Dynamic ranking of multi-dimensional uncertain information based on fuzzy order on the fuzzy ellipsoid number space

Abstract

In this paper, the problem of ranking multi-dimensional uncertain information based on fuzzy order is studied. At first, a specific fuzzy order on the fuzzy ellipsoid number space is defined, with which we can set up a dynamic ranking method which changes with the change of the level r for some objects which can be characterized by some multichannel uncertain digital information. Then its properties are investigated, some related results are obtained. And then, we give a practical example to show how to set up a dynamic ranking method which changes with the change of the level r for some objects which can be characterized by some multichannel uncertain digital information by using the fuzzy order proposed by us in the paper.

Keywords

Fuzzy number fuzzy ellipsoid numbers order fuzzy order ranking multi-dimensional uncertain information

1. Introduction

It is well known that the information processing is becoming more and more important with the development of science and society, many experts and scholars have been studying this field.

In a precise or certain environment, single or multi channel digital signals can be respectively represented by elements of single or multi dimensional Euclidean space, i.e., real numbers or multi dimensional real number vectors. If however we wish to study single or multi channel digital signals in an imprecise or uncertain environment, then the signals themselves are imprecise or uncertain, and it becomes unwise to use crisp real numbers or multi dimensional real number vectors to represent them. We know that 1 or n dimensional fuzzy numbers are good mean to express respectively imprecise or uncertain single or multi channel digital information.

The concept of fuzzy numbers was introduced by Chang and Zadeh [4] in 1972 with the consideration of the properties of probability functions. Since then both the numbers and the problems in relation to them have been widely studied, see for example, [2 , 15] and the references therein. With the development of theories and applications of fuzzy numbers, this concept becomes more and more important [3 , 11]. Recently, there are still a lot of work in this area or related to this area. For example, in [17], Moreno-Garcia, Linares, Rodriguez-Benitez and Castillo proposed a method to construct trapezoidal fuzzy number approximations from raw discrete data; In [14], Hong considered The law of large numbers and renewal process for T-related weighted fuzzy numbers on R^p; In [1], Arotaritei and Ionescu introduced ą $\overset{ˇ}{r}$ fuzzy Voronoiąś diagrams for fuzzy numbers of dimension two by extension of Voronoi diagrams for fuzzy numbers; In [5], Coroianu, Gagolewski and Grzegorzewski studied the problem of the nearest approximation of fuzzy numbers by piecewise linear 1-knot fuzzy numbers; In [25], Yazdi, GhasemiGol and Effati et al presented a new hierarchical tree approach to clustering fuzzy numbers.

For general n dimensional fuzzy numbers, due to their structural complexity, they can not be used conveniently in some fields of applications and some researches of theory [19 , 24]. In [19 , 24], we proposed a special type of n dimensional fuzzy numbers which are called fuzzy n-cell numbers and can be used much more conveniently than general n dimensional fuzzy numbers in theoretical investigations and in applications, and recommended using fuzzy n-cell numbers to represent imprecise or uncertain multi-channel digital signals. However, fuzzy n-cell numbers have some defects in structure, it will bring about some bad effects in applications [23]. In order to overcome the weakness, and make fuzzy numbers can better be applied, in [23], a new special type of n dimensional fuzzy numbers whose r-level sets (for all r ∈ [0, 1]) are all n-ellipsoids are introduced and called fuzzy n-ellipsoid numbers. And for the sake of the application, and some methods of constructing such fuzzy ellipsoid numbers to represent imprecise or uncertain multi-channel digital information are proposed in [23].

On the ranking of fuzzy numbers, there are also many researchers who have been engaged in or are doing research in this area. For example, in [22], we set up some weak order on fuzzy cell number space, and use them to rank some objets which are characterized by some multichannel uncertain digital information; In [10], Eslamipoor, Haji and Sepehriar proposed a new revised method for ranking fuzzy numbers; In [6], Deng presented a new approach for comparing and ranking fuzzy numbers in a simple manner in decision making under uncertainty; In [13], Haji, Zare, Eslamipoor and Sepehriar presented a new method for ranking fuzzy numbers based on the left and right using distance method and level set; In [16], Kim, Moon, Jeong and Hong also did the research for ranking methods for fuzzy numbers; In [12], Gu and Xuan introduced a new method of ranking generalized L-R fuzzy numbers based on possibility theory and the implication of possibilistic mean and possibilistic standard deviation; In [18], we also set up a method of ranking multi-dimensional uncertain information fuzzy via setting up specific metrics on fuzzy ellipsoid number space.

In this paper, we try to set up a method of dynamic ranking multi-dimensional uncertain digital information via setting up specific fuzzy order on fuzzy ellipsoid number space. Specific arrangements are as follows: In Section 2, we briefly review some basic notions, definitions and results about fuzzy numbers. In Section 3, we define a specific fuzzy binary relation on the fuzzy ellipsoid number space, show that it has the attributes of order (so we call it a fuzzy order), and obtain some related results; In Section 4, we give a practical example to show how to set up a dynamic ranking method which changes with the change of the level r for some objects which can be characterized by some multichannel uncertain digital information by using the fuzzy order proposed by us in the paper, and a remark to illustrate the advantage of the dynamic ranking method.

2. Basic definition and notation

Let n be a natural number, R be the real number set, and Rⁿ be the n-dimensional Euclidean space. A fuzzy subset (for short, a fuzzy set) of Rⁿ is a function u : Rⁿ → [0, 1]. For each such fuzzy set u, we denote by [u] ^r = {x ∈ Rⁿ : u (x) ≥ r} for any r ∈ (0, 1], is r-level set. By suppu we denote the support of u, i.e., the set {x ∈ Rⁿ : u (x) >0}. By [u] ⁰ we denote the closure of the set suppu, i.e., $[u]^{0} = \bar{{x \in R^{n} : u (x) > 0}}$ .

If u is a normal and fuzzy convex fuzzy set of Rⁿ, u (x) is upper semi-continuous, and [u] ⁰ is compact, then u is called a n-dimensional fuzzy number, and the collection of all n-dimensional fuzzy numbers is denoted by Eⁿ.

It is known that if u ∈ Eⁿ, then for each r ∈ [0, 1], [u] ^r is a non-empty compact set of Rⁿ.

Let K (Rⁿ) be the collection of all non-empty compact subsets of Rⁿ. For any A, B ∈ K (Rⁿ), the Hausdorff metric of A and B defined as $d (A, B) = max {sup_{a \in A} inf_{b \in B} | | a - b | |, sup_{b \in B} inf_{a \in A} | | a - b | |}$ where || · || is the Euclidean norm.

Let 1≤ p < ∞. The mappings D and ρ_p : Eⁿ × Eⁿ → [0, + ∞) are defined as $\begin{matrix} D (u, v) = sup_{r \in [0, 1]} d ([u]^{r}, [v]^{r}) \\ ρ_{p} (u, v) = (\int_{0}^{1} [d ([u]^{r}, [v]^{r})]^{p} dr)^{\frac{1}{p}} \end{matrix}$ for any u, v ∈ Eⁿ, respectively. Then D and ρ_p are two complete metrics on Eⁿ [7].

If u ∈ Eⁿ, and for each r ∈ [0, 1], [u] ^r is a n-ellipsoid, i.e., exist $\underline{u_{i}} (r), \bar{u_{i}} (r) \in R$ with $\underline{u_{i}} (r) \leq \bar{u_{i}} (r)$ for any i = 1, 2, ⋯, n, such that $[u]^{r} = {(x_{1}, x_{2}, \dots, x_{n}) \in R^{n} | \sum_{i = 1}^{n} \frac{(x_{i} - \frac{\bar{u_{i}} (r) + \underline{u_{i}} (r)}{2})^{2}}{(\frac{\bar{u_{i}} (r) - \underline{u_{i}} (r)}{2})^{2}} \leq 1}$ (where, $\underline{u_{i}} (r) = \bar{u_{i}} (r)$ can be allowed for some i = 1, 2, ⋯, n, and r ∈ [0, 1], the meaning of $\sum_{i = 1}^{n} \frac{(x_{i} - \frac{\bar{u_{i}} (r) + \underline{u_{i}} (r)}{2})^{2}}{(\frac{\bar{u_{i}} (r) - \underline{u_{i}} (r)}{2})^{2}} \leq 1$ can be see Remark 1 in [23]), then we call u a fuzzy n-ellipsoid number [23]. And we denote the collection of all closed n-ellipsoids by E_C (Rⁿ), and denote the collection of all fuzzy n-ellipsoid numbers by E (Eⁿ).

