0.21 \\ 0 \cdot u_{1} + 0.2 \cdot u_{2} + 0.1 \cdot u_{3} + 0.2 \cdot u_{4} + 0.3 \cdot u_{5} + 0.2 \cdot u_{6} = 0.14 \\ 0 \cdot u_{1} + 0 \cdot u_{2} + 0.2 \cdot u_{3} + 0.1 \cdot u_{4} + 0.2 \cdot u_{5} + 0.3 \cdot u_{6} = 0.14 \\ 0 \cdot u_{1} + 0 \cdot u_{2} + 0 \cdot u_{3} + 0.2 \cdot u_{4} + 0.2 \cdot u_{5} + 0.3 \cdot u_{6} = 0.09 \\ 0 \cdot u_{1} + 0 \cdot u_{2} + 0 \cdot u_{3} + 0 \cdot u_{4} + 0.2 \cdot u_{5} + 0.1 \cdot u_{6} = 0.01 \\ 0 \cdot u_{1} + 0 \cdot u_{2} + 0 \cdot u_{3} + 0 \cdot u_{4} + 0 \cdot u_{5} + 0.2 \cdot u_{6} = 0.02 \\ u_{1} + u_{2} + u_{3} + u_{4} + u_{5} + u_{6} = 1 \end{array}$ (7)

It is obvious that the coefficient matrix A and augmented matrix $\bar{A}$ of the linear algebraic equations (7) have same rank, i.e. $rank (A) = rank (\bar{A}) = 6$ , where $A = [\begin{matrix} 0 & 0 & 0 & 0 & 0 & 0 \\ 0.2 & 0 & 0 & 0 & 0 & 0 \\ 0.3 & 0.2 & 0 & 0 & 0 & 0 \\ 0.2 & 0.3 & 0.2 & 0 & 0 & 0 \\ 0.1 & 0.2 & 0.3 & 0.2 & 0 & 0 \\ 0.2 & 0.1 & 0.2 & 0.3 & 0.2 & 0 \\ 0 & 0.2 & 0.1 & 0.2 & 0.3 & 0.2 \\ 0 & 0 & 0.2 & 0.1 & 0.2 & 0.3 \\ 0 & 0 & 0 & 0.2 & 0.1 & 0.2 \\ 0 & 0 & 0 & 0 & 0.2 & 0.1 \\ 0 & 0 & 0 & 0 & 0 & 0.2 \\ 1 & 1 & 1 & 1 & 1 & 1 \end{matrix}]$ and $\bar{A} = [\begin{matrix} 0 & 0 & 0 & 0 & 0 & 0 \\ 0.2 & 0 & 0 & 0 & 0 & 0 \\ 0.3 & 0.2 & 0 & 0 & 0 & 0 \\ 0.2 & 0.3 & 0.2 & 0 & 0 & 0 \\ 0.1 & 0.2 & 0.3 & 0.2 & 0 & 0 \\ 0.2 & 0.1 & 0.2 & 0.3 & 0.2 & 0 \\ 0 & 0.2 & 0.1 & 0.2 & 0.3 & 0.2 \\ 0 & 0 & 0.2 & 0.1 & 0.2 & 0.3 \\ 0 & 0 & 0 & 0.2 & 0.1 & 0.2 \\ 0 & 0 & 0 & 0 & 0.2 & 0.1 \\ 0 & 0 & 0 & 0 & 0 & 0.2 \\ 1 & 1 & 1 & 1 & 1 & 1 \end{matrix} \begin{matrix} 0 \\ 0.04 \\ 0.06 \\ 0.1 \\ 0.17 \\ 0.21 \\ 0.14 \\ 0.14 \\ 0.09 \\ 0.03 \\ 0.02 \\ 1 \end{matrix}]$ Through calculation, we get p₁₂ as the exact solution of (7): $p_{12} = \frac{0.2}{6} + \frac{0}{5} + \frac{0.3}{4} + \frac{0.3}{3} + \frac{0.1}{2} + \frac{0.1}{1} .$ In what follows we should verify whether p₁₂ exactly satisfies (4). When the variables on the left side of the equation (4) are replaced with p₁₂, we found that the numerical value on the right side is p₂ exactly. Thus the considered p₁ and p₂, there exists p₁₂ such that p₂ = p₁ ∘ p₁₂. Furthermore, we point out there is not a unique p₁₂ such that (4) for the considered p₁ and any other p₂. For any distribution p₂, in most case, there exists a unique p₁ which exactly satisfies (4) exists. The corresponding exact solutions p₁₂ of (4) can be found by using the same procedures for all the distributions p₂. At the third step, we compute membership degrees μ_{p
₁} (p₁) and μ_{p
₂} (p₂) for the fuzzy restrictions over distributions p₁ and p₂, That is, $\begin{matrix} μ_{p_{1}} (p_{1}) & = & μ_{{\tilde{b}}_{1}} (\sum_{k = 1}^{6} μ_{{\tilde{a}}_{1}} (x_{1 k}) p_{1} (x_{1 k})) \\ = & μ_{{\tilde{b}}_{1}} (0.2 \cdot 0 + 0.6 \cdot 0.2 + 1 \cdot 0.3 \\ + 0.9 \cdot 0.2 + 0.5 \cdot 0.1 + 0.1 \cdot 0.2) \\ = & μ_{{\tilde{b}}_{1}} (0.67) = 0.5 \end{matrix}$ Similarly, we have $μ_{p_{2}} (p_{2}) = μ_{{\tilde{b}}_{1}} (0.66) = 0.6 .$

