The relationship of three difference operations for fuzzy numbers to three kinds of derivative

Abstract

The difference operation for fuzzy number is an essential concept for the fuzzy set theory. There are several differences proposed: generalized difference, generalized Hukuhara difference and granule difference. Based on these differences, generalized differentiability, generalized Hukuhara differentiability and granule differentiability are also proposed, respectively. In this paper, the relations among these three kinds of differences and that of related three kinds of differentiability are clarified.

Keywords

Generalized differences generalized differentiability granule differentiability fuzzy-number-valued function

1 Introduction

1.1 Related works

Zadeh [30] put forward the fuzzy set theory in 1965, and soon it attracted many scientists who were committed to apply it to the field of mathematics. Up to now, there are still a large number of literatures using fuzzy set theory to expand the traditional four arithmetic operations and differentiability. In most cases, the results of information obtained are often uncertain due to the errors brought about by the inherent subjectivity of human thinking and the complex and changing environment. Quantifying uncertain information with clear and accurate numbers is therefore a challenge. According to fuzzy set theory, we express this uncertainty as fuzzy number, that is, a class of possible sets, and then analyze the information or model with uncertainty. Furthermore, using fuzzy set theory one can investigate fuzzy dynamical systems control [16, 17] and fuzzy differential equations [2, 7]. In terms of practical application, [29] combined the fuzzy set theory and a technique for order preference by similarity to ideal solution (TOPSIS), the Fuzzy-Rough-TOPSIS model of project risk assessment is established. And [31] introduced a new rough fuzzy set and studied its application in TOPSIS multi-attribute decision making. Moreover, Sarwar et al. [21] proposed rough ELECTRE II approach which can be effectively applied to enhance the efficiency of working conditions and handle the uncertainty in real world.

The Hukuhara [6] difference (H-difference, for short) is defined in such a way that the "difference" between two identical intervals is zero, but in general the results are not the same with the traditional difference between two different intervals. And it requires that the interval length of the subtractor is not less than the interval length of the minuend, which limits its practical application (see [22, 23]). This restriction continued until 2008, the appearance of a new difference—generalized Hukuhara difference (gH-difference, for short) has been broken that restriction, which was introduced by Stefanini [22]. It is also proved that the gH-difference has much wider application range than that of H-difference, and that gH-difference is equal to the H-difference when H-difference exists. Based on the gH-difference between two fuzzy numbers Bede in 2013 [1] defined generalized Hukuhara derivative (gH-derivative, for short) of fuzzy-number-valued functions. One of the advantage of gH-difference is that there is no need to limit the length of the interval, that is to say, the gH-difference always exists between two intervals. However, this is not the case for any two fuzzy numbers, there are also some limitations to gH-difference of fuzzy numbers. The μ-level sets of fuzzy numbers x and y are denoted as $[x]_{μ} = [x_{μ}^{-}, x_{μ}^{+}]$ and $[y]_{μ} = [y_{μ}^{-}, y_{μ}^{+}]$ , and $L (\cdot)$ denotes the interval length of "·". The gH-difference between x and y exists if and only if it satisfies one of the following two cases for all μ ∈ [0, 1] [23]: $1) L ([x]_{μ}) \geq L ([y]_{μ})$ , then $x_{μ}^{-} - y_{μ}^{-}$ is increasing and $x_{μ}^{+} - y_{μ}^{+}$ is decreasing with respect to μ, or $2) L ([x]_{μ}) \leq L ([y]_{μ})$ , then $x_{μ}^{-} - y_{μ}^{-}$ is decreasing and $x_{μ}^{+} - y_{μ}^{+}$ is increasing with respect to μ.

It can be seen that gH-difference between two FNs may not be obtained except in some simple cases. Similarly, the same limitation exists for gH-derivative that based on gH-difference. Thus, generalized difference (g-difference for short), a much wider difference, was proposed by Stefanini [22, 23] and Bede [1] with a corresponding derivative called generalized derivative (g-derivative, for short). Although it was originally proved that g-difference exists between any two fuzzy numbers, a counterexample presented in 2015 [5] by Gomes demonstrated that a little modification is necessary for the definition of g-difference such that the g-difference is invariably guaranteed to be a fuzzy number between two fuzzy numbers. Moreover, this modification does not change the corresponding properties and theorems that proposed with g-derivative.

It should be noted that the gH-difference is calculated on the basis of the α-level sets, by only considering the endpoint values of the α-level sets. Unfortunately, if someone want to acquire the full information of the points in this interval, it cannot get the desired result by applying the gH-difference.

In the early 21st century, Untiedt [10, 11] had presented the Constraint Fuzzy Interval (CFI, for short) representation which Piegat et al. [18 , 27] called the horizontal membership function. Mazandarani et al. [13] proposed the granule difference (gr-difference, for short) based on horizontal membership function. It should be noted that the α-level sets are originated from the result granule which is the full result. The result granule is of great help if someone is interested in complete information result which including lower and upper points of uncertainty. But if someone just prefers to consider the uncertainty information that only include the maximum and minimum values, then the α-level sets is enough for their work. Similarly, Mazandarani et al. [13] also proposed the granule derivative (gr-derivative, for short) based on gr-difference.

1.2 Innovative contribution

What are the relations between these kinds of differences and the corresponding derivatives? Mazandarani et al. in [13] discussed the relations among these derivatives. But in this paper, we will analyze the relations of these derivatives more specifically. We will discuss the following problems to help people reasonably select the corresponding difference operation and derivative in practice.

1) Although we have already known that there exists the g-difference and the gr-difference between any two fuzzy numbers, do they get the same results? Do they have the same derivative with the corresponding differentiability?

2) It is also known that the g-differentiability is more general than the gH-differentiability and have the same derivatives if the gH-derivative exists, then does the gH-differentiability have the same relation with gr-differentiability or not?

1.3 Framework of the paper

We divided this paper into four sections. In Section 2, we will introduce some basic concepts and definitions that we will use throughout the paper. The Section 3 is divided into two parts including Differences 3.1 and Differentiability 3.2, and proposed the main results on these three differences and differentiability. The Section 4 gives the conclusions.

2 Basic Concepts

In this section, we will introduce the necessary notations and give some definitions that will be used in the later sequel.

The sets of all real numbers, positive real numbers, bounded closed intervals of real numbers and fuzzy numbers on real numbers are denoted by $ℝ$ , $ℝ^{+}$ , K_c and $F_{1}$ , respectively. For any $x \in F_{1}$ , we use [x] _μ to represent the μ-level set of x, and the upper and lower endpoints of [x] _μ are denoted as $x_{μ}^{+}$ and $x_{μ}^{-}$ for all μ ∈ [0, 1]. We use the triple (a₁, a₂, a₃) with a₁ ≤ a₂ ≤ a₃ represents the triangular fuzzy number x as x = (a₁, a₂, a₃). Additionally, the trapezoidal fuzzy numbers y is denoted as y = (b₁, b₂, b₃, b₄) with b₁ ≤ b₂ ≤ b₃ ≤ b₄.

Definition 2.1.[22] Let X, Y ∈ K_c, we define the gH-difference of X and Y as the interval Z ∈ K_c such that

$X \underset{gH}{⊖} Y = Z \Leftrightarrow {\begin{matrix} (i) X = Y + Z, \\ or \\ (ii) Y = X + (- 1) Z . \end{matrix}$ (1)

According to Definition 2.1, the gH-difference always exists between any two intervals X = [x^-, x⁺] , Y = [y^-, y⁺] and it is equal to $\begin{matrix} X \underset{gH}{⊖} Y & = [min {x^{+} - y^{+}, x^{-} - y^{-}}, \\ max {x^{+} - y^{+}, x^{-} - y^{-}}] . \end{matrix}$ (2)

However, the gH-difference between two fuzzy numbers does not exists all the time. Please refer to [1] for more details. If their exists the gH-difference between $x, y \in F_{1}$ , then in the form of an μ-level set, we have, for each μ ∈ [0, 1], $\begin{matrix} [x \underset{gH}{⊖} y]_{μ} \\ = [min {x_{μ}^{+} - y_{μ}^{+}, x_{μ}^{-} - y_{μ}^{-}}, \\ max {x_{μ}^{+} - y_{μ}^{+}, x_{μ}^{-} - y_{μ}^{-}}] . \end{matrix}$ (3) To overcome this weakness, Bede and Stefanini [1] proposed a new difference—g-difference. Gomes and Barros [8] demonstrated that it is necessary to add convexity to ensure that the new difference is always be a fuzzy number.