In this paper, we still use the notation in [23] $\begin{matrix} (\begin{matrix} a_{1} & b_{1} \\ a_{2} & b_{2} \\ ⋮ & ⋮ \\ a_{n} & b_{n} \end{matrix}) (in short, {(a_{i}, b_{i})}_{i = 1}^{n}) \\ = {(x_{1}, x_{2}, \dots, x_{n}) \in R^{n} | \sum_{i = 1}^{n} \frac{(x_{i} - \frac{b_{i} + a_{i}}{2})^{2}}{(\frac{b_{i} - a_{i}}{2})^{2}} \leq 1} \end{matrix}$ So, for u ∈ E (Eⁿ), we can denote $[u]^{r} = (\underline{u_{i}} (r), \bar{u_{i}} (r))_{i = 1}^{n}$ or $[u]^{r} = (\begin{matrix} \underline{u_{1}} (r) & \bar{u_{1}} (r) \\ \underline{u_{2}} (r) & \bar{u_{2}} (r) \\ ⋮ & ⋮ \\ \underline{u_{n}} (r) & \bar{u_{n}} (r) \end{matrix})$ for any r ∈ [0, 1].

For u, v ∈ E (Eⁿ), we define u ≤ v or v ≥ u if and only if $\underline{u_{i}} (r) \leq \underline{v_{i}} (r)$ and $\bar{u_{i}} (r) \leq \bar{v_{i}} (r)$ for any r ∈ [0, 1] and i = 1, 2, ⋯, n.

For u ∈ E (Eⁿ) (with $[u]^{r} = (\underline{u_{i}} (r), \bar{u_{i}} (r))_{i = 1}^{n}$ ), if there exist a_i, b_i, c_i ∈ R with a_i ≤ b_i ≤ c_i (i = 1, 2, ⋯, n) such that $\begin{matrix} \underline{u_{i}} (r) = {\begin{matrix} a_{i} + (b_{i} - a_{i}) r & if & r \in [\frac{\underline{d_{i}} - a_{i}}{b_{i} - a_{i}}, 1] \\ \underline{d_{i}} & if & r \in [0, \frac{\underline{d_{i}} - a_{i}}{b_{i} - a_{i}}) \end{matrix} \\ \bar{u_{i}} (r) = {\begin{matrix} c_{i} - (c_{i} - b_{i}) r & if & r \in [\frac{c_{i} - \bar{d_{i}}}{c_{i} - b_{i}}, 1] \\ \bar{d_{i}} & if & r \in [0, \frac{c_{i} - \bar{d_{i}}}{c_{i} - b_{i}}) \end{matrix} \end{matrix}$ for r ∈ [0, 1] and i = 1, 2, ⋯, n, where $\underline{d_{i}} \in [a_{i}, b_{i}]$ , $\bar{d_{i}} \in [b_{i}, c_{i}]$ (i = 1, 2, ⋯, n), then we call u a cone-type fuzzy ellipsoid number (for short, CT-fuzzy ellipsoid number), and denote it as $\begin{matrix} CT (\begin{matrix} a_{1} |_{\underline{d_{1}}} & b_{1} & c_{1} |_{\bar{d_{1}}} \\ ⋮ & ⋮ & ⋮ \\ a_{n} |_{\underline{d_{n}}} & b_{n} & c_{n} |_{\bar{d_{n}}} \end{matrix}) \\ (for short CT (a_{i} |_{\underline{d_{i}}}, b_{i}, c_{i} |_{\bar{d_{i}}})_{i = 1}^{n}) \end{matrix}$

3. Fuzzy order

Let u ∈ E (Eⁿ) with $[u]^{r} = {(\underline{u_{i}} (r), \bar{u_{i}} (r))}_{i = 1}^{n}$ for any r ∈ [0, 1], p = (p₁, p₂, …, p_n) ∈ Rⁿ with p_i ≥ 0 (i = 1, 2, …, n) and $\sum_{i = 1}^{n} p_{i} = 1$ . We denote ${LM}_{p} (u) = \sum_{i = 1}^{n} p_{i} \int_{0}^{1} 2 r \underline{u_{i}} (r) dr$ ( $= 2 \sum_{i = 1}^{n} p_{i} \int_{0}^{1} r \underline{u_{i}} (r) dr$ ), $R M_{p} (u) = \sum_{i = 1}^{n} p_{i} \int_{0}^{1} 2 r \bar{u_{i}} (r) dr$ ( $= 2 \sum_{i = 1}^{n} p_{i} \int_{0}^{1} r \bar{u_{i}} (r) dr$ ), $M_{p} (u) = \frac{{LM}_{p} (u) + {RM}_{p} (u)}{2} = \sum_{i = 1}^{n} p_{i} \int_{0}^{1} r (\underline{u_{i}} (r) + \bar{u_{i}} (r)) dr$ , and call LM_p (u), RM_p (u) and M_p (u) left mean, right mean and mean of u (with respect to weight p), respectively.

Let u, v ∈ E (Eⁿ) with $[u]^{r} = (\underline{u_{i}} (r), \bar{u_{i}} (r))_{i = 1}^{n}$ and $[v]^{r} = (\underline{v_{i}} (r), \bar{v_{i}} (r))_{i = 1}^{n}$ for any r ∈ [0, 1], p = (p₁, p₂, …, p_n) ∈ Rⁿ with p_i ≥ 0 (i = 1, 2, …, n) and $\sum_{i = 1}^{n} p_{i} = 1$ . We denote LD_p (u, v) $= \sum_{i = 1}^{n} p_{i} \int_{0}^{1} 2 r | \underline{u_{i}} (r) - \underline{v_{i}} (r) | dr$ ( $= 2 \sum_{i = 1}^{n} p_{i} \int_{0}^{1} r | \underline{u_{i}} (r) - \underline{v_{i}} (r) | dr$ ), RD_p $(u, v) = \sum_{i = 1}^{n} p_{i} \int_{0}^{1} 2 r | \bar{u_{i}} (r) - \bar{v_{i}} (r) | dr$ ( $= 2 \sum_{i = 1}^{n} p_{i}$ $\int_{0}^{1} r | \bar{u_{i}} (r) - \bar{v_{i}} (r) | dr$ ), $D_{p} (u, v) = \frac{{LD}_{p} (u, v) + {RD}_{p} (u, v)}{2}$ $= \sum_{i = 1}^{n} p_{i} \int_{0}^{1} r (| \underline{u_{i}} (r) - \underline{v_{i}} (r) | + | \bar{u_{i}} (r) - \bar{v_{i}} (r) |) dr$ , and call LD_p (u, v), RD_p (u, v) and D_p (u, v) left absolute difference, right absolute difference and absolute difference of u (with respect to weight p), respectively.

Remark 1. As weight p = (p₁, p₂, …, p_n) satisfies p_i > 0 (i = 1, 2, …, n), the mapping D_p : E (Eⁿ) × E (Eⁿ) → R, (u, v) → D_p (u, v) for any (u, v) ∈ E (Eⁿ) × E (Eⁿ) satisfies

(1) D_p (u, v) ≥0 for any (u, v) ∈ E (Eⁿ) × E (Eⁿ);

(2) For (u, v) ∈ E (Eⁿ) × E (Eⁿ), D_p (u, v) =0 if and only if u = v;

(3) D_p (u, v) = D_p (v, u) for any (u, v) ∈ E (Eⁿ) × E (Eⁿ);

(4) D_p (u, v) ≤ D_p (u, w) + D_p (w, u) for any u, v and w ∈ E (Eⁿ), i.e., D_p is a metric on E (Eⁿ)

Proof. The proofs of (1) and (3) can be directly completed by the definition of D_p, we only need to show (2) and (4) of the remark.

By the definition of D_p and the left continuity (see Theorem 2 of [23]), we have

$\begin{matrix} D_{p} (u, v) = 0 \\ \Leftrightarrow \sum_{i = 1}^{n} p_{i} \int_{0}^{1} r (| \underline{u_{i}} (r) - \underline{u_{i}} (r) | + | \bar{u_{i}} (r) - \bar{u_{i}} (r) |) dr = 0 \\ \Leftrightarrow p_{i} \int_{0}^{1} r (| \underline{u_{i}} (r) - \underline{u_{i}} (r) | + | \bar{u_{i}} (r) - \bar{u_{i}} (r) |) = 0, \\ i = 1, 2, \dots, n \\ \Leftrightarrow \int_{0}^{1} r (| \underline{u_{i}} (r) - \underline{u_{i}} (r) | + | \bar{u_{i}} (r) - \bar{u_{i}} (r) |) = 0, \\ i = 1, 2, \dots, n \\ \Leftrightarrow r (| \underline{u_{i}} (r) - \underline{u_{i}} (r) | + | \bar{u_{i}} (r) - \bar{u_{i}} (r) |) = 0 \\ holds for r almost everywhere on [0, 1], \\ i = 1, 2, \dots, n \\ \Leftrightarrow | \underline{u_{i}} (r) - \underline{u_{i}} (r) | = 0 and | \bar{u_{i}} (r) - \bar{u_{i}} (r) | = 0 \\ holds for r almost everywhere on [0, 1], \\ i = 1, 2, \dots, n \\ \Leftrightarrow \underline{u_{i}} (r) = \underline{u_{i}} (r) and \bar{u_{i}} (r) = \bar{u_{i}} (r) \\ holds for r almost everywhere on [0, 1], \\ i = 1, 2, \dots, n \\ \Leftrightarrow \underline{u_{i}} (r) = \underline{u_{i}} (r) and \bar{u_{i}} (r) = \bar{u_{i}} (r) \\ for any r \in [0, 1], i = 1, 2, \dots, n \\ \Leftrightarrow u = v \end{matrix}$ so (2) is correct.