At the fourth step, we compute membership degrees of the convolutions p₁₂. It is known that p₁₂ is obtained and does not depend on a type of arithmetic operation. What is more, only depend on the degrees and of distributions p₁ and p₂. Therefore, we have $μ_{p_{12}} (p_{12}) = \min {μ_{p_{1}} (p_{1}), μ_{p_{2}} (p_{2})} = 0.5.$

At the fifth step, we should proceed to construction of ${\tilde{b}}_{12}$ as a soft constraint on a probability measure $P ({\tilde{a}}_{12})$ based on (5)-(6). We need to compute values of probability measure $P ({\tilde{a}}_{12})$ based on Definition 2.4 by using the obtained convolutions p₁₂. For example, $P ({\tilde{a}}_{12})$ computed based on p₁₂ considered above is $P ({\tilde{a}}_{12}) = \sum_{i = 1}^{6} μ_{{\tilde{a}}_{12}} (x_{i}) p_{12} (x_{i}) = 0.776$ Thus, one basic value of ${\tilde{b}}_{12}$ is found as b₁₂ = 0.5 with $μ_{{\tilde{b}}_{12}} (b_{12} = 0.776) = 0.5$ . Therefore, by carrying out analogous computations based on (5)-(6), the constructed ${\tilde{b}}_{12}$ is given below: $b_{12} = \frac{0.2}{0.52} + \frac{0.6}{0.66} + \frac{1}{0.77} + \frac{0.8}{0.85} + \frac{0.3}{0.94} + \frac{0.1}{0.98} .$ In conclusion, the result of gH-difference $Z_{12} = ({\tilde{a}}_{12}, {\tilde{b}}_{12})$ is obtained.

4. g-difference of continuous Z-numbers

The gH-difference of two fuzzy numbers exists under much less restrictive conditions, however it does not always exist [12, 13]. The g-difference proposed in [14] overcomes these shortcomings of the above discussed concepts and the g-difference of two fuzzy numbers always exists. As observed in [8], a convexification is required for the new difference to be always a fuzzy number. In this section, we firstly introduce the definition of the generalized difference. Then we will use the concept of generalized difference [14] to define the difference of two continuous Z-numbers. Finally, we will give an example to show the g-difference of continuous Z-numbers.

Definition 4.1. [8 , 13] The generalized difference (g-difference) of two fuzzy numbers $\tilde{a}, \tilde{b} \in F$ is a fuzzy number $\tilde{a} \underset{g}{⊖} \tilde{b} = \tilde{c}$ , which is defined by its level sets as ${[\tilde{a} ⊝_{g} \tilde{b}]}^{α} = cocl \underset{β \geq α}{\cup} ({[\tilde{a}]}^{β} ⊝_{gH} {[\tilde{b}]}^{β}),$ for all α ∈ [0, 1], where coclA denotes the connex hull of the closure of a crisp set A and the gH-difference _⊝gH is with interval operands $[\tilde{a}]^{β}$ and $[\tilde{b}]^{β}$ .