Definition 2.2.[8] The g-difference between any two $x, y \in F_{1}$ is given levelwise by

$[x \underset{g}{⊖} y]_{μ} = cl (conv ⋃_{α \geq μ} ([x]_{α} \underset{gH}{⊖} [y]_{α})) .$ (4)

Proposition 2.1. [8] The g-difference can also be represented by $\begin{matrix} [x \underset{g}{⊖} y]_{μ} & = [inf_{α ⩾ μ} min (x_{α}^{+} - y_{α}^{+}, x_{α}^{-} - y_{α}^{-}), \\ sup_{α ⩾ μ} max (x_{α}^{+} - y_{α}^{+}, x_{α}^{-} - y_{α}^{-})] . \end{matrix}$ (5)

Bede and Stefanini proposed the following definitions of derivatives according to the gH-difference and the g-difference for fuzzy-number-valued function in [1, 25].

Definition 2.3. [25] Let f be a fuzzy-number-valued function such that $f : (a, b) \to F_{1}$ . We define the gH-derivative of f at t₀ ∈ (a, b) as

$f_{gH}^{'} (t_{0}) = lim_{h \to 0} \frac{f (t_{0} + h) ⊖_{gH} f (t_{0})}{h},$ (6) where t₀ + h ∈ (a, b). We say that f is generalized Hukuhara differentiable (gH-differentiable, for short) at t₀ if $f_{gH}^{'} (t_{0}) \in F_{1}$ satisfying (2.5) exists.

Chalco-Cano and Romĺćn-Flores [4], Chalco-Cano and Rodrĺłguez-Lĺőpez [3] and Stefanini and Arana-Jim|nez [24] proposed the sufficient and necessary conditions of gH-differentiability. Furthermore Qiu [20] clarified the relations of gH-differentiability for a fuzzy-number-valued function and for these functions given a complete characterization of the gH-differentiability at a point.

Theorem 2.1. [20] Let $f : (a, b) \to F_{1}$ be a fuzzy-number-valued function, $[f (x)]_{α} = [f_{α}^{-} (x), f_{α}^{+} (x)]$ . f is gH-differentiable at x₀ ∈ (a, b) if and only if one of the following four conditions holds: (i) The right derivatives and the left derivatives $(f_{α}^{-})_{+}^{'} (x_{0}), (f_{α}^{+})_{+}^{'} (x_{0}), (f_{α}^{-})_{-}^{'} (x_{0})$ and $(f_{α}^{+})_{-}^{'} (x_{0})$ exist and $\begin{matrix} [f^{'} (x_{0})]_{α} \\ = [min {(f_{α}^{-})_{+}^{'} (x_{0}), (f_{α}^{+})_{+}^{'} (x_{0})}, \\ max {(f_{α}^{-})_{+}^{'} (x_{0}), (f_{α}^{+})_{+}^{'} (x_{0})}] \\ = [min {(f_{α}^{-})_{-}^{'} (x_{0}), (f_{α}^{+})_{-}^{'} (x_{0})}, \\ max {(f_{α}^{-})_{-}^{'} (x_{0}), (f_{α}^{+})_{-}^{'} (x_{0})}] . \end{matrix}$ (ii) f^- (x) and f⁺ (x) are right differentiable at x₀. Slope functions φ_{f
^-} (h) and φ_{f
⁺} (h) are left complementary at 0, i.e., C_L(0) (φ_{f
^-}) = C_L(0) (φ_{f
⁺}) = {a^-, a⁺}, where $a^{-}, a^{+} \in ℝ$ and a^- < a⁺. Moreover, $\begin{matrix} [f^{'} (x_{0})]_{α} \\ = [min {(f_{α}^{-})_{+}^{'} (x_{0}), (f_{α}^{+})_{+}^{'} (x_{0})}, \\ max {(f_{α}^{-})_{+}^{'} (x_{0}), (f_{α}^{+})_{+}^{'} (x_{0})}] \\ = [a^{-}, a^{+}] . \end{matrix}$ (iii) f^- (x) and f⁺ (x) are left differentiable at x₀. φ_{f
^-} (h) and φ_{f
⁺} (h) are right complementary at 0, i.e., C_R(0) (φ_{f
^-}) = C_R(0) (φ_{f
⁺}) = {a^-, a⁺}, where $a^{-}, a^{+} \in ℝ$ and a^- < a⁺. Moreover, $\begin{matrix} [f^{'} (x_{0})]_{α} \\ = [min {(f_{α}^{-})_{-}^{'} (x_{0}), (f_{α}^{+})_{-}^{'} (x_{0})}, \\ max {(f_{α}^{-})_{-}^{'} (x_{0}), (f_{α}^{+})_{-}^{'} (x_{0})}] \\ = [a^{-}, a^{+}] . \end{matrix}$ (iv) φ_{f
^-} (h) and φ_{f
⁺} (h) are both left complementary and right complementary at 0, i.e., C_R(0) (φ_{f
^-}) = C_R(0) (φ_{f
⁺}) = C_L(0) (φ_{f
^-}) = C_L(0) (φ_{f
⁺}) = {a^-, a⁺}, where $a^{-}, a^{+} \in ℝ$ and a^- < a⁺. Moreover, $\begin{matrix} [f^{'} (x_{0})]_{α} = [a^{-}, a^{+}] . \end{matrix}$

Definition 2.4. [1] Let f be a fuzzy-number-valued function such that $f : (a, b) \to F_{1}$ . We define the g-derivative of f at t₀ ∈ (a, b) for h as

$f_{g}^{'} (t_{0}) = lim_{h \to 0} \frac{f (t_{0} + h) ⊖_{g} f (t_{0})}{h},$ (7) where t₀ + h ∈ (a, b). We say that f is generalized differentiable (g-differentiable, for short) at t₀ if $f_{g}^{'} (t_{0}) \in F_{1}$ satisfying (7) exists.

The following theorem indicates a practical formula for the g-derivative.

Theorem 2.2. [1] Let $f : (a, b) \to F_{1}$ be such that $[f (t)]_{μ} = [f_{μ}^{-} (t), f_{μ}^{+} (t)]$ . If $f_{μ}^{-} (t)$ and $f_{μ}^{+} (t)$ are differentiable real-valued functions with respect to t, uniformly for μ ∈ [0, 1], then f (t) is g-differentiable and we have $\begin{matrix} [f_{g}^{'} (t)]_{μ} \\ = [inf_{α \geq μ} min {(f_{α}^{+})^{'} (t), (f_{α}^{-})^{'} (t)}, \\ sup_{α \geq μ} max {(f_{α}^{+})^{'} (t), (f_{α}^{-})^{'} (t)}] . \end{matrix}$ (8)

Next, we will introduce the horizontal membership functions which Piegat, Landowski and Tomaszewska [18 , 27] showed that without using Zadeh’s extension principle [30] they are great of direct introduction of fuzzy numbers in conventional mathematical formulas.

Definition 2.5. [16, 22] Let w : [a, b] → [0, 1] be a fuzzy number. Then w (x) is represented as w^gr (β, α_w) by the horizontal membership function w^gr : [0, 1] × [0, 1] → [a, b]. The ągrąaś means the granule of information included in x ∈ [a, b] and β ∈ [0, 1] represents the membership degree of x in w (x). We called α_w ∈ [0, 1] relative-distance-measure (RDM, for short) variable, and $w^{gr} (β, α_{w}) = w_{β}^{-} + (w_{β}^{+} - w_{β}^{-}) α_{w}$ .

Remark 2.1. [16, 22] We can also denote the horizontal membership function of $w (x) \in F_{1}$ by $H (w) ≜ w^{gr} (β, α_{w})$ . Moreover, using $\begin{matrix} H^{- 1} (w^{gr} (β, α_{w})) = [w]_{β} \\ = [inf_{μ ⩾ β} min_{α_{w}} w^{gr} (μ, α_{w}), sup_{μ ⩾ β} max_{α_{w}} w^{gr} (μ, α_{w})] \end{matrix}$ (9) can obtain the β-level sets of the span of the information granule of w (x), which is called vertical membership function.

Next is the definition of four arithmetic operations between two fuzzy numbers based on horizontal membership functions.

Definition 2.6. [13] For any $x, y \in F_{1}$ , their horizontal membership functions are represented as x^gr (β, α_x) and y^gr (β, α_y), respectively. Then, the gr-difference $x \underset{gr}{⊖} y$ is a fuzzy number z such that $H (z) ≜ x^{gr} (β, α_{x}) - y^{gr} (β, α_{y})$ .