By the definition of D_p, we see $\begin{matrix} D_{p} (u, v) \\ = \sum_{i = 1}^{n} p_{i} \int_{0}^{1} r (| \underline{u_{i}} (r) - \underline{u_{i}} (r) | + | \bar{u_{i}} (r) - \bar{u_{i}} (r) |) dr \\ = \sum_{i = 1}^{n} p_{i} \int_{0}^{1} r (| [\underline{u_{i}} (r) - \underline{w_{i}} (r)] + [\underline{w_{i}} (r) - \underline{u_{i}} (r)] | \\ + | [\bar{u_{i}} (r) - \bar{w_{i}} (r)] + [\bar{w_{i}} (r) - \bar{u_{i}} (r)] |) dr \\ \leq \sum_{i = 1}^{n} p_{i} \int_{0}^{1} r (| \underline{u_{i}} (r) - \underline{w_{i}} (r) | + | \underline{w_{i}} (r) - \underline{u_{i}} (r) | \\ + | \bar{u_{i}} (r) - \bar{w_{i}} (r) | + | \bar{w_{i}} (r) - \bar{u_{i}} (r) |) dr \\ = \sum_{i = 1}^{n} p_{i} \int_{0}^{1} r (| \underline{u_{i}} (r) - \underline{w_{i}} (r) | + | \bar{u_{i}} (r) - \bar{w_{i}} (r) |) dr \\ + \sum_{i = 1}^{n} p_{i} \int_{0}^{1} r (| \underline{w_{i}} (r) - \underline{u_{i}} (r) | + | \bar{w_{i}} (r) - \bar{u_{i}} (r) |) dr \\ = D_{p} (u, w) + D_{p} (w, v) \end{matrix}$ so (4) also holds.

In the following, we set up a specific fuzzy binary relation (in fact, it is a fuzzy order relationship, see Remark 2) on E (Eⁿ).

Let p = (p₁, p₂, …, p_n) ∈ Rⁿ with p_i ≥ 0 (i = 1, 2, …, n) and $\sum_{i = 1}^{n} p_{i} = 1$ . For any u, v ∈ E (Eⁿ) with $[u]^{r} = {(\underline{u_{i}} (r), \bar{u_{i}} (r))}_{i = 1}^{n}$ and $[v]^{r} = {(\underline{v_{i}} (r), \bar{v_{i}} (r))}_{i = 1}^{n}$ , r ∈ [0, 1], from the definition of M_p, we see that $\begin{matrix} | M_{p} (u) - M_{p} (v) | \\ = | \sum_{i = 1}^{n} p_{i} \int_{0}^{1} r (\underline{u_{i}} (r) - \underline{v_{i}} (r) + \bar{u_{i}} (r) - \bar{v_{i}} (r)) dr | \\ \leq \sum_{i = 1}^{n} p_{i} | \int_{0}^{1} r (\underline{u_{i}} (r) - \underline{v_{i}} (r) + \bar{u_{i}} (r) - \bar{v_{i}} (r)) dr | \\ \leq \sum_{i = 1}^{n} p_{i} \int_{0}^{1} r | \underline{u_{i}} (r) - \underline{v_{i}} (r) + \bar{u_{i}} (r) - \bar{v_{i}} (r) | dr \\ \leq \sum_{i = 1}^{n} p_{i} \int_{0}^{1} r | \underline{u_{i}} (r) - \underline{v_{i}} (r) | + | \bar{u_{i}} (r) - \bar{v_{i}} (r) | dr \\ = D_{p} (u, v) \end{matrix}$ so $| \frac{M_{p} (u) - M_{p} (v)}{D_{p} (u, v)} | \leq 1$ , i.e., $- 1 \leq \frac{M_{p} (u) - M_{p} (v)}{D_{p} (u, v)} \leq 1$ .

From this, we have that $0 \leq \frac{1}{2} (1 + \frac{M_{p} (u) - M_{p} (v)}{D_{p} (u, v)}) \leq 1$

Therefore, we can give the following definition: □

Definition 1. Let p = (p₁, p₂, …, p_n) ∈ Rⁿ with p_i ≥ 0 and $\sum_{i = 1}^{n} p_{i} = 1$ . We define a fuzzy binary relation ${\tilde{≺}}_{p}$ on L (Eⁿ), i.e., a mapping ${\tilde{≺}}_{p} : L (E^{n}) \times L (E^{n}) \to [0, 1]$ as following: ${\tilde{≺}}_{p} (u, v) = {\begin{matrix} \frac{1}{2} (1 + \frac{M_{p} (v) - M_{p} (u)}{D_{p} (u, v)}), & u \neq v \\ \frac{1}{2}, & u = v \end{matrix}$ for any u, v ∈ E (Eⁿ) with $[u]^{r} = {(\underline{u_{i}} (r), \bar{u_{i}} (r))}_{i = 1}^{n}$ and $[v]^{r} = {(\underline{v_{i}} (r), \bar{v_{i}} (r))}_{i = 1}^{n}$ , r ∈ [0, 1].

In order to investigate the properties of fuzzy binary relation “ ${\tilde{≺}}_{p}$ ”, we first give the following lemma:

Lemma 1. If a, b, c and d ∈ R and c, d > 0, then $\frac{a + b}{c + d} \geq min {\frac{a}{c}, \frac{b}{d}}$

Proof. In the following, we prove the lemma in two cases: $\frac{a}{c} \geq \frac{b}{d}$ and $\frac{a}{c} < \frac{b}{d}$ .

(1) As $\frac{a}{c} \geq \frac{b}{d}$ , from c, d > 0, we see that $ad \geq bc$ It implies $ad + bd \geq bc + bd$ i.e., $(a + b) d \geq b (c + d)$ Therefore, we have that $\frac{a + b}{c + d} \geq \frac{b}{d} = min {\frac{a}{c}, \frac{b}{d}}$

(2) Likewise, as $\frac{a}{c} < \frac{b}{d}$ , we can obtain that $\frac{a + b}{c + d} \geq \frac{a}{c} = min {\frac{a}{c}, \frac{b}{d}}$ the proof of the lemma is completed. □

Theorem 1. Let p = (p₁, p₂, …, p_n) ∈ Rⁿ with p_i ≥ 0 and $\sum_{i = 1}^{n} p_{i} = 1$ . Then

(1) ${\tilde{≺}}_{p} (u, v) + {\tilde{≺}}_{p} (v, u) = 1$ for any u, v ∈ E (Eⁿ);

(2) ${\tilde{≺}}_{p} (u, w) \geq min {{\tilde{≺}}_{p} (u, v), {\tilde{≺}}_{p} (v, w)}$ (it is equivalent to that ${\tilde{≺}}_{p} (u, v) \geq r$ and ${\tilde{≺}}_{p} (v, w) \geq r$ $\Rightarrow {\tilde{≺}}_{p} (u, w) \geq r$ for any r ∈ [0, 1]) for any u, v, w ∈ E (Eⁿ);

(3) u ≤ v and u ≠ v ⇒ ${\tilde{≺}}_{p} (u, v) = 1$ (it is equivalent to that u ≤ v and u ≠ v ⇒ ${\tilde{≺}}_{p} (v, u) = 0$ ) for any u, v ∈ E (Eⁿ);

(4) ${\tilde{≺}}_{p} (u \oplus w, v \oplus w) = {\tilde{≺}}_{p} (u, v)$ for any u, v, w ∈ E (Eⁿ), where the addition operation “⊕” is defined by Definition 1 in [20];

(5) ${\tilde{≺}}_{p} (u, v) = \frac{1}{2}$ ⇔ M_p (u) = M_p (v) for any u, v ∈ E (Eⁿ).

Proof. The proof of (1):

If u = v, then, by the definition of ${\tilde{≺}}_{p}$ , we see that ${\tilde{≺}}_{p} (u, v) + {\tilde{≺}}_{p} (v, u) = \frac{1}{2} + \frac{1}{2} = 1$ .