A more useful expression for the g-difference is given in [2].

Lemma 4.2. [2] The g-difference between two fuzzy numbers $\tilde{a}$ and $\tilde{b}$ is given levelwise by $\begin{matrix} {[\tilde{a} ⊝_{g} \tilde{b}]}^{α} = \\ [inf_{β \geq α} min {{\tilde{a}}_{L} (β) - {\tilde{b}}_{L} (β), {\tilde{a}}_{R} (β) - {\tilde{b}}_{R} (β)}, \\ sup_{β \geq α} max {{\tilde{a}}_{L} (β) - {\tilde{b}}_{L} (β), {\tilde{a}}_{R} (β) - {\tilde{b}}_{R} (β)}], \end{matrix}$ for all α ∈ [0, 1].

The following example will show when the gH-difference does not exist, while the g-difference exists.

Example 4.3. We consider two trapezoidal fuzzy numbers $\tilde{1} = 〈 0, 2, 2, 4 〉$ and $\tilde{2} = 〈 0, 1, 2, 3 〉$ , their α-level set is ${[\tilde{1}]}^{α} = [2 α, 4 - 2 α]$ and ${[\tilde{2}]}^{α} = [α, 3 - α]$ respectively. By Definition 4.1 their g-difference is the trapezoidal fuzzy number 〈0, 0, 1, 1〉, i.e., the [0, 1] interval. However, their gH-difference does not exist.

Considering the advantage of g-difference, we can discuss g-difference of continuous Z-numbers. Let $Z_{1} = ({\tilde{a}}_{1}, {\tilde{b}}_{1})$ and $Z_{2} = ({\tilde{a}}_{2}, {\tilde{b}}_{2})$ be continuous Z-numbers. Consider the problem of computation of g-difference $Z_{12} = Z_{1} \underset{g}{⊖} Z_{2}$ , where $Z_{12} = ({\tilde{a}}_{12}, {\tilde{b}}_{12})$ .

First, computing g-difference Z₁₂ can starts with the computation over the corresponding Z⁺-number. The Z⁺-number $Z_{12}^{+} = Z_{1}^{+} \underset{g}{⊖} Z_{2}^{+}$ is determined as follows: $Z_{1}^{+} \underset{g}{⊖} Z_{2}^{+} = ({\tilde{a}}_{12}, R_{1} - R_{2}) = ({\tilde{a}}_{1} \underset{g}{⊖} {\tilde{a}}_{2}, R_{1} - R_{2}) .$ Normally, we take broad family of probability distributions R₁ and R₂ into consideration. To simplicity, we consider normal distributions: $p_{1} = \frac{1}{\sqrt{2 π σ_{1}^{2}}} e^{- \frac{{(x - μ_{1})}^{2}}{2 σ_{1}^{2}}},$ (8) $p_{2} = \frac{1}{\sqrt{2 π σ_{2}^{2}}} e^{- \frac{{(x - μ_{2})}^{2}}{2 σ_{2}^{2}}},$ (9) The g-difference ${\tilde{a}}_{12}$ can be obtained by Lemma 4.2 and R₁ - R₂ is a convolution p₁₂ = p₁ ∘ p₂ of continuous probability distributions. Thus we have $p_{12} = \frac{1}{\sqrt{2 π (σ_{1}^{2} + σ_{2}^{2})}} e^{- \frac{{(x - μ_{1} - μ_{2})}^{2}}{2 (σ_{1}^{2} + σ_{2}^{2})}} .$ (10)

Therefore, we will complete the first step in the calculation Z₁₂ as the result of computation with $Z_{12}^{+} = ({\tilde{a}}_{1} \underset{g}{⊖} {\tilde{a}}_{2}, p_{12})$ came out.