Remark 2.2. [13] Let $z = x \underset{gr}{⊖} y$ . Then, the β-level sets of z is always be presented as $[z]_{β} = H^{- 1} (x^{gr} (β, α_{x}) - y^{gr} (β, α_{y}))$ .

Before introducing gr-differentiability we first review a related definition.

Definition 2.7. [16] Let $x, y \in F_{1}$ and the function $D_{gr} : F_{1} \times F_{1} \to ℝ^{+} \cup {0}$ . Then, the distance between two fuzzy numbers x and y is denoted as $D_{gr} (x, y) = sup_{β} max_{α_{x}, α_{y}} ∣ x^{gr} (β, α_{x}) - y^{gr} (β, α_{y}) ∣$ which is a metric on $F_{1}$ .

Definition 2.8. [13] Let f be a fuzzy-number-valued function that $f : (a, b) \to F_{1}$ . We define the gr-derivative of f at t₀ ∈ (a, b) as

$f_{gr}^{'} (x_{0}) = lim_{h \to 0} \frac{f (x_{0} + h) ⊖_{gr} f (x_{0})}{h},$ (10) where t₀ + h ∈ (a, b) and the limit is taken in the metric space $(F_{1}, D_{gr})$ . If there exists a fuzzy number $f_{gr}^{'} (x_{0}) \in F_{1}$ , we say that f is granular differentiable (gr-differentiable, for short) at x₀.

Theorem 2.3. = [13] Let $f : [a, b] \to F_{1}$ be a fuzzy function. The horizontal membership function f^gr (x, β, α_f) of f is differentiable with respect to x at x₀ ∈ (a, b) if and only if f is gr-differentiable at x₀, Moreover, $\begin{matrix} H (f_{gr}^{'} (x_{0})) = \frac{\partial f^{gr} (x, β, α_{f})}{\partial x} ∣_{x = x_{0}} . \end{matrix}$

3 Main Results

This section is divided into two parts. First in subsection 3.1 we will clarify the relations among gH-difference, g-difference and gr-difference. Moreover, we give some properties when two trapezoidal fuzzy numbers making these subtractions. Next in the subsection 3.2, we propose the relations among gH-differentiability, g-differentiability and gr-differentiability, and some examples are given to understand them better.

3.1 Differences

For any two fuzzy numbers the g-difference and gr-difference always exist, but they are not always equivalent. Let $x, y \in F_{1}$ be as follows, $x (u) {\begin{matrix} 1, i f u \in 0, \\ 0.5, i f u \in [- 2, 0) \cup (0, 2] \\ 0, o t h e r w i s e \end{matrix},$ $x (u) {\begin{matrix} 1, i f u \in 0, \\ 0.5, i f u \in [- 2, 0) \cup (0, 2] \\ 0, o t h e r w i s e \end{matrix},$ Gomes in [8] calculated the g-difference between x and y as

${[x ⊖_{g} y]}_{μ} = {\begin{matrix} 0, i f 0.5 < μ \leq 1, \\ [0, 1], i f 0 \leq μ \leq 0.5. \end{matrix}$ However, Mazandarani and Pariz calculated the gr-difference in [13] between x and y, and get $\begin{matrix} [x \underset{gr}{⊖} y]_{μ} & = H^{- 1} (x^{gr} (β, α_{x}) - y^{gr} (β, α_{y})) \\ = {\begin{matrix} 0, & if 0.5 < μ \leq 1, \\ [- 3, 5], & if 0 \leq μ \leq 0.5 . \end{matrix} \end{matrix}$

Next, we will give the relations of these differences for fuzzy numbers. For any $x, y \in F_{1}$ , let $[x]_{μ} = [x_{μ}^{-}, x_{μ}^{+}], [y]_{μ} = [y_{μ}^{-}, y_{μ}^{+}]$ be their μ-level sets. The g-difference $x \underset{g}{⊖} y = z^{g}$ and the gr-difference $x \underset{gr}{⊖} y = z^{gr}$ having μ-level sets as $[z^{g}]_{μ} = [z_{μ}^{g -}, z_{μ}^{g +}]$ and $[z^{gr}]_{μ} = [z_{μ}^{gr -}, z_{μ}^{gr +}]$ , respectively.

Theorem 3.1.For any $x, y \in F_{1}$ , the gr-difference between x and y is given levelwise by

$[x \underset{gr}{⊖} y]_{α} = [inf_{μ ⩾ α} (x_{μ}^{-} - y_{μ}^{+}), sup_{μ ⩾ α} (x_{μ}^{+} - y_{μ}^{-})] .$ (11)

Proof. According to (2.11) we get the α-level set of gr-difference as follows, $\begin{matrix} [z^{gr}]_{α} = [x \underset{gr}{⊖} y]_{α} \\ = [inf_{μ ⩾ α} min_{α_{z}} z^{gr} (μ, α_{z}), sup_{μ ⩾ α} max_{α_{z}} z^{gr} (μ, α_{z})] . \end{matrix}$ (12)

Based on Definition 2.9, the horizontal membership functions of x, y, z can be written as $\begin{matrix} x^{gr} (μ, α_{x}) = x_{μ}^{-} + (x_{μ}^{+} - x_{μ}^{-}) α_{x}, \\ y^{gr} (μ, α_{y}) = y_{μ}^{-} + (y_{μ}^{+} - y_{μ}^{-}) α_{y}, \\ z^{gr} (μ, α_{z}) = z^{gr} (μ, α_{x}, α_{y}) = x^{gr} (μ, α_{x}) - y^{gr} (μ, α_{y}) \\ = x_{μ}^{-} - y_{μ}^{-} + (x_{μ}^{+} - x_{μ}^{-}) α_{x} - (y_{μ}^{+} - y_{μ}^{-}) α_{y} . \end{matrix}$ (13)

It is easy to see that z^gr (μ, α_x, α_y) is monotonically increasing with respect to α_x or α_y. Therefore the extreme values of z^gr (μ, α_z) can be achieved at the following four cases: If α_x = 0, α_y = 0, then $\begin{matrix} z^{gr} (μ, α_{z}) = x_{μ}^{-} - y_{μ}^{-} . \end{matrix}$ If α_x = 0, α_y = 1, then $\begin{matrix} z^{gr} (μ, α_{z}) = x_{μ}^{-} - y_{μ}^{+} . \end{matrix}$ If α_x = 1, α_y = 0, then $\begin{matrix} z^{gr} (μ, α_{z}) = x_{μ}^{+} - y_{μ}^{-} . \end{matrix}$ If α_x = 1, α_y = 1, then $\begin{matrix} z^{gr} (μ, α_{z}) = x_{μ}^{+} - y_{μ}^{+} . \end{matrix}$ It is obvious that $\begin{matrix} min_{α_{z}} z^{gr} (μ, α_{z}) = x_{μ}^{-} - y_{μ}^{+}, \\ max_{α_{z}} z^{gr} (μ, α_{z}) = x_{μ}^{+} - y_{μ}^{-} . \end{matrix}$ Consequently, the α-level sets of gr-difference in (3.2) is equal to $\begin{matrix} [z^{gr}]_{α} = [x \underset{gr}{⊖} y]_{α} \\ = [inf_{μ ⩾ α} (x_{μ}^{-} - y_{μ}^{+}), sup_{μ ⩾ α} (x_{μ}^{+} - y_{μ}^{-})] . \end{matrix}$ □

Theorem 3.2. For all α ∈ [0, 1] we have $[x \underset{g}{⊖} y]_{α} \subset [x \underset{gr}{⊖} y]_{α}$ for any $x, y \in F_{1}$ .