If u ≠ v, then, by the definition of ${\tilde{≺}}_{p}$ and the (3) of Remark 1, we have that $\begin{matrix} {\tilde{≺}}_{p} (u, v) + {\tilde{≺}}_{p} (v, u) \\ = \frac{1}{2} (1 + \frac{M_{p} (v) - M_{p} (u)}{D_{p} (v, u)}) + \frac{1}{2} (1 + \frac{M_{p} (u) - M_{p} (v)}{D_{p} (u, v)}) \\ = \frac{1}{2} (1 + \frac{M_{p} (v) - M_{p} (u)}{D_{p} (u, v)}) + 1 + \frac{M_{p} (u) - M_{p} (v)}{D_{p} (u, v)} \\ = \frac{1}{2} (2 + \frac{M_{p} (v) - M_{p} (u) + M_{p} (u) - M_{p} (v)}{D_{p} (u, v)}) \\ = 1 \end{matrix}$ so Conclusion (1) holds.

The proof of (2):

It is obvious that ${\tilde{≺}}_{p} (u, w) \geq min {{\tilde{≺}}_{p} (u, v), {\tilde{≺}}_{p} (v, w)}$ is equivalent to that ${\tilde{≺}}_{p} (u, v) \geq r$ and ${\tilde{≺}}_{p} (v, w) \geq r$ $\Rightarrow {\tilde{≺}}_{p} (u, w) \geq r$ for any r ∈ [0, 1], so we only need to prove ${\tilde{≺}}_{p} (u, w) \geq min {{\tilde{≺}}_{p} (u, v), {\tilde{≺}}_{p} (v, w)}$ .

If u = w, then by the definition of ${\tilde{≺}}_{p}$ , we see that ${\tilde{≺}}_{p} (u, w) = \frac{1}{2}$ . On the other hand, by Conclusion (1), we have that ${\tilde{≺}}_{p} (u, v) + {\tilde{≺}}_{p} (v, w) = 1$ , so $min {{\tilde{≺}}_{p} (u, v), {\tilde{≺}}_{p} (v, w)} \leq \frac{{\tilde{≺}}_{p} (u, v) + {\tilde{≺}}_{p} (v, w)}{2} = \frac{1}{2}$ . Thus we obtain ${\tilde{≺}}_{p} (u, w) \geq min {{\tilde{≺}}_{p} (u, v), {\tilde{≺}}_{p} (v, w)}$ .

If u ≠ w and v = u, then ${\tilde{≺}}_{p} (u, w) = {\tilde{≺}}_{p} (v, w) \geq min {{\tilde{≺}}_{p} (u, v), {\tilde{≺}}_{p} (v, w)}$ .

If u ≠ w and v = w, then ${\tilde{≺}}_{p} (u, w) = {\tilde{≺}}_{p} (u, v) \geq min {{\tilde{≺}}_{p} (u, v), {\tilde{≺}}_{p} (v, w)}$ .

If u ≠ w, v ≠ u and v ≠ w, then, by the definitions of M_p and D_p and Lemma 1, we have that

$\begin{matrix} \frac{M_{p} (w) - M_{p} (u)}{D_{p} (u, w)} \\ = \frac{\sum_{i = 1}^{n} p_{i} \int_{0}^{1} r (\underline{w_{i}} (r) + \bar{w_{i}} (r)) dr - \sum_{i = 1}^{n} p_{i} \int_{0}^{1} r (\underline{u_{i}} (r) + \bar{u_{i}} (r)) dr}{\sum_{i = 1}^{n} p_{i} \int_{0}^{1} r (| \underline{u_{i}} (r) - \underline{w_{i}} (r) | + | \bar{u_{i}} (r) - \bar{w_{i}} (r) |) dr} \\ = \frac{\sum_{i = 1}^{n} p_{i} \int_{0}^{1} r (\underline{w_{i}} (r) - \underline{u_{i}} (r) + \bar{w_{i}} (r) - \bar{u_{i}} (r)) dr}{\sum_{i = 1}^{n} p_{i} \int_{0}^{1} r (| \underline{u_{i}} (r) - \underline{w_{i}} (r) | + | \bar{u_{i}} (r) - \bar{w_{i}} (r) |) dr} \\ = \frac{\sum_{i = 1}^{n} p_{i} \int_{0}^{1} r (\underline{w_{i}} (r) - \underline{v_{i}} (r) + \underline{v_{i}} (r) - \underline{u_{i}} (r) + \bar{w_{i}} (r) - \bar{v_{i}} (r) + \bar{v_{i}} (r) - \bar{u_{i}} (r)) dr}{\sum_{i = 1}^{n} p_{i} \int_{0}^{1} r (| \underline{u_{i}} (r) - \underline{v_{i}} (r) + \underline{v_{i}} (r) - \underline{w_{i}} (r) | + | \bar{u_{i}} (r) - \bar{v_{i}} (r) + \bar{v_{i}} (r) - \bar{w_{i}} (r) |) dr} \\ \geq \frac{\sum_{i = 1}^{n} p_{i} \int_{0}^{1} r (\underline{w_{i}} (r) - \underline{v_{i}} (r) + \bar{w_{i}} (r) - \bar{v_{i}} (r) + \underline{v_{i}} (r) - \underline{u_{i}} (r) + \bar{v_{i}} (r) - \bar{u_{i}} (r)) dr}{\sum_{i = 1}^{n} p_{i} \int_{0}^{1} r (| \underline{u_{i}} (r) - \underline{v_{i}} (r) | + | \bar{u_{i}} (r) - \bar{v_{i}} (r) | + | \underline{v_{i}} (r) - \underline{w_{i}} (r) | + | \bar{v_{i}} (r) - \bar{w_{i}} (r) |) dr} \\ \geq min {\frac{\sum_{i = 1}^{n} p_{i} \int_{0}^{1} r (\underline{w_{i}} (r) + \bar{w_{i}} (r) - \underline{v_{i}} (r) - \bar{v_{i}} (r)) dr}{\sum_{i = 1}^{n} p_{i} \int_{0}^{1} r (| \underline{v_{i}} (r) - \underline{w_{i}} (r) | + | \bar{v_{i}} (r) - \bar{w_{i}} (r) |) dr}, \\ \frac{\sum_{i = 1}^{n} p_{i} \int_{0}^{1} r (\underline{v_{i}} (r) + \bar{v_{i}} (r) - \underline{u_{i}} (r) - \bar{u_{i}} (r)) dr}{\sum_{i = 1}^{n} p_{i} \int_{0}^{1} r (| \underline{u_{i}} (r) - \underline{v_{i}} (r) | + | \bar{u_{i}} (r) - \bar{v_{i}} (r) |) dr}} \\ = min {\frac{\sum_{i = 1}^{n} p_{i} \int_{0}^{1} r (\underline{w_{i}} (r) + \bar{w_{i}} (r)) dr - \sum_{i = 1}^{n} p_{i} \int_{0}^{1} r (\underline{v_{i}} (r) + \bar{v_{i}} (r)) dr}{\sum_{i = 1}^{n} p_{i} \int_{0}^{1} r (| \underline{v_{i}} (r) - \underline{w_{i}} (r) | + | \bar{v_{i}} (r) - \bar{w_{i}} (r) |) dr}, \\ \frac{\sum_{i = 1}^{n} p_{i} \int_{0}^{1} r (\underline{v_{i}} (r) + \bar{v_{i}} (r)) dr - \sum_{i = 1}^{n} p_{i} \int_{0}^{1} r (\underline{u_{i}} (r) - \bar{u_{i}} (r)) dr}{\sum_{i = 1}^{n} p_{i} \int_{0}^{1} r (| \underline{u_{i}} (r) - \underline{v_{i}} (r) | + | \bar{u_{i}} (r) - \bar{v_{i}} (r) |) dr}} \\ = min {\frac{M_{p} (w) - M_{p} (v)}{D_{p} (v, w)}, \frac{M_{p} (v) - M_{p} (u)}{D_{p} (u, v)}} \\ = min {\frac{M_{p} (v) - M_{p} (u)}{D_{p} (u, v)}, \frac{M_{p} (w) - M_{p} (v)}{D_{p} (v, w)}} \end{matrix}$ It implies that $\begin{matrix} \frac{1}{2} (1 + \frac{M_{p} (w) - M_{p} (u)}{D_{p} (u, w)}) \\ \geq min {\frac{1}{2} (1 + \frac{M_{p} (v) - M_{p} (u)}{D_{p} (u, v)}), \frac{1}{2} (1 + \frac{M_{p} (w) - M_{p} (v)}{D_{p} (v, w)})} \end{matrix}$ by the definition of ${\tilde{≺}}_{p}$ , we know that ${\tilde{≺}}_{p} (u, w) \geq min {{\tilde{≺}}_{p} (u, v), {\tilde{≺}}_{p} (v, w)}$ .