At the second step, Let us mention that the true probability distributions p₁ and p₂ in Z-number Z₁ and Z₂ are not exactly known. The fuzzy restrictions, by comparison, can be used to describe the information available, $\int_{R} μ_{{\tilde{a}}_{1}} (x_{1}) p_{R_{1}} (x_{1}) d x_{1} i s {\tilde{b}}_{1},$ $\int_{R} μ_{{\tilde{a}}_{2}} (x_{2}) p_{R_{2}} (x_{2}) d x_{2} i s {\tilde{b}}_{2},$ And then, the information available are represented in accordance with the membership functions as: $μ_{{\tilde{b}}_{1}} (\int_{R} μ_{{\tilde{a}}_{1}} (x_{1}) p_{R_{1}} (x_{1}) d x_{1}),$ $μ_{{\tilde{b}}_{2}} (\int_{R} μ_{{\tilde{a}}_{2}} (x_{2}) p_{R_{2}} (x_{2}) d x_{2}),$ which have the fuzzy sets of probability distributions p₁ and p₂ with the membership functions defined as $μ_{p_{R 1}} (p_{R 1}) = μ_{{\tilde{b}}_{1}} (\int_{R} μ_{{\tilde{a}}_{1}} (x_{1}) p_{R_{1}} (x_{1}) d x_{1},$ (11) $μ_{p_{R 2}} (p_{R 2}) = μ_{{\tilde{b}}_{2}} (\int_{R} μ_{{\tilde{a}}_{2}} (x_{2}) p_{R_{2}} (x_{2}) d x_{2} .$ (12)

In this case basic values $b_{jl} \in supp ({\tilde{b}}_{j}), j = 1, 2; l = 1, \dots, m$ of a discrete fuzzy number ${\tilde{b}}_{j}, j = 1, 2$ are values of a probability measure of ${\tilde{a}}_{j}$ , $b_{jl} = P ({\tilde{a}}_{j})$ . Thus, given b_jk, we can get probability distribution p_jl such that $b_{jl} = μ_{{\tilde{a}}_{j}} (x_{j 1}) p_{jl} (x_{j 1}) + \dots + μ_{{\tilde{a}}_{j}} (x_{{jn}_{j}}) p_{j 2} (x_{{jn}_{j}}) .$ what is more, it is easy to find that p_jl satisfy: $\sum_{k = 1}^{n_{j}} p_{j l} (x_{j k}) = 1, p_{j l} (x_{j k}) \geq 0 .$

Thus, p_j can be find by solving the following goal programming problem $μ_{{\tilde{a}}_{j}} (x_{j 1}) p_{jl} (x_{j 1}) + \dots + μ_{{\tilde{a}}_{j}} (x_{{jn}_{j}}) p_{j 2} (x_{{jn}_{j}}) \to b_{jl}$ (13) subject to ${\begin{array}{l} p_{j l} (x_{j 1}) + p_{j l} (x_{j 2}) + \dots + p_{j l} (x_{j n}) = 1 \\ p_{j l} (x_{j 1}), p_{j l} (x_{j 2}), \dots, p_{j l} (x_{j n}) \geq 0 \end{array}$ (14) Then, let $u_{k} = μ_{{\tilde{a}}_{j}} (x_{jk})$ and w_k = p_j (x_jk) , k = 1, …, n . The problem (13)-(14) can be written as the goal linear programming problem $u_{1} w_{1} + u_{2} w_{2} + \dots + u_{n} w_{n} \to b_{jl}$ (15) subject to ${\begin{array}{l} w_{1} + w_{2} + \dots + w_{n} = 1 \\ u_{1}, u_{2}, \dots, u_{n} \geq 0 \end{array}$ (16) The probability distribution p_jl can be obtained by having obtained the solution w₁, w₂, …, w_n of problems (15)-(16) for each l = 1, 2, …, m since w_k = p_j (x_jk) , k = 1, …, n . In what follows given b_jl, the desired degree is

$μ_{p_{j l}} (p_{j l}) = μ_{{\tilde{b}}_{j}} (b_{j l}) = μ_{b_{j}} (\sum_{k = 1}^{n_{j}} μ_{{\tilde{a}}_{j}} (x_{j k}) p_{j l} (x_{j k})) .$

Constructing a fuzzy set of probability distributions p_jl needs to solve n simple goal linear programming problems (15)-(16). The problem is reduced to optimization problem with only one parameter for normal random variables. Then we can use some simple optimization methods to solve this problem.