Proof. According to (2.4) and (3.4) we give the α-level sets of the g-difference and gr-difference as follows, $\begin{matrix} [z^{g}]_{α} = [x \underset{g}{⊖} y]_{α} \\ = [inf_{μ ⩾ α} min {x_{μ}^{+} - y_{μ}^{+}, x_{μ}^{-} - y_{μ}^{-}}, \\ sup_{μ ⩾ α} max {x_{μ}^{+} - y_{μ}^{+}, x_{μ}^{-} - y_{μ}^{-}}], \\ [z^{gr}]_{α} = [x \underset{gr}{⊖} y]_{α} \\ = [inf_{μ ⩾ α} (x_{μ}^{-} - y_{μ}^{+}), sup_{μ ⩾ α} (x_{μ}^{+} - y_{μ}^{-})] . \end{matrix}$ For any fixed μ ∈ [0, 1], since $x_{μ}^{-} \leq x_{μ}^{+}, y_{μ}^{-} \leq y_{μ}^{+}$ we have $\begin{matrix} x_{μ}^{-} - y_{μ}^{+} \leq x_{μ}^{-} - y_{μ}^{-} \leq x_{μ}^{+} - y_{μ}^{-}, \\ x_{μ}^{-} - y_{μ}^{+} \leq x_{μ}^{+} - y_{μ}^{+} \leq x_{μ}^{+} - y_{μ}^{-}, \end{matrix}$ which means that $\begin{matrix} x_{μ}^{-} - y_{μ}^{+} & \leq min {x_{μ}^{+} - y_{μ}^{+}, x_{μ}^{-} - y_{μ}^{-}} \\ \leq max {x_{μ}^{+} - y_{μ}^{+}, x_{μ}^{-} - y_{μ}^{-}} \\ \leq x_{μ}^{+} - y_{μ}^{-} . \end{matrix}$ Thus, we can get $[x \underset{g}{⊖} y]_{α} \subset [x \underset{gr}{⊖} y]_{α}$ . □

Remark 3.1. The relations among gH-difference, g-difference and gr-difference between $x, y \in F_{1}$ have been shown in Table 1. Moreover, we can get different figures in terms of existence (Fig. 1) and the result (Fig. 2) if gH-difference exists.

Fig. 1

IGSO-ISSCTA framework. In terms of existence.

Fig. 2

In terms of result if gH-difference exists.

The following proposition can be obtained.

Proposition 3.1.Let trapezoidal fuzzy numbers u_i = (a_i, b_i, c_i, d_i) , a_i ≤ b_i ≤ c_i ≤ d_i, i = 1, 2.

If $u_{1} \underset{gH}{⊖} u_{2}$ exists, then w^g = w^gH, and we have $w_{β}^{gH -} - w_{β}^{gr -} = w_{β}^{gr +} - w_{β}^{gH +} = min {L ([u_{1}]_{β}), L ([u_{2}]_{β})}$ . Moreover, w^gr = (a₁ - d₂, b₁ - c₂, c₁ - b₂, d₁ - a₂), $\begin{matrix} w^{g} = w^{gH} \\ = (min {a_{1} - a_{2}, d_{1} - d_{2}}, min {c_{1} - c_{2}, b_{1} - b_{2}}, \\ max {c_{1} - c_{2}, b_{1} - b_{2}}, max {a_{1} - a_{2}, d_{1} - d_{2}}) . \end{matrix}$ If and only if one of u₁, u₂ is a real number, then $w_{β}^{gH -} - w_{β}^{gr -} = w_{β}^{gr +} - w_{β}^{gH +} = 0$ , which means that w^gH = w^gr.

Specially, if u_i = (a_i, b_i, c_i) are triangular fuzzy numbers, then w^gr = (a₁ - c₂, b₁ - b₂, c₁ - a₂), w^gH = (min {a₁ - a₂, c₁ - c₂} , b₁ - b₂, max {a₁ - a₂, c₁ - c₂}).

If the gH-difference between u₁ and u₂ do not exists, then we have $\begin{matrix} w^{g} = ( & min {a_{1} - a_{2}, b_{1} - b_{2}}, b_{1} - b_{2}, \\ c_{1} - c_{2}, max {c_{1} - c_{2}, d_{1} - d_{2}}), \end{matrix}$ and $w_{α}^{g -} - w_{α}^{gr -} = w_{α}^{gr +} - w_{α}^{g +}$ if and only if c₁ - d₁ = c₂ - d₂ and b₁ - a₁ = b₂ - a₂ hold simultaneously.

Moreover, $w_{α}^{g -} - w_{α}^{gr -} = w_{α}^{gr +} - w_{α}^{g +} = 0$ , i.e., w^g = w^gr if and only if one of the following case holds:

(1) u₂ is a real number and c₁ = d₁, or

(2) u₂ is a real number and a₁ = b₁.

Specially, if u_i = (a_i, b_i, c_i) are triangular fuzzy numbers, then $\begin{matrix} w^{g} = ( & min {a_{1} - a_{2}, b_{1} - b_{2}}, b_{1} - b_{2}, \\ max {b_{1} - b_{2}, c_{1} - c_{2}}) . \end{matrix}$

Next, the Table 2 shows the laws of different differences between two trapezoidal fuzzy numbers u₁ and u₂ in Proposition 3.1. Detailed prove of Proposition 3.1 is in the Appendix A.

Table 2

The relations between w^gH, w^g and w^gr according to Proposition

	gH-difference exists	gH-difference does not exists
w ^gr	w^gr = (a₁ - d₂, b₁ - c₂, c₁ - b₂, d₁ - a₂)	w^gr = (a₁ - d₂, b₁ - c₂, c₁ - b₂, d₁ - a₂)
	w^gH = (min {a₁ - a₂, d₁ - d₂} ,
w ^gH	min {c₁ - c₂, b₁ - b₂} ,
	max {c₁ - c₂, b₁ - b₂} ,	∖
	max {a₁ - a₂, d₁ - d₂})
w ^g	w^gH = w^g	w^g = (min {a₁ - a₂, b₁ - b₂} , b₁ - b₂,
		c₁ - c₂, max {c₁ - c₂, d₁ - d₂})
$w_{α}^{gH -} - w_{α}^{gr -} =$	holds all the time	∖
$w_{α}^{gr +} - w_{α}^{gH +}$
w^gH = w^gr	if and only if u₁ or(and) u₂
	k is(are) real number(s)	∖
$w_{α}^{g -} - w_{α}^{gr -} =$	w^gH = w^g	if and only if c₁ - d₁ = c₂ - d₂ and
$w_{α}^{gr +} - w_{α}^{g +}$		b₁ - a₁ = b₂ - a₂ hold simultaneously
w^g = w^gr	w^gH = w^g	if and only if u₂ is a real number and
		c₁ = d₁ or a₁ = b₁ holds

Example 3.1. (a) Let u = (1, 2, 3) , v = (0.5, 1, 1.5). According to Proposition 3.1. we can get $\begin{matrix} w^{g} = w^{gH} \\ = (min {1 - 0.5, 3 - 1.5}, 1, max {1 - 0.5, 3 - 1.5}) \\ = (0.5, 1, 1.5), \\ w^{gr} = (1 - 1.5, 2 - 1, 3 - 0.5) = (- 0.5, 1, 2.5) . \end{matrix}$

(b) Let u = (2, 4, 7) , v = (1, 8, 9). After a simple calculate we can find the gH-difference between u and v do not exists, then from Proposition 3.1. we have $\begin{matrix} w^{g} & = (min {2 - 1, 4 - 8}, 4 - 8, max {4 - 8, 7 - 9}) \\ = (- 4, - 4, - 2), \end{matrix}$ w^gr = (2 - 9, 4 - 8, 7 - 1) = (-7, - 4, 6).

(c) Let u = (1, 2, 3, 7) , v = (-1.5, - 1, 0, 3.5). By Proposition 3.1. we get $\begin{matrix} w^{g} = w^{gH} \\ = (min {1 + 1.5, 7 - 3, 5}, min {2 + 1, 3 - 0}, \\ max {2 + 1, 3 - 0}, max {1 + 1.5, 7 - 3, 5}) \\ = (2.5, 3, 3, 3.5), \\ w^{gr} = (1 - 3.5, 2 - 0, 3 + 1, 7 + 1.5) \\ = (- 2.5, 2, 4, 8.5) . \end{matrix}$

We have already known the existence conditions of gH-difference in [23]. As for the other two differences, as is observed, if one want more information dates for the research, gr-difference is the best chose. However, if one want relatively accurate dates then g-difference will be more helpful.

3.2 Differentiability

It is necessary to study the characterizations of gr-differentiability since this differentiability becomes more and more popular in the literature [8 , 28]. First we give a necessary and sufficient condition of gr-differentiability and discuss the relation between g-differentiability and gr-differentiability.

Theorem 3.3.Let $g : [a, b] \to F_{1}$ and $[g (x)]_{α} = [g_{α}^{-} (x), g_{α}^{+} (x)]$ are the α-level sets of g. Then g (x) is gr-differentiable if and only if $g_{α}^{-} (x)$ and $g_{α}^{+} (x)$ are differentiable real-valued functions with respect to x, uniformly for α ∈ [0, 1], and we have $\begin{matrix} [g_{gr}^{'} (x)]_{α} \\ = [inf_{β \geq α} min {(g_{β}^{-})^{'} (x), (g_{β}^{+})^{'} (x)}, \\ sup_{β \geq α} max {(g_{β}^{-})^{'} (x), (g_{β}^{+})^{'} (x)}] . \end{matrix}$ (14) Moreover, if the fuzzy-number-valued function g is gr-differentiable then it is g-differentiable as well, and $g_{gr}^{'} = g_{g}^{'}$ .