All the above proof tell us that Conclusion (2) holds.

The proof of (3):

By conclusion (1), we can easily see that ${\tilde{≺}}_{p} (u, v) = 1$ is equivalent to ${\tilde{≺}}_{p} (v, u) = 0$ , so we only need to prove that u ≤ v and u ≠ v ⇒ ${\tilde{≺}}_{p} (u, v) = 1$ .

By the definition of u ≤ v, we know that u ≤ v and u ≠ v ⇒ u ≠ v and $\underline{u_{i}} (r) \leq \underline{v_{i}} (r)$ and $\bar{u_{i}} (r) \leq \bar{v_{i}} (r)$ for any r ∈ [0, 1] and i = 1, 2, ⋯, n. Therefore, by the definition of ${\tilde{≺}}_{p}$ , we have that $\begin{matrix} {\tilde{≺}}_{p} (u, v) \\ = \frac{1}{2} (1 + \frac{M_{p} (v) - M_{p} (u)}{D_{p} (u, v)}) \\ = \frac{1}{2} (1 + \frac{\sum_{i = 1}^{n} p_{i} \int_{0}^{1} r (\underline{v_{i}} (r) + \bar{v_{i}} (r)) dr - \sum_{i = 1}^{n} p_{i} \int_{0}^{1} r (\underline{u_{i}} (r) + \bar{u_{i}} (r)) dr}{\sum_{i = 1}^{n} p_{i} \int_{0}^{1} r (| \underline{u_{i}} (r) - \underline{v_{i}} (r) | + | \bar{u_{i}} (r) - \bar{v_{i}} (r) |) dr}) \\ = \frac{1}{2} (1 + \frac{\sum_{i = 1}^{n} p_{i} \int_{0}^{1} r (\underline{v_{i}} (r) + \bar{v_{i}} (r) - \underline{u_{i}} (r) - \bar{u_{i}} (r)) dr}{\sum_{i = 1}^{n} p_{i} \int_{0}^{1} r (\underline{v_{i}} (r) - \underline{u_{i}} (r) + \bar{v_{i}} (r) - \bar{u_{i}} (r)) dr}) \\ = \frac{1}{2} (1 + 1) \\ = 1 \end{matrix}$ so, Conclusion (3) holds.

The proof of (4):

Let u, v, w ∈ E (Eⁿ). If u = v, then u ⊕ w = v ⊕ w, so ${\tilde{≺}}_{p} (u \oplus w, v \oplus w) = \frac{1}{2} = {\tilde{≺}}_{p} (u, v)$ ; If u ≠ v, then u ⊕ w ≠ v ⊕ w, so by the definition of ${\tilde{≺}}_{p}$ and Theorem 5 in [20], we have that

$\begin{matrix} {\tilde{≺}}_{p} (u \oplus w, v \oplus w) \\ = \frac{1}{2} (1 + \frac{M_{p} (v \oplus w) - M_{p} (u \oplus w)}{D_{p} (u \oplus w, v \oplus w)}) \\ = \frac{1}{2} (1 + \frac{\sum_{i = 1}^{n} p_{i} \int_{0}^{1} r (\underline{(v \oplus w) i} (r) + \bar{(v \oplus w)_{i}} (r)) dr - \sum_{i = 1}^{n} p_{i} \int_{0}^{1} r (\underline{(u \oplus w)_{i}} (r) + \bar{(u \oplus w)_{i}} (r)) dr}{\sum_{i = 1}^{n} p_{i} \int_{0}^{1} r (| \underline{(u \oplus w)_{i}} (r) - \underline{(v \oplus w)_{i}} (r) | + | \bar{(u \oplus w)_{i}} (r) - \bar{(v \oplus w)_{i}} (r) |) dr}) \\ = \frac{1}{2} (1 + \frac{\sum_{i = 1}^{n} p_{i} \int_{0}^{1} r (\underline{v_{i}} (r) + \underline{w_{i}} (r) + \bar{v_{i}} (r) + \bar{w_{i}} (r) - \underline{u_{i}} (r) - \underline{w_{i}} (r) - \bar{u_{i}} (r) - \bar{w_{i}} (r)) dr}{\sum_{i = 1}^{n} p_{i} \int_{0}^{1} r (| \underline{u_{i}} (r) - \underline{v_{i}} (r) | + | \bar{u_{i}} (r) - \bar{v_{i}} (r) |) dr}) \\ = \frac{1}{2} (1 + \frac{\sum_{i = 1}^{n} p_{i} \int_{0}^{1} r (\underline{v_{i}} (r) + \bar{v_{i}} (r) - \underline{u_{i}} (r) - \bar{u_{i}} (r)) dr}{\sum_{i = 1}^{n} p_{i} \int_{0}^{1} r (| \underline{u_{i}} (r) - \underline{v_{i}} (r) | + | \bar{u_{i}} (r) - \bar{v_{i}} (r) |) dr}) \\ = \frac{1}{2} (1 + \frac{\sum_{i = 1}^{n} p_{i} \int_{0}^{1} r (\underline{v_{i}} (r) + \bar{v_{i}} (r)) dr - \sum_{i = 1}^{n} p_{i} \int_{0}^{1} r (\underline{u_{i}} (r) + \bar{u_{i}} (r)) dr}{\sum_{i = 1}^{n} p_{i} \int_{0}^{1} r (| \underline{u_{i}} (r) - \underline{v_{i}} (r) | + | \bar{u_{i}} (r) - \bar{v_{i}} (r) |) dr}) \\ = \frac{1}{2} (1 + \frac{M_{p} (v) - M_{p} (u)}{D_{p} (u, v)}) \\ = {\tilde{≺}}_{p} (u, v) \end{matrix}$

so, Conclusion (4) holds.

The proof of (5):

Let u, v ∈ E (Eⁿ) and ${\tilde{≺}}_{p} (u, v) = \frac{1}{2}$ . In the following, we use reduction to absurdity to prove that M_p (u) = M_p (v): Suppose M_p (u) ≠ M_p (v), then u ≠ v and $\frac{M_{p} (v) - M_{p} (u)}{D_{p} (u, v)} \neq 0$ , so by the definition of ${\tilde{≺}}_{p}$ , we have that ${\tilde{≺}}_{p} (u, v) = \frac{1}{2}$ $(1 + \frac{M_{p} (v) - M_{p} (u)}{D_{p} (u, v)}) \neq \frac{1}{2} (1 + 0)) = \frac{1}{2}$ , this is in contradiction with ${\tilde{≺}}_{p} (u, v) = \frac{1}{2}$ . Therefore, ${\tilde{≺}}_{p} (u, v) = \frac{1}{2}$ ⇒ M_p (u) = M_p (v) is right.

Conversely, let u, v ∈ E (Eⁿ) and M_p (u) = M_p (v), i.e., M_p (v) - M_p (u) =0. If u = v, by the definition of ${\tilde{≺}}_{p}$ , we can directly see ${\tilde{≺}}_{p} (u, v) = \frac{1}{2}$ ; If u ≠ v, then by the definition of ${\tilde{≺}}_{p}$ , we have that ${\tilde{≺}}_{p} (u, v) = \frac{1}{2} (1 + \frac{M_{p} (v) - M_{p} (u)}{D_{p} (u, v)}) = \frac{1}{2} (1 + 0) = \frac{1}{2}$ , so M_p (u) = M_p (v) ⇒ ${\tilde{≺}}_{p} (u, v) = \frac{1}{2}$ is also right.

Thus, we have done the proof of the theorem.

Remark 2. From Conclusion (2) of Theorem 1, we know that the fuzzy binary relation ${\tilde{≺}}_{p}$ on L (Eⁿ) is transitive. And from the meaning of the definition expression of ${\tilde{≺}}_{p} (u, v)$ and Conclusion (3), we see that ${\tilde{≺}}_{p} (u, v)$ can reasonably reflect the credibility in which fuzzy number v is “larger” than fuzzy number u. So we say ${\tilde{≺}}_{p}$ to be a fuzzy order onL (Eⁿ).

Remark 3. For u, v ∈ E (Eⁿ), if ${\tilde{≺}}_{p} (u, v) \geq r_{0}$ , then we say that v is larger than u in level r = r₀, and denote as $u {\tilde{≺}}_{p} v$ in level r = r₀. Especially, we denote ${\tilde{≺}}_{p} (u, v) = r_{0}$ by $u {\tilde{≺}}_{p}^{r_{0}} v$ .