It is known that the fuzzy sets of probability distributions p₁ and p₂ obtained from approximation of calculated p_jl (x_jk) by a normal distribution. Therefore, μ_{p
₁₂} (p₁₂) should be found as

$μ_{p_{12}} (p_{12}) = max_{{p_{1}, p_{2} | p_{1} = p_{2} \circ p_{12}}} min {μ_{p_{1}} (p_{1}), μ_{p_{2}} (p_{2})} .$

At the third step, given p₁₂, the probability measure of ${\tilde{a}}_{12} = {\tilde{a}}_{1} ⊝_{g} {\tilde{a}}_{2}$ can be obtained from $P ({\tilde{a}}_{12}) = \int_{R} μ_{{\tilde{a}}_{12}} (u) p_{12} (u) d u .$

At the fourth step, we construct ${\tilde{b}}_{12}$ . It is well known that $P ({\tilde{a}}_{12})$ is a fuzzy set ${\tilde{b}}_{12}$ with the membership function $μ_{{\tilde{b}}_{12}}$ defined as follows:

$μ_{{\tilde{b}}_{12}} (b_{12, s}) = sup (μ_{p_{12, s}} (p_{12, s}))$ (17) subject to $b_{12, s} = \sum_{k} p_{12, s} (x_{k}) μ_{{\tilde{a}}_{12}} (x_{k}) .$ (18) By solving the problem (17)-(18), ${\tilde{b}}_{12}$ can be obtained.

In conclusion, we have the result of Z₁₂ = Z₁_⊝gZ₂ as $Z_{12} = ({\tilde{a}}_{12}, {\tilde{b}}_{12})$ .

Example 4.4. Consider computation of Z₁₂ = Z₁_⊝gZ₂ for the Z-numbers $Z_{1} = ({\tilde{a}}_{1}, {\tilde{b}}_{1})$ and $Z_{2} = ({\tilde{a}}_{2}, {\tilde{b}}_{2})$ , where ${\tilde{a}}_{1} = 〈 0, 3, 5 〉 {\tilde{b}}_{1} = 〈 0, 0.4, 0.7 〉;$ ${\tilde{a}}_{2} = 〈 0, 2, 3, 4 〉 {\tilde{b}}_{2} = 〈 0.2, 0.4, 0.8 〉 .$ Apparently, ${\tilde{a}}_{1}$ , ${\tilde{b}}_{1}$ and ${\tilde{b}}_{2}$ are triangular fuzzy numbers. ${\tilde{a}}_{1}$ is a trapezoid fuzzy number.

At the first step, let us consider the case when the fuzzy sets of probability distributions underlying the considered Z-numbers are normal probability distributions. The corresponding Z⁺-number $Z_{12}^{+} = Z_{1}^{+} ⊝_{g} Z_{2}^{+}$ is determined as follows: $Z_{12}^{+} = ({\tilde{a}}_{12}, R_{1} - R_{2}) = ({\tilde{a}}_{1} \underset{g}{⊖} {\tilde{a}}_{2}, R_{1} - R_{2}),$ where R₁ and R₂ are the following normal probability distributions: $\begin{matrix} p_{1} & = & N (2, 9^{2}), \\ p_{2} & = & N (1.5, (3.3)^{2}) . \end{matrix}$ ${\tilde{a}}_{12} = {\tilde{a}}_{1} \underset{g}{⊖} {\tilde{a}}_{2}$ can be obtained based on Lemma 4.2, and ${\tilde{a}}_{12} = 〈 0, 0, 1, 1 〉$ . Then, we can get the probability distribution p₁₂ = N (0.5, 91.89) based on (10), in turn, R₁ - R₂ can be described by p₁₂. Thus, we obtain $Z_{12}^{+} = ({\tilde{a}}_{12}, R_{1} - R_{2}) = ({\tilde{a}}_{12}, p_{12})$ .