Proof. Necessity. If g is gr-differentiable at x ∈ (a, b), according to Theorem if and only if $g^{gr} (x, α, α_{g}) = g_{α}^{-} (x) + (g_{α}^{+} (x) - g_{α}^{-} (x)) α_{g}$ is differentiable with respect to x, which means that $g_{α}^{-} (x)$ and $g_{α}^{+} (x)$ are differentiable real-valued functions with respect to x. Actually, based on Theorem we have

$H (g_{gr}^{'} (x)) = \frac{\partial g^{gr} (x, α, α_{g})}{\partial x},$ (15) that is to say, holds for any fixed α, α_g ∈ [0, 1]. Especially we have $\begin{matrix} \frac{\partial g^{gr} (x, α, 0)}{\partial x} = (g_{α}^{-})^{'} (x), \\ \frac{\partial g^{gr} (x, α, 1)}{\partial x} = (g_{α}^{+})^{'} (x), \end{matrix}$ which implies that $g_{α}^{-} (x)$ and $g_{α}^{+} (x)$ are differentiable real-valued functions with respect to x. Since (h’) by (2.11) and the the monotonicity of $(g_{β}^{-})^{'} (x) + [(g_{β}^{+})^{'} (x) - (g_{β}^{-})^{'} (x)] α_{g}$ we get that $\begin{matrix} [g_{gr}^{'} (x)]_{α} = H^{- 1} (\frac{\partial g^{gr} (x, α, α_{g})}{\partial x}) \\ = [inf_{β ⩾ α} min_{α_{g}} \frac{\partial g^{gr} (x, β, α_{g})}{\partial x}, \\ sup_{β ⩾ α} max_{α_{g}} \frac{\partial g^{gr} (x, β, α_{g})}{\partial x}] \\ = [inf_{β ⩾ α} min_{α_{g}} {(g_{β}^{-})^{'} (x) + [(g_{β}^{+})^{'} (x) - (g_{β}^{-})^{'} (x)] α_{g}}, \\ sup_{β ⩾ α} max_{α_{g}} {(g_{β}^{-})^{'} (x) + [(g_{β}^{+})^{'} (x) - (g_{β}^{-})^{'} (x)] α_{g}}], \\ = [inf_{β \geq α} min {(g_{β}^{-})^{'} (x), (g_{β}^{+})^{'} (x)}, \\ sup_{β \geq α} max {(g_{β}^{-})^{'} (x), (g_{β}^{+})^{'} (x)}] . \end{matrix}$

Sufficiency. If $g_{α}^{-} (x)$ and $g_{α}^{+} (x)$ are differentiable real-valued functions with respect to x, since $g^{gr} (x, α, α_{g}) = g_{α}^{-} (x) + (g_{α}^{+} (x) - g_{α}^{-} (x)) α_{g}$ we have g^gr (x, α, α_g) is differentiable with respect to x. Moreover, $\begin{matrix} H (g_{gr}^{'} (x)) = \frac{\partial g^{gr} (x, α, α_{g})}{\partial x} \\ = \frac{\partial (g_{α}^{-} (x) + (g_{α}^{+} (x) - g_{α}^{-} (x)) α_{g})}{\partial x}, \\ = (g_{α}^{-})^{'} (x) (1 - α_{g}) + (g_{α}^{+})^{'} (x) α_{g} . \end{matrix}$ According to theorem we have that g^gr (x, α, α_g) is differentiable with respect to x, which implies that g is gr-differentiable.

It is easy to get that if g is gr-differentiable then it is g-differentiable as well and $g_{gr}^{'} = g_{g}^{'}$ according to Theorem □

Next, we illustrate the theorem with an example.

Example 3.2. Define $f (u) : [- 2, 2] \to F_{1}$ as follows, $\begin{matrix} f (u) = (\frac{u^{3}}{3}, \frac{u^{3}}{3} + u + 3, \frac{2 u^{3}}{3} + 4) . \end{matrix}$ The β-level sets of f (u) are $\begin{matrix} [f (u)]_{β} = [f_{β}^{-} (u), f_{β}^{+} (u)] \\ = [\frac{u^{3}}{3} + (u + 3) β, \frac{2 u^{3}}{3} + 4 - (\frac{u^{3}}{3} - u + 1) β] . \end{matrix}$ We can see that $f_{β}^{-} (u)$ and $f_{β}^{+} (u)$ are differentiable with respect to u. Then based on (3.6) we have $\begin{matrix} (f_{β}^{-})^{'} (u) = u^{2} + β, \end{matrix}$ $\begin{matrix} (f_{β}^{+})^{'} (u) = 2 u^{2} - u^{2} β + β = 2 u^{2} + β (1 - u^{2}), \end{matrix}$ where β ∈ [α, 1]. Thus we obtain, ${[f_{g r}^{'}]}_{α} = {\begin{matrix} [u^{2} + α, u^{2} + 1] i f | u | \leq 1, \\ [u^{2} + α, 2 u^{2} - u^{2} α + α], i f | u | > 1. \end{matrix}$ According to Theorem 2, we have the g-derivative as follows, $\begin{matrix} {[f_{g r}^{'}]}_{α} = [\inf_{β \geq α} (u^{2} + α), \sup_{β \geq α} (2 u^{2} - u^{2} α + α)] \\ = {\begin{matrix} [u^{2} + α, u^{2} + 1], i f | u | \leq 1, \\ [u^{2}_α, 2 u^{2} - u^{2} α + α], i f | u | > 1. \end{matrix} \end{matrix}$ It can be easily seen that $f_{gr}^{'} = f_{g}^{'}$ .

In the end of this subsection, we will give the relationship between gr-differentiability and gH-differentiability. Using the function in Example , we have the α-level sets of gH-derivative as follows, $\begin{matrix} [f_{gH}^{'}]_{α} = lim_{h \to 0} \frac{1}{h} \\ [min {f_{α}^{+} (u + h) - f_{α}^{+} (u), f_{α}^{-} (u + h) - f_{α}^{-} (u)}, \\ max {f_{α}^{+} (u + h) - f_{α}^{+} (u), f_{α}^{-} (u + h) - f_{α}^{-} (u)}] \\ = [u^{2} + α, 2 u^{2} - u^{2} α + α] . \end{matrix}$

It can be seen that if ∣u ∣ <1, then the resulting intervals $[f_{gH}^{'}]_{α} = [u^{2} + α, 2 u^{2} - u^{2} α + α]$ are not the α-level sets of a fuzzy number. Thus this function is not gH-differentiable in ∣u ∣ <1. It is possible to draw the wrong conclusion that gr-differentiable is more general than gH-differentiable. In fact, based on Theorem 2.1, differentiability of the endpoint functions is only the sufficient condition of gH-differentiable. If the endpoint functions are not differentiable but the left or right derivative of endpoint functions exists and satisfying certain conditions, the gH-derivative still exists. For more details, please refer to [9]. According to Theorem 3.3 we can find the fuzzy-number-valued function is not gr-differentiable if the endpoint functions are not differentiable, see the next Example 3.3.

Example 3.3. Define the fuzzy-number-valued function f (x) = (-1, 0, 1) |x|. The β-level sets are $[f]_{β} = [f_{β}^{-}, f_{β}^{+}] = [- | x | + | x | β, | x | - | x | β]$ . Since $f_{β}^{-}$ and $f_{β}^{+}$ are not differentiable at 0, we can get that this function is not gr-differentiable. But after a simple calculation we see that f is gH-differentiable and $f_{gH}^{'} ∣_{x = 0} = (- 1, 0, 1)$ .

We can find from Example 3.2 (gr-differentiable but not gH-differentiable) and Example 3.3 (gH-differentiable but not gr-differentiable) that these two derivatives are incomparable.

Just for the sake of convenience, the case (i) in Theorem 2.1 is denoted by gH_i-differentiable. Then we can get the following corollary.