For the fuzzy order ${\tilde{≺}}_{p}$ , we can make the following explanation:

If ${\tilde{≺}}_{p} (u, v) = r$ , i.e., $u {\tilde{≺}}_{p}^{r} v$ (r ∈ [0, 1]), then we can think that the credibility in which v is “larger” than u is r. For example, Conclusion (3) tell us that if u ≤ v and u ≠ v, then the credibility in which v is “larger” than u is 1 (one hundred percent), i.e., v is definitely “larger” than u. However Conclusion (5) tell us that if the mean of u and mean of v are same, then the credibility in which v is “larger” than u and the credibility in which u is “larger” than v are all $\frac{1}{2}$ , i.e., that v is “larger” than u and that u is “larger” than v all have half the credibility. Conversely, if ${\tilde{≺}}_{p} (u, v) = \frac{1}{2}$ i.e., $u {\tilde{≺}}_{p}^{\frac{1}{2}} v$ , then we can think that there is no big difference between u and v.

From the above discussion we also see that as ${\tilde{≺}}_{p} (u, v) > \frac{1}{2}$ , i.e., $u {\tilde{≺}}_{p} v$ in level $r = \frac{1}{2}$ , we can can think that v is “larger” than u; as ${\tilde{≺}}_{p} (u, v) < \frac{1}{2}$ , we can can think that u is “larger” than v; as ${\tilde{≺}}_{p} (u, v) = \frac{1}{2}$ , i.e., $u {\tilde{≺}}_{p}^{\frac{1}{2}} v$ , we can think that u and v are almost the same, no one is “larger” than anyoneelse.

Remark 4. Compared with general (classical) order, the advantage of fuzzy order is that ${\tilde{≺}}_{p} (u, v)$ not only can tell us fuzzy number u and fuzzy number v who is big and “small”, but also tell us the credibility in which v is “larger” than u or u is “larger” than v. However, a classical order is not to do this.

4. Dynamic ranking

In the section, we give the following example (the problem to be considered and the some data sets come from Example 3.1 in [22], or Example 4 in [18]) to show how to set up a dynamic ranking method which changes with the change of the level r for some objects which can be characterized by some multichannel uncertain digital information by using the fuzzy order proposed by us in thepaper.

Example 1. Let us to rank five cities $A, B, C, D, E$ according to the pros and cons of the following five aspects of the cities as evaluating evidences:

Evidence 1 (denoted by E₁): Families’ annual average income per capita (the unit: thousand CNY) in the city;

Evidence 2 (denoted by E₂): Living space per person (the unit: m²) in the city;

Evidence 3 (denoted by E₃): Years spent in education per person aged between 18 and 60 in the city;

Evidence 4 (denoted by E₄): Satisfaction of merchandise availability (graded by percentage points) in the city;

Evidence 5 (denoted by E₅): Satisfaction of traffic conditions (graded by percentage points) in the city.

As we explained in example 3.1 in [22], for a certain family in a certain city, of course, E₁, E₂ ought to be definite crisp numbers, for a certain person in a certain city, E₃ - E₅ ought to be definite crisp numbers, too. But as for all the families or all the persons concerned in the city, E₁, E₂ and E₃ - E₅ are no definite crisp numbers, because E₁, E₂ and E₃ - E₅ vary in different families or persons. If we are forced to use definite crisp numbers (such as mean values) to represent respectively uncertain quantities E₁ - E₅, the information of reflecting the degrees of polarization would be lost, which is one of the important factors to separate superior from inferior. Therefore, we think it is reasonable to use the fuzzy 5-dimension numbers to represent uncertain quantities.

For simplicity, in the following, liking Example 4 in [18], we also only give the construction process of the fuzzy 5-ellipsoid number $u_{A}$ to represent City $A$ ( $u_{B}, u_{C}, u_{D}, u_{E}$ can be similarly constructed) according to the Steps 1, 2 and 3 (see Page 623 in [18]) of the constructing method (of fuzzy ellipsoid numbers) proposed in [18]:

Suppose that the following set of data are the statistical values of the twenty samples that are arbitrarily taken from city $A$ : $A = {(a_{ij})}_{20 \times 5} : \begin{matrix} E_{1} & E_{2} & E_{3} & E_{4} & E_{5} \\ Sample 1 & 21.3 & 32 & 19 & 85 & 70 \\ Sample 2 & 33.5 & 43 & 16 & 90 & 80 \\ Sample 3 & 0.5 & 20 & 12 & 75 & 75 \\ Sample 4 & 82.3 & 60 & 12 & 90 & 55 \\ Sample 5 & 26.6 & 26 & 11 & 80 & 60 \\ Sample 6 & 5.3 & 12 & 6 & 80 & 90 \\ Sample 7 & 0.9 & 35 & 19 & 70 & 65 \\ Sample 8 & 15.6 & 28 & 9 & 82 & 65 \\ Sample 9 & 11.3 & 25 & 12 & 85 & 65 \\ Sample 10 & 20.6 & 26 & 22 & 60 & 65 \\ Sample 11 & 13.8 & 24 & 12 & 85 & 70 \\ Sample 12 & 17.9 & 30 & 6 & 86 & 50 \\ Sample 13 & 23.6 & 31 & 16 & 86 & 70 \\ Sample 14 & 28.1 & 33 & 12 & 88 & 75 \\ Sample 15 & 11.5 & 28 & 9 & 88 & 55 \\ Sample 16 & 21.4 & 27 & 16 & 90 & 75 \\ Sample 17 & 19.7 & 25 & 19 & 80 & 80 \\ Sample 18 & 15.6 & 18 & 16 & 75 & 60 \\ Sample 19 & 13.1 & 23 & 9 & 78 & 80 \\ Sample 20 & 52.6 & 108 & 16 & 95 & 60 \end{matrix}$

In the following, we use the method introduced in [18] (see Page 623 in [18]) to construct a fuzzy 5-ellipsoid number to express city $A$ :

Step 1: By $μ_{A_{jitsc}} = \frac{1}{20} \sum_{i = 1}^{20} a_{ij}$ , j = 1, 2, 3, 4, 5 (see Formula (7) in [18]), we can obtain $μ_{A_{1}} = 21.775$ , $μ_{A_{2}} = 32.7$ , $μ_{A_{3}} = 13.45$ , $μ_{A_{4}} = 82.4$ , $μ_{A_{5}} =$ 68.25.

Step 2: By $L σ_{A_{jitsc}} = \frac{1}{N_{L_{A_{jitsc}}}} \sum_{a_{ij} < μ_{A_{jitsc}}} (μ_{A_{jitsc}} - a_{ij})$ and $R σ_{A_{jitsc}} = \frac{1}{N_{R_{A_{jitsc}}}} \sum_{a_{ij} > μ_{A_{jitsc}}} (a_{ij} - μ_{A_{jitsc}})$ , j = 1, 2, 3, 4, 5 (see Formula (8) in [18]), and where $N_{L_{A_{jitsc}}}$ and $N_{R_{A_{jitsc}}}$ are the number of the character values a_ij which satisfy $a_{ij} < μ_{A_{jitsc}}$ and the number of the character values a_ij which satisfy $a_{ij} > μ_{A_{jitsc}}$ , respectively), we can obtain $L σ_{A_{1}} = 8.318, R σ_{A_{1}} = 19.392$ , $L σ_{A_{2}} = 7.7, R σ_{A_{2}} = 23.1$ , $L σ_{A_{3}} = 3.45, R σ_{A_{3}} = 4.22$ , $L σ_{A_{4}} = 6.84, R σ_{A_{4}} = 4.22$ , $L σ_{A_{5}} = 8.25, R σ_{A_{5}} = 8.25$ .