At the second step, we need to construct distributions p₁ and p₂ and compute membership degrees μ_{p
₁} (p₁) and μ_{p
₂} (p₂). Considering that u_i (j = 1, 2) are fixed, the determination of $p_{j} = (u_{j}, σ_{j}^{2}) (j = 1, 2)$ are written as: $\int_{s u p p (a_{1})} μ_{a_{1}} (x) \frac{1}{\sqrt{2 π σ_{j l}^{2}}} e^{\frac{{(x - u_{1})}^{2}}{2 σ_{j l}^{2}}} d x \to b_{j l}$ (19) subject to $σ_{jl} > 0 .$ (20)

It is easy to solve this simple nonlinear optimization problem with respect to σ_j. What is more, we have solved this problem and obtained the results as follows:

b _1l	σ _1l	b _2l	σ _2l
0.4219	2	0.7291	0.5
0.3072	3	0.6603	0.9
0.2382	4	0.5702	1.3
0.1937	5	0.4864	1.7
0.1628	6	0.4180	2.1
0.1403	7	0.3638	2.5
0.1232	8	0.3208	2.9
0.1098	9	0.2863	3.3
0.0990	10	0.2581	3.7
0.0901	11	0.2348	4.1

Now, we compute membership degrees μ_{p
₁} (p₁) and μ_{p
₂} (p₂) for the fuzzy restrictions (11)- (12). Thus, $\int_{R} μ_{{\tilde{a}}_{1}} (x) \frac{1}{\sqrt{2 π \cdot 9^{2}}} e^{\frac{{(x - 2)}^{2}}{2 \cdot 9^{2}}} d x \approx 0.1098$ $\int_{R} μ_{{\tilde{a}}_{2}} (x) \frac{1}{\sqrt{2 π \cdot {(3.3)}^{2}}} e^{\frac{{(x - 1.5)}^{2}}{2 \cdot {(3.3)}^{2}}} d x \approx 0.2863 .$ Furthermore, we have $μ_{p_{1}} (p_{1}) = μ_{{\tilde{b}}_{1}} (0.1098) \approx 0.2745$ $μ_{p_{2}} (p_{2}) = μ_{{\tilde{b}}_{2}} (0.1098) \approx 0.4315 .$ Homoplastically, the membership degrees of all the considered p₁ and p₂ can be obtained.

At the third step, we compute the membership degrees μ_{p
₁₂} (p₁₂) of the convolutions p₁₂. Given μ_{p
₁} () and μ_{p
₂} (), the membership degree of this restriction for the convolutions p₁₂ obtained above is $μ_{p_{12}} (p_{12}) = min {μ_{p_{1}} (p_{1}), μ_{p_{2}} (p_{2})} = 0.2745 .$ Homoplastically, the degrees for all the considered p₁₂ can be obtained.

At the fourth step, we proceed to construction of ${\tilde{b}}_{12}$ . The values of probability measure $P ({\tilde{a}}_{12})$ with respect to the known convolutions p₁₂ = (0.5, 91.89) can be computed by $P ({\tilde{a}}_{12}) = \int_{R} μ_{{\tilde{a}}_{12}} (x) \frac{1}{\sqrt{2 π \cdot 91.89}} e^{\frac{{(x - 0.5)}^{2}}{2 \cdot 991.89}} d x \approx 0.0416 .$

Therefore, one basic value of ${\tilde{b}}_{12}$ is found as b₁₂ = 0.0416 such that $μ_{\tilde{b_{12}}} (b_{12} = 0.0416) = 0.2745$ . Furthermore, the ${\tilde{b}}_{12}$ can be constructed by carrying out analogous computations. The obtained result approximated as a triangular fuzzy number is given below: ${\tilde{b}}_{12} = 〈 0.0028, 0.1487, 0.3812 〉 .$

In conclusion, $Z_{12} = ({\tilde{a}}_{12}, {\tilde{b}}_{12})$ is obtained as the result of g-difference.

5. Conclusions

Z-number is a new notion which has a flexible presentation of uncertain information to simulate the knowledge or behavior of human being. There is a significant need in development of a background of arithmetic of Z-numbers. Taking into account that the definition of Z-number, in this paper, we have researched the generalized Hukuhara difference of discrete Z-numbers and presented the procedures underlying computation of generalized Hukuhara difference of discrete Z-numbers. More calculation process details are given in a relevant example. Then, we also have shown the computing process of generalized difference of continuous Z-numbers. Similarly, the reliability of the computation will be illustrated by a relevant example. We also hope that our results in this paper may lead to significant, new and innovative results in other related fields.

Footnotes

Acknowledgements

This work was supported by The National Natural Science Foundation of China (Grant no. 11671001).

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