Corollary 3.1. Let $f : [a, b] \to F_{1}$ be a fuzzy-number-valued function such that $[f (x)]_{α} = [f_{α}^{-} (x), f_{α}^{+} (x)]$ . If f is gH_i-differentiable and gr-differentiable at the same time then we have $\begin{matrix} [f_{{gH}_{i}}^{'} (x_{0})]_{α} = [min {(f_{α}^{+})^{'} (x_{0}), (f_{α}^{-})^{'} (x_{0})}, \\ max {(f_{α}^{+})^{'} (x_{0}), (f_{α}^{-})^{'} (x_{0})}], \end{matrix}$ (16) and $f_{{gH}_{i}}^{'} = f_{gr}^{'}$ .

Proof. Since f is gH_i-differentiable and gr-differentiable at the same time we have $(f_{α}^{-})^{'} (x_{0}) = (f_{α}^{-})_{+}^{'} (x_{0}) = (f_{α}^{-})_{-}^{'} (x_{0})$ and $(f_{α}^{+})^{'} (x_{0}) = (f_{α}^{+})_{+}^{'} (x_{0}) = (f_{α}^{+})_{-}^{'} (x_{0})$ .

Moreover, by Theorem 2.1, we have $\begin{matrix} [f_{{gH}_{i}}^{'} (x_{0})]_{α} \\ = [min {(f_{α}^{-})_{+}^{'} (x_{0}), (f_{α}^{+})_{+}^{'} (x_{0})}, \\ max {(f_{α}^{-})_{+}^{'} (x_{0}), (f_{α}^{+})_{+}^{'} (x_{0})}] \\ = [min {(f_{α}^{-})^{'} (x_{0}), (f_{α}^{+})^{'} (x_{0})}, \\ max {(f_{α}^{-})^{'} (x_{0}), (f_{α}^{+})^{'} (x_{0})}] . \end{matrix}$ Moreover, if f is gH_i-differentiable, then f is g-differentiable and $f_{{gH}_{i}}^{'} = f_{g}^{'}$ [1]. According to Theorem 3.3 we have $f_{gr}^{'} = f_{g}^{'}$ , i.e., $f_{{gH}_{i}}^{'} = f_{gr}^{'}$ . □

The following example will illustrate this corollary.

Example 3.4. Change the domain of the function f (u) in Example 3.4 as $f (u) : [- 1, 1] \to F_{1}$ . We get f is gH_i-differentiable and gr-differentiable with $\begin{matrix} [f_{gH}^{'}]_{α} = [f_{gr}^{'}]_{α} = [u^{2} + α, 2 u^{2} - u^{2} α + α] \\ = [min {(f_{α}^{+})^{'} (u), (f_{α}^{-})^{'} (u)}, \\ max {(f_{α}^{+})^{'} (u), (f_{α}^{-})^{'} (u)}] . \end{matrix}$

If a function is gH-differentiable (or gr-differentiable) at a point, then it may not be gr-differentiable (or gH-differentiable) at that point. However, if a function is g-differentiable at a point, then it may not be gH-differentiable or gr-differentiable at that point. The following example will illustrate this conclusion.

Example 3.5. Consider the fuzzy function $f : [- 1, 1] \to F_{1}$ , $\begin{matrix} f (t) = {\begin{matrix} (1 - t^{2}) [- 1, 1] + | t |, & if \frac{1}{2} < α \leq 1, \\ (1 + t^{2}) [- 1, 1] + | t |, & if 0 \leq α \leq \frac{1}{2}, \end{matrix} \end{matrix}$ in other words, $\begin{matrix} f^{-} (t) = {\begin{matrix} t^{2} - 1 + | t |, & if \frac{1}{2} < α \leq 1, \\ - 1 - t^{2} + | t |, & if 0 \leq α \leq \frac{1}{2}, \end{matrix} \end{matrix}$ and $\begin{matrix} f^{+} (t) = {\begin{matrix} 1 - t^{2} + | t |, & if \frac{1}{2} < α \leq 1, \\ 1 + t^{2} + | t |, & if 0 \leq α \leq \frac{1}{2} . \end{matrix} \end{matrix}$

It is easy to check that gH-difference does not exist at t = 0, thus f is not gH-differentiable at t = 0. Indeed, for $\frac{1}{2} < α \leq 1$ , we have diam (f (t)) =2 - 2t² < diam (f (0)) =2, for $0 \leq α \leq \frac{1}{2}$ , we have diam (f (t)) =2 + 2t² > diam (f (0)) =2. That is to say, there does not exist a fuzzy number ρ such that f (t) = ρ + f (0) or f (0) = f (t) ⊖ (- ρ). According to Theorem 3.3, f is not gr-differentiable at t = 0 since f⁺ (t) and f^- (t) are not differentiable at t = 0. In the following, we show that f is g-differentiable at t = 0. According to (2.6), we have $\begin{matrix} f_{g}^{'} (0) & = lim_{h \to 0} \frac{f (0 + h) ⊖_{g} f (0)}{h} \\ = lim_{h \to 0} \frac{1}{h} [inf_{β ⩾ α} min (| h | - h^{2}, | h | + h^{2}), \\ sup_{β ⩾ α} max (| h | - h^{2}, | h | + h^{2})] \\ = [0, 0] . \end{matrix}$

Remark 3.2. The relations among gH-differentiability, g-differentiability and gr-differentiability have been shown in Table 3 and Fig. 3.

Table 3

The relations between $f_{gH}^{'}, f_{g}^{'}$ and $f_{gr}^{'}$

	gH-derivative	g-derivative	gr-derivative
gH-derivative exists	∖	g-derivative exists and $f_{g}^{'} = f_{gH}^{'}$	gr-derivative does not necessarily exist
		(see [1])	(Example 3.3)
g-derivative exists	gH-derivative does not necessarily exist	∖	gr-derivative does not necessarily exist
	(see[1])		(Example 3.3)
gr-derivative exists	gH-derivative does not necessarily exist	g-derivative exists and $f_{g}^{'} = f_{gr}^{'}$	∖
	(Example 3.2)	(Theorem 3.2)

Fig. 3

In terms of existence. The relations among gH-differentiability, g-differentiability and gr-differentiability.

4 Conclusions

By studying the relations among three kinds of fuzzy differences we find that the gr-difference exists between any two fuzzy numbers just like g-difference, but the result of g-difference is contained in that of gr-difference. See Table 1. Specially, we also find some special propositions about the gH-difference, g-difference and gr-difference between two fuzzy numbers in Table 2. By studying these three kinds of derivative relations, we get that the gr-derivative is not more general than g-derivative and it is incomparable between gr-derivative and gH-derivative. Mazandarani et al. [13] also discussed the relations among some derivatives and get that the gr-derivative is more reasonable. The relations of these three derivatives is pointed out in this paper more specifically.

Table 1

The relations between z^gH, z^g and z^gr

	gH-difference	g-difference	gr-difference
gH-difference exists	∖	z^g = z^gH (see [1])	z^gH = z^g ⊂ z^gr (Theorem contain)
g-difference exists	gH-difference does not necessarily exist	∖	gr-difference exists and z^g ⊂ z^gr
	(see [1])		(Theorem 3.1)
gr-difference exists	gH-difference does not necessarily exist	g-difference exists and z^g ⊂ z^gr	∖
	(see [1])	(Theorem 3.1)

It should be noted that since the practical problems are different, we can choose a reasonable derivative to deal with the specific issues. Therefore, in this sense, the three derivatives have their respective applicable scope and are meaningful. In the further, we can use different approaches such as gH-derivative, g-derivative and gr-derivative dealing with fuzzy differential equations, such as fuzzy linear differential equations of motion for the Boeing 747 in longitudinal direction. Fuzzy dynamical systems control [16, 17] with the concept of gr-differentiability is also an interesting potential application.

Acknowledgements

The authors would like to thank the referee for his valuable comments. This work was supported by the National Natural Science Foundations of China (Grant no. 12171065, 11671001 and 61901074) and the Science and Technology Research Program of Chongqing Municipal Education Commission (Grant No. KJQN201900636).