Step 3: For practicality, we take $[α_{A_{1}}, β_{A_{1}}] = [0.5, + \infty]$ (0.5 is the minimum subsistence allowance of the City $A$ ), $[α_{A_{2}}, β_{A_{2}}] = [0, + \infty]$ , $[α_{A_{3}}, β_{A_{3}}] = [0, 23]$ (from the primary school, gets the highest degree (doctoral) is 23 years), $[α_{A_{4}},$ $β_{A_{4}}] = [0, 100]$ and $[α_{A_{5}}, β_{A_{5}}] = [0, 100]$ . Taking λ = 4, then we see $\begin{matrix} μ_{A_{1}} - λ L σ_{A_{1}} = - 11.497, μ_{A_{1}} + λ R σ_{A_{1}} = 99.343 \\ μ_{A_{2}} - λ L σ_{A_{2}} = 1.9, μ_{A_{2}} + λ R σ_{A_{2}} = 125.1 \\ μ_{A_{3}} - λ L σ_{A_{3}} = - 0.35, μ_{A_{3}} + λ R σ_{A_{3}} = 30.33 \\ μ_{A_{4}} - λ L σ_{A_{4}} = 55.04, μ_{A_{4}} + λ R σ_{A_{4}} = 104.8 \\ μ_{A_{5}} - λ L σ_{A_{5}} = 35.25, μ_{A_{5}} + λ R σ_{A_{5}} = 101.25 \end{matrix}$ and $\begin{matrix} (max_{_})_{A_{1}} = max {μ_{A_{1}} - λ L σ_{A_{1}}, α_{A_{1}}} = 0.5 \\ (min^{¯})_{A_{1}} = min {μ_{A_{1}} + λ R σ_{A_{1}}, β_{A_{1}}} = 99.343 \\ (max_{_})_{A_{2}} = max {μ_{A_{2}} - λ L σ_{A_{2}}, α_{A_{2}}} = 1.9 \\ (min^{¯})_{A_{2}} = min {μ_{A_{2}} + λ R σ_{A_{2}}, β_{A_{2}}} = 125.1 \\ (max_{_})_{A_{3}} = max {μ_{A_{3}} - λ L σ_{A_{3}}, α_{A_{3}}} = 0 \\ (min^{¯})_{A_{3}} = min {μ_{A_{3}} + λ R σ_{A_{3}}, β_{A_{3}}} = 23 \\ (max_{_})_{A_{4}} = max {μ_{A_{4}} - λ L σ_{A 4}, α_{A 4}} = 55.04 \\ (min^{¯})_{A_{4}} = min {μ_{A_{4}} + λ R σ_{A_{4}}, β_{A_{4}}} = 100 \\ (max_{_})_{A_{5}} = max {μ_{A_{5}} - λ L σ_{A_{5}}, α_{A_{5}}} = 35.25 \\ (min^{¯})_{A_{5}} = min {μ_{A_{5}} + λ R σ_{A_{5}}, β_{A_{5}}} = 100 \end{matrix}$

It is also obvious that the five components (E_j, j = 1, 2, 3, 4, 5) should be all of Two-sided type. Therefore, by $\begin{array}{l} u_{A_{j}} \\ = C T {( ( μ_{ A_{j}} - λ L σ_{ A_{ j}}) |_{{( \max_{_})}_{ A_{ j}}}, μ_{ A_{ j}}, (μ_{ A_{ j}} + λ R σ_{ A_{ j}}) |_{{( \min^{¯})}_{ A_{ j}}})}_{j = 1}^{5} \end{array}$ (see Formulae (9) in [18]), we can construct CT-fuzzy 5-ellipsoid number $u_{A}$ as $u_{A} = CT (\begin{matrix} - 11.494 |_{0.5} & 21.775 & 99.343 |_{99.343} \\ 1.9 |_{1.9} & 32.7 & 125.1 |_{125.1} \\ - 0.35 |_{0} & 13.45 & 30.33 |_{23} \\ 55.04 |_{55.04} & 82.4 & 104.8 |_{100} \\ 35.25 |_{35.25} & 68.25 & 101.25 |_{100} \end{matrix})$ to express the City $A$ .

By using the same methods, we can get $u_{B}$ , $u_{C}$ , $u_{D}$ and $u_{E}$ to respectively represent City $B$ , City $C$ , City $D$ and City $E$ . Suppose that $u_{B} = CT (\begin{matrix} - 4.81 |_{0.60} & 26.60 & 113.35 |_{113.35} \\ 0.26 |_{0.26} & 25.57 & 106.09 |_{106.09} \\ 0.10 |_{0.10} & 14.32 & 28.88 |_{23} \\ 37.63 |_{37.63} & 78.13 & 106.97 |_{100} \\ 29.42 |_{29.42} & 66.63 & 99.76 |_{99.76} \end{matrix})$ $u_{C} = CT (\begin{matrix} - 2.04 |_{0.67} & 25.34 & 103.65 |_{103.65} \\ 3.89 |_{3.89} & 35.26 & 125.56 |_{125.56} \\ 1.98 |_{1.98} & 17.08 & 31.08 |_{23} \\ 59.10 |_{59.10} & 89.31 & 102.50 |_{100} \\ 28.18 |_{28.18} & 60.35 & 92.52 |_{92.52} \end{matrix})$ $u_{D} = CT (\begin{matrix} - 2.93 |_{0.65} & 26.44 & 107.25 |_{107.25} \\ 5.24 |_{5.24} & 25.33 & 104.65 |_{104.65} \\ 0.75 |_{0.75} & 14.56 & 30.46 |_{23} \\ 53 . 71_{53.71} & 80.01 & 104.6 |_{100} \\ 27.15 |_{27.15} & 61.27 & 95.61 |_{95.61} \end{matrix})$ $u_{E} = CT (\begin{matrix} 0.92 |_{0.92} & 31.53 & 131.98 |_{131.98} \\ - 2.19 |_{0} & 31.27 & 133.58 |_{133.58} \\ 1.63 |_{1.63} & 14.81 & 29.91 |_{23} \\ 49.39 |_{49.39} & 75.31 & 99.68 |_{99.68} \\ 23.89 |_{23.89} & 56.21 & 88.53 |_{88.53} \end{matrix})$ By the above expressions of $u_{A}$ , $u_{B}$ , $u_{C}$ , $u_{D}$ and $u_{E}$ , we can obtain that $\begin{matrix} \underline{u_{A_{1}}} (r) = {\begin{matrix} 0.5, & r \in [0, 0.36) \\ 33.27 r - 11.5, & r \in [0.36, 1] \end{matrix} \\ \bar{u_{A_{1}}} (r) = 99.34 - 77.57 r, r \in [0, 1] \\ \underline{u_{A_{2}}} (r) = 30.80 r + 1.90, r \in [0, 1] \\ \bar{u_{A_{2}}} (r) = 125.10 - 92.40 r, r \in [0, 1] \\ \underline{u_{A_{3}}} (r) = {\begin{matrix} 0, & r \in [0, 0.03) \\ 13.8 r - 0.35, & r \in [0.03, 1] \end{matrix} \\ \bar{u_{A_{3}}} (r) = {\begin{matrix} 23, & r \in [0, 0.43) \\ 30.33 - 16.88 r, & r \in [0.43, 1] \end{matrix} \\ \underline{u_{A_{4}}} (r) = 27.36 r + 55.04, r \in [0, 1] \\ \bar{u_{A_{4}}} (r) = {\begin{matrix} 100, & r \in [0, 0.21) \\ 104.8 - 22.40 r, & r \in [0.21, 1] \end{matrix} \\ \underline{u_{A_{5}}} (r) = 33.00 r + 35.25, r \in [0, 1] \\ \bar{u_{A_{5}}} (r) = {\begin{matrix} 100, & r \in [0, 0.04) \\ 101.25 - 33.00 r, & r \in [0.04, 1] \end{matrix} \end{matrix}$ $\begin{matrix} \underline{u_{B_{1}}} (r) = {\begin{matrix} 0.6, & r \in [0, 0.17) \\ 31.41 r - 4.81, & r \in [0.17, 1] \end{matrix} \\ \bar{u_{B_{1}}} (r) = 113.35 - 86.75 r, r \in [0, 1] \\ \underline{u_{B_{2}}} (r) = 25.31 r + 0.26, r \in [0, 1] \end{matrix}$ $\begin{matrix} \bar{u_{B_{2}}} (r) = 106.09 - 80.52 r, r \in [0, 1] \\ \underline{u_{B_{3}}} (r) = 13.80 r + 0.10, r \in [0, 1] \\ \bar{u_{B_{3}}} (r) = {\begin{matrix} 23, & r \in [0, 0.44) \\ 29.88 - 15.56 r, & r \in [0.44, 1] \end{matrix} \end{matrix}$ $\begin{matrix} \underline{u_{B_{4}}} (r) = 40.50 r + 37.63, r \in [0, 1] \\ \bar{u_{B_{4}}} (r) = {\begin{matrix} 100, & r \in [0, 0.24) \\ 106.97 - 28.84 r, & r \in [0.24, 1] \end{matrix} \\ \underline{u_{B_{5}}} (r) = 37.21 r + 29.42, r \in [0, 1] \\ \bar{u_{B_{5}}} (r) = 99.76 - 33.13 r, r \in [0, 1] \end{matrix}$ $\begin{matrix} \underline{u_{C_{1}}} (r) = {\begin{matrix} 0.67, & r \in [0, 0.1) \\ 27.38 r - 2.04, & r \in [0.1, 1] \end{matrix} \\ \bar{u_{C_{1}}} (r) = 103.65 - 78.31 r, r \in [0, 1] \\ \underline{u_{C_{2}}} (r) = 31.37 r + 3.89, r \in [0, 1] \\ \bar{u_{C_{2}}} (r) = 125.56 - 90.30 r, r \in [0, 1] \\ \underline{u_{C_{3}}} (r) = 15.1 r + 1.98, r \in [0, 1] \end{matrix}$ $\begin{matrix} \bar{u_{C_{3}}} (r) = {\begin{matrix} 23, & r \in [0, 0.58) \\ 31.08 - 14 r, & r \in [0.58, 1] \end{matrix} \\ \underline{u_{C_{4}}} (r) = 30.21 r + 59.10, r \in [0, 1] \\ \bar{u_{C_{4}}} (r) = {\begin{matrix} 100, & r \in [0, 0.19) \\ 102.5 - 13.19 r, & r \in [0.19, 1] \end{matrix} \\ \underline{u_{C_{5}}} (r) = 32.17 r + 28.18, r \in [0, 1] \\ \bar{u_{C_{5}}} (r) = 92.52 - 32.17 r, r \in [0, 1] \end{matrix}$ $\begin{matrix} \underline{u_{D_{1}}} (r) = {\begin{matrix} 0.65, & r \in [0, 0.12) \\ 29.37 r - 2.93, & r \in [0.12, 1] \end{matrix} \\ \bar{u_{D_{1}}} (r) = 107.25 - 80.81 r, r \in [0, 1] \\ \underline{u_{D_{2}}} (r) = 20.09 r + 5.24, r \in [0, 1] \\ \bar{u_{D_{2}}} (r) = 104.65 - 79.32 r, r \in [0, 1] \\ \underline{u_{D_{3}}} (r) = 13.81 r + 0.75, r \in [0, 1] \\ \bar{u_{D_{3}}} (r) = {\begin{matrix} 23, & r \in [0, 0.47) \\ 30.46 - 15.9 r, & r \in [0.47, 1] \end{matrix} \\ \underline{u_{D_{4}}} (r) = 26.3 r + 53.71, r \in [0, 1] \\ \bar{u_{D_{4}}} (r) = {\begin{matrix} 100, & r \in [0, 0.19) \\ 104.6 - 24.59 r, & r \in [0.19, 1] \end{matrix} \\ \underline{u_{D_{5}}} (r) = 34.12 r + 27.15, r \in [0, 1] \\ \bar{u_{D_{5}}} (r) = 95.61 - 34.34 r, r \in [0, 1] \end{matrix}$ $\begin{matrix} \underline{u_{E_{1}}} (r) = 30.61 r + 0.92, r \in [0, 1] \\ \bar{u_{E_{1}}} (r) = 131.98 - 100.45 r, r \in [0, 1] \\ \underline{u_{E_{2}}} (r) = {\begin{matrix} 0, & r \in [0, 0.07) \\ 33.46 r - 2.19, & r \in [0.07, 1] \end{matrix} \\ \bar{u_{E_{2}}} (r) = 133.58 - 102.31 r, r \in [0, 1] \\ \underline{u_{E_{3}}} (r) = 13.18 r + 1.63, r \in [0, 1] \\ \bar{u_{E_{3}}} (r) = {\begin{matrix} 23, & r \in [0, 0.46) \\ 29.91 - 15.1 r, & r \in [0.46, 1] \end{matrix} \\ \underline{u_{E_{4}}} (r) = 25.91 r + 49.39, r \in [0, 1] \\ \bar{u_{E_{4}}} (r) = 99.68 - 24.38 r, r \in [0, 1] \\ \underline{u_{E_{5}}} (r) = 32.32 r + 23.89, r \in [0, 1] \\ \bar{u_{E_{5}}} (r) = 88.53 - 32.32 r, r \in [0, 1] \end{matrix}$