Appendix A the detailed prove of Proposition 3.1

Let u_i = (a_i, b_i, c_i, d_i) and having β-level sets $[u_{i}]_{β} = [(u_{i}^{-})_{β}, (u_{i}^{+})_{β}] = [a_{i} + (b_{i} - a_{i}) β, d_{i} + (c_{i} - d_{i}) β]$ , where a_i ≤ b_i ≤ c_i ≤ d_i and i = 1, 2. The horizonal membership functions are $H (u_{i}) = a_{i} + (b_{i} - a_{i}) β + [(d_{i} - a_{i}) + (c_{i} - d_{i} - b_{i} + a_{i}) β] α_{u_{i}}$ . Then $H (w) = u_{1}^{gr} (β, α_{u_{1}}) - u_{2}^{gr} (β, α_{u_{2}})$ = a₁ - a₂ + (b₁ - a₁ - b₂ + a₂) β + [(d₁ - a₁) + (c₁ - d₁ - b₁ + a₁) β] α_{u
₁} - [(d₂ - a₂) + (c₂ - d₂ - b₂ + a₂) β] α_{u
₂}. By comparing the values when α_{u
₁}, α_{u
₂} are equal to 0 or 1, we can get the extreme values in (2.11) as follows: If α_{u
₁} = 0, α_{u
₂} = 0, then $\begin{matrix} w^{gr} (β, α_{w}) = (u_{1}^{-})_{β} - (u_{2}^{-})_{β} \\ = a_{1} - a_{2} + (b_{1} - a_{1} - b_{2} + a_{2}) β . \end{matrix}$ If α_{u
₁} = 0, α_{u
₂} = 1, then $\begin{matrix} w^{gr} (β, α_{w}) = (u_{1}^{-})_{β} - (u_{2}^{+})_{β} \\ = a_{1} - d_{2} + (b_{1} - a_{1} - c_{2} + d_{2}) β . \end{matrix}$ If α_{u
₁} = 1, α_{u
₂} = 0, then $\begin{matrix} w^{gr} (β, α_{w}) = (u_{1}^{+})_{β} - (u_{2}^{-})_{β} \\ = d_{1} - a_{2} + (c_{1} - d_{1} - b_{2} + a_{2}) β . \end{matrix}$ If α_{u
₁} = 1, α_{u
₂} = 1, then $\begin{matrix} w^{gr} (β, α_{w}) = (u_{1}^{+})_{β} - (u_{2}^{+})_{β} \\ = d_{1} - d_{2} + (c_{1} - d_{1} - c_{2} + d_{2}) β . \end{matrix}$ According to Theorem 3.1 and b₁ - a₁ - c₂ + d₂ > 0, c₁ - d₁ - b₂ + a₂ < 0, we obtain that the endpoints of α-level set of gr-difference is $w_{α}^{gr -} = a_{1} - d_{2} + (b_{1} - a_{1} - c_{2} + d_{2}) α$ and $w_{α}^{gr +} = d_{1} - a_{2} + (c_{1} - d_{1} - b_{2} + a_{2}) α$ . In other words, $\begin{matrix} [w^{gr}]_{α} = [u_{1} \underset{gr}{⊖} u_{2}]_{α} \\ = [a_{1} - d_{2} + (b_{1} - a_{1} - c_{2} + d_{2}) α, \\ d_{1} - a_{2} + (c_{1} - d_{1} - b_{2} + a_{2}) α] . \end{matrix}$ Hence the gr-difference between two trapezoidal fuzzy numbers is w^gr = (a₁ - d₂, b₁ - c₂, c₁ - d₂, d₁ - a₂).

We have already known that $u_{1} \underset{g}{⊖} u_{2}$ always exists, moreover, if $u_{1} \underset{gH}{⊖} u_{2}$ exists then $u_{1} \underset{g}{⊖} u_{2} = u_{1} \underset{gH}{⊖} u_{2}$ . Next, we divide it into two cases: Case 1. If $u_{1} \underset{gH}{⊖} u_{2}$ exists, then we have $[u_{1} ⊖_{g} H u_{2}] β = [\min {u_{β}^{+} - v_{β}^{+}, u_{β}^{-} - v_{β}^{-}}, \max {u_{β}^{+} - v_{β}^{+}, u_{β}^{-} - v_{β}^{-}}] .$

If $[u_{1} \underset{gH}{⊖} u_{2}]_{β} = [u_{β}^{-} - v_{β}^{-}, u_{β}^{+} - v_{β}^{+}]$ which means that $L ([u_{1}]_{β}) \geq L ([u_{2}]_{β})$ , then we have $w_{β}^{gH -} - w_{β}^{gr -} = d_{2} - a_{2} + (a_{2} - b_{2} + c_{2} - d_{2}) β$ , $w_{β}^{gr +} - w_{β}^{gH +} = d_{2} - a_{2} + (a_{2} - b_{2} + c_{2} - d_{2}) β$ .

It is obvious that $w_{β}^{gH -} - w_{β}^{gr -} = w_{β}^{gr +} - w_{β}^{gH +} = L ([u_{2}]_{β})$ , the gH-difference between two trapezoidal fuzzy numbers is w^gH = (a₁ - a₂, b₁ - b₂, c₁ - c₂, d₁ - d₂) in this case.

If $[u_{1} \underset{gH}{⊖} u_{2}]_{β} = [u_{β}^{+} - v_{β}^{+}, u_{β}^{-} - v_{β}^{-}]$ which means that $L ([u_{1}]_{β}) < L ([u_{2}]_{β})$ , we have $w_{β}^{gH -} - w_{β}^{gr -} = d_{1} - a_{1} + (a_{1} - b_{1} + c_{1} - d_{1}) β$ , $w_{β}^{gr +} - w_{β}^{gH +} = d_{2} - a_{2} + (a_{1} - b_{1} + c_{1} - d_{1}) β$ .

We can see $w_{β}^{gH -} - w_{β}^{gr -} = w_{β}^{gr +} - w_{β}^{gH +} = L ([u_{1}]_{β})$ , the gH-difference between two trapezoidal fuzzy numbers is w^gH = (d₁ - d₂, c₁ - c₂, b₁ - b₂, a₁ - a₂) in this case.

Case 2. If the gH-difference between u₁ and u₂ do not exists, then considering their g-difference, the α-level sets are as follows, $\begin{matrix} [w^{g}]_{α} = [u_{1} \underset{g}{⊖} u_{2}]_{α} \\ = [inf_{β ⩾ α} min {u_{β}^{+} - v_{β}^{+}, u_{β}^{-} - v_{β}^{-}}, \\ sup_{β ⩾ α} max {u_{β}^{+} - v_{β}^{+}, u_{β}^{-} - v_{β}^{-}}] . \end{matrix}$ If $u_{β}^{-} - v_{β}^{-} < u_{β}^{+} - v_{β}^{+}$ , we have $\begin{matrix} [u_{1} \underset{g}{⊖} u_{2}]_{α} = [inf_{β ⩾ α} u_{β}^{-} - v_{β}^{-}, sup_{β ⩾ α} u_{β}^{+} - v_{β}^{+}] \\ = [inf_{β ⩾ α} a_{1} - a_{2} + (b_{1} - a_{1} - b_{2} + a_{2}) β, \\ sup_{β ⩾ α} d_{1} - d_{2} + (c_{1} - d_{1} - c_{2} + d_{2}) β] . \end{matrix}$ Next we show the monotonicity of $u_{β}^{-} - v_{β}^{-}$ and $u_{β}^{+} - v_{β}^{+}$ , i) If b₁ - a₁ - b₂ + a₂ < 0 then $u_{β}^{-} - v_{β}^{-}$ is monotonically decreasing with the respect to β, w^g- = b₁ - b₂ when β = 1, ii) If b₁ - a₁ - b₂ + a₂ > 0 then $u_{β}^{-} - v_{β}^{-}$ is monotonically increasing with the respect to β, w^g- = a₁ - a₂ + (b₁ - a₁ - b₂ + a₂) α when β = α, iii) If b₁ - a₁ - b₂ + a₂ = 0 then w^g- = a₁ - a₂, iv) If c₁ - d₁ - c₂ + d₂ < 0 then $u_{β}^{+} - v_{β}^{+}$ is monotonically decreasing with the respect to β, w^g+ = d₁ - d₂ + (c₁ - d₁ - c₂ + d₂) α when β = α, v) If c₁ - d₁ - c₂ + d₂ > 0 then $u_{β}^{+} - v_{β}^{+}$ is monotonically increasing with the respect to β, w^g+ = c₁ - c₂ when β = 1, vi) If c₁ - d₁ - c₂ + d₂ = 0 then w^g+ = d₁ - d₂. Hence the α-level sets of g-difference can be divided into the following nine cases.

I) : [w^g] _α = [b₁ - b₂, d₁ - d₂ + (c₁ - d₁ - c₂ + d₂) α], then w^g = (b₁ - b₂, b₁ - b₂, c₁ - c₂, d₁ - d₂), the difference between the endpoints of α-level sets of g-difference and gr-difference are

$w_{α}^{g -} - w_{α}^{gr -} = b_{1} - a_{1} + d_{2} - b_{2} - (b_{1} - a_{1} - c_{2} + d_{2}) α$ ,

$w_{α}^{gr +} - w_{α}^{g +} = d_{2} - a_{2} + (a_{2} - b_{2} + c_{2} - d_{2}) α$ . $w_{α}^{g -} - w_{α}^{gr -} = w_{α}^{gr +} - w_{α}^{g +}$ holds if and only if b₁ - a₁ = b₂ - a₂. Moreover, $w_{α}^{g -} - w_{α}^{gr -} = w_{α}^{gr +} - w_{α}^{g +} = 0$ if and only if u₂ is a real number and a₁ = b₁.

II) : [w^g] _α = [b₁ - b₂, c₁ - c₂], then w^g = (b₁ - b₂, b₁ - b₂, c₁ - c₂, c₁ - c₂), and $w_{α}^{g -} - w_{α}^{gr -} = b_{1} - a_{1} + d_{2} - b_{2} - (b_{1} - a_{1} - c_{2} + d_{2}) α$ ,

$w_{α}^{gr +} - w_{α}^{g +} = d_{1} - c_{1} + c_{2} - a_{2} + (a_{2} - b_{2} + c_{1} - d_{1}) α$ . $w_{α}^{g -} - w_{α}^{gr -} = w_{α}^{gr +} - w_{α}^{g +}$ holds if and only if b₁ - a₁ + c₁ - d₁ = b₂ - a₂ + c₂ - d₂. Moreover, $w_{α}^{g -} - w_{α}^{gr -} = w_{α}^{gr +} - w_{α}^{g +} = 0$ if and only if u₂ and u₂ are both real numbers.

III) : [w^g] _α = [b₁ - b₂, d₁ - d₂], then w^g = (b₁ - b₂, b₁ - b₂, c₁ - c₂ (= d₁ - d₂) , d₁ - d₂), and $w_{α}^{g -} - w_{α}^{gr -} = b_{1} - a_{1} + d_{2} - b_{2} - (b_{1} - a_{1} - c_{2} + d_{2}) α$ ,

$w_{α}^{gr +} - w_{α}^{g +} = d_{2} - a_{2} + (a_{2} - b_{2} + c_{1} - d_{1}) α$ . $w_{α}^{g -} - w_{α}^{gr -} = w_{α}^{gr +} - w_{α}^{g +}$ holds if and only if b₁ - a₁ = b₂ - a₂ and c₁ - d₁ = c₂ - d₂. Moreover, $w_{α}^{g -} - w_{α}^{gr -} = w_{α}^{gr +} - w_{α}^{g +} = 0$ if and only if u₂ and u₂ are both real numbers.

IV) : [w^g] _α = [a₁ - a₂ + (b₁ - a₁ - b₂ + a₂) α, d₁ - d₂ + (c₁ - d₁ - c₂ + d₂) α], then w^g = (a₁ - a₂, b₁ - b₂, c₁ - c₂, d₁ - d₂), and $w_{α}^{g -} - w_{α}^{gr -} = d_{2} - a_{2} + (a_{2} - b_{2} + c_{2} - d_{2}) α$ ,

$w_{α}^{gr +} - w_{α}^{g +} = d_{2} - a_{2} + (a_{2} - b_{2} + c_{2} - d_{2}) α$ . Then $w_{α}^{g -} - w_{α}^{gr -} = w_{α}^{gr +} - w_{α}^{g +}$ holds unconditionally. Moreover, $w_{α}^{g -} - w_{α}^{gr -} = w_{α}^{gr +} - w_{α}^{g +} = 0$ if and only if u₂ is a real number.

V) : [w^g] _α = [a₁ - a₂ + (b₁ - a₁ - b₂ + a₂) α, c₁ - c₂], then w^g = (a₁ - a₂, b₁ - b₂, c₁ - c₂, c₁ - c₂), and $w_{α}^{g -} - w_{α}^{gr -} = d_{2} - a_{2} + (a_{2} - b_{2} + c_{2} - d_{2}) α$ ,

$w_{α}^{gr +} - w_{α}^{g +} = d_{1} - c_{1} + c_{2} - a_{2} + (a_{2} - b_{2} + c_{1} - d_{1}) α$ . $w_{α}^{g -} - w_{α}^{gr -} = w_{α}^{gr +} - w_{α}^{g +}$ holds if and only if c₁ - d₁ = c₂ - d₂. Moreover, $w_{α}^{g -} - w_{α}^{gr -} = w_{α}^{gr +} - w_{α}^{g +} = 0$ if and only if u₂ is a real number and c₁ = d₁.

VI) : [w^g] _α = [a₁ - a₂ + (b₁ - a₁ - b₂ + a₂) α, d₁ - d₂], then w^g = (a₁ - a₂, b₁ - b₂, c₁ - c₂ (= d₁ - d₂) , d₁ - d₂), and $w_{α}^{g -} - w_{α}^{gr -} = d_{2} - a_{2} + (a_{2} - b_{2} + c_{2} - d_{2}) α$ ,

$w_{α}^{gr +} - w_{α}^{g +} = d_{2} - a_{2} + (a_{2} - b_{2} + c_{1} - d_{1}) α$ . $w_{α}^{g -} - w_{α}^{gr -} = w_{α}^{gr +} - w_{α}^{g +}$ holds if and only if c₁ - d₁ = c₂ - d₂. Moreover, $w_{α}^{g -} - w_{α}^{gr -} = w_{α}^{gr +} - w_{α}^{g +} = 0$ if and only if u₂ is a real number and c₁ = d₁.

VII) : [w^g] _α = [a₁ - a₂, d₁ - d₂ + (c₁ - d₁ - c₂ + d₂) α], then w^g = (a₁ - a₂, b₁ - b₂ (= a₁ - a₂) , c₁ - c₂, d₁ - d₂), and $w_{α}^{g -} - w_{α}^{gr -} = d_{2} - a_{2} - (b_{1} - a_{1} - c_{2} + d_{2}) α$ ,

$w_{α}^{gr +} - w_{α}^{g +} = d_{2} - a_{2} - (b_{2} - a_{2} - c_{2} + d_{2}) α$ . $w_{α}^{g -} - w_{α}^{gr -} = w_{α}^{gr +} - w_{α}^{g +}$ holds if and only if b₁ - a₁ = b₂ - a₂. Moreover, $w_{α}^{g -} - w_{α}^{gr -} = w_{α}^{gr +} - w_{α}^{g +} = 0$ if and only if u₂ is a real number and a₁ = b₁.

VIII) : [w^g] _α = [a₁ - a₂, c₁ - c₂], then w^g = (a₁ - a₂, b₁ - b₂ (= a₁ - a₂) , c₁ - c₂, c₁ - c₂), and $w_{α}^{g -} - w_{α}^{gr -} = d_{2} - a_{2} - (b_{1} - a_{1} - c_{2} + d_{2}) α$ ,

$w_{α}^{gr +} - w_{α}^{g +} = d_{1} - a_{2} - c_{1} + c_{2} + (a_{2} - b_{2} + c_{1} - d_{1}) α$ . $w_{α}^{g -} - w_{α}^{gr -} = w_{α}^{gr +} - w_{α}^{g +}$ holds if and only if c₁ - d₁ = c₂ - d₂ and b₁ - a₁ = b₂ - a₂. Moreover, $w_{α}^{g -} - w_{α}^{gr -} = w_{α}^{gr +} - w_{α}^{g +} = 0$ if and only if u₂ and u₂ are both real numbers.

IX) : [w^g] _α = [a₁ - a₂, d₁ - d₂], then w^g = (a₁ - a₂, b₁ - b₂ (= a₁ - a₂) , c₁ - c₂ (= d₁ - d₂) , d₁ - d₂), and $w_{α}^{g -} - w_{α}^{gr -} = d_{2} - a_{2} - (b_{1} - a_{1} - c_{2} + d_{2}) α$ ,

$w_{α}^{gr +} - w_{α}^{g +} = d_{2} - a_{2} - (b_{2} - a_{2} - c_{1} + d_{1}) α$ . $w_{α}^{g -} - w_{α}^{gr -} = w_{α}^{gr +} - w_{α}^{g +}$ holds if and only if c₁ - d₁ = c₂ - d₂. Moreover, $w_{α}^{g -} - w_{α}^{gr -} = w_{α}^{gr +} - w_{α}^{g +} = 0$ if and only if u₂ and u₂ are both real numbers. We can use the same method to the case when $u_{β}^{-} - v_{β}^{-} > u_{β}^{+} - v_{β}^{+}$ .

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