Taking $p = (\frac{1}{5}, \frac{1}{5}, \frac{1}{5}, \frac{1}{5}, \frac{1}{5})$ , by the definitions of M_p and D_p, we get that $\begin{matrix} M_{p} (u_{A}) = 47.17, M_{p} (u_{B}) = 45.38 \\ M_{p} (u_{C}) = 48.43, M_{p} (u_{D}) = 45.17 \\ M_{p} (u_{E}) = 46.41 \end{matrix}$ $\begin{matrix} D_{p} (u_{A}, u_{B}) = D_{p} (u_{B}, u_{A}) = 5.36 \\ D_{p} (u_{A}, u_{C}) = D_{p} (u_{C}, u_{A}) = 5.11 \\ D_{p} (u_{A}, u_{D}) = D_{p} (u_{D}, u_{A}) = 4.64 \\ D_{p} (u_{A}, u_{E}) = D_{p} (u_{E}, u_{A}) = 7.15 \\ D_{p} (u_{B}, u_{C}) = D_{p} (u_{C}, u_{B}) = 6.07 \\ D_{p} (u_{B}, u_{D}) = D_{p} (u_{D}, u_{B}) = 2.14 \\ D_{p} (u_{B}, u_{E}) = D_{p} (u_{E}, u_{B}) = 5.89 \\ D_{p} (u_{C}, u_{D}) = D_{p} (u_{D}, u_{C}) = 4.15 \\ D_{p} (u_{C}, u_{E}) = D_{p} (u_{E}, u_{C}) = 6.00 \\ D_{p} (u_{D}, u_{E}) = D_{p} (u_{E}, u_{D}) = 5.30 \end{matrix}$

By the definition of ${\tilde{≺}}_{p}$ , we can work out $\begin{matrix} {\tilde{≺}}_{p} (u_{A}, u_{B}) = 0.33 (i . e ., {\tilde{≺}}_{p} (u_{B}, u_{A}) = 0.67) \\ {\tilde{≺}}_{p} (u_{A}, u_{C}) = 0.62 (i . e ., {\tilde{≺}}_{p} (u_{C}, u_{A}) = 0.38) \\ {\tilde{≺}}_{p} (u_{A}, u_{D}) = 0.28 (i . e ., {\tilde{≺}}_{p} (u_{D}, u_{A}) = 0.72) \\ {\tilde{≺}}_{p} (u_{A}, u_{E}) = 0.45 (i . e ., {\tilde{≺}}_{p} (u_{E}, u_{A}) = 0.55) \\ {\tilde{≺}}_{p} (u_{B}, u_{C}) = 0.75 (i . e ., {\tilde{≺}}_{p} (u_{C}, u_{B}) = 0.25) \\ {\tilde{≺}}_{p} (u_{B}, u_{D}) = 0.45 (i . e ., {\tilde{≺}}_{p} (u_{D}, u_{B}) = 0.55) \\ {\tilde{≺}}_{p} (u_{B}, u_{E}) = 0.59 (i . e ., {\tilde{≺}}_{p} (u_{E}, u_{B}) = 0.41) \\ {\tilde{≺}}_{p} (u_{C}, u_{D}) = 0.11 (i . e ., {\tilde{≺}}_{p} (u_{D}, u_{C}) = 0.89) \\ {\tilde{≺}}_{p} (u_{C}, u_{E}) = 0.33 (i . e ., {\tilde{≺}}_{p} (u_{E}, u_{C}) = 0.67) \\ {\tilde{≺}}_{p} (u_{D}, u_{E}) = 0.62 (i . e ., {\tilde{≺}}_{p} (u_{E}, u_{D}) = 0.38) \end{matrix}$

From the above results worked out by us, we have that $u_{D} {\tilde{≺}}_{p}^{0.55} u_{B} {\tilde{≺}}_{p}^{0.59} u_{E} {\tilde{≺}}_{p}^{0.55} u_{A} {\tilde{≺}}_{p}^{0.62} u_{C}$ From this we can obtain a ranking (from good to bad) of the five cities: City $C$ , City $A$ , City $E$ , City $B$ , City $D$ , and see that the degrees to which City $A$ is superior to City $E$ and City $B$ is superior to City $D$ are relatively small, they are all 0.55, however, the degree to which City $C$ is superior to City $A$ is relatively large, it is 0.62.

And if we take the level r = 0.55, r = 0.59 and r = 0.62, we can respectively get the following dynamic rankings (from big to small) which change with the change of the level r: $u_{D} {\tilde{≺}}_{p} u_{B} {\tilde{≺}}_{p} u_{E} {\tilde{≺}}_{p} u_{A} {\tilde{≺}}_{p} u_{C} in the level r = 0.55$ $u_{D}, u_{B} {\tilde{≺}}_{p} u_{E}, u_{A} {\tilde{≺}}_{p} u_{C} in the level r = 0.59$ and $u_{D}, u_{B}, u_{E}, u_{A} {\tilde{≺}}_{p} u_{C} in the level r = 0.62$ i.e., we can obtain the following three dynamic rankings (from good to bad) which change with the change of the level r for the five cities: (1) First: City $C$ , Second: City $A$ , Third: City $E$ , Fourth: City $B$ , Fifth: City $D$ in the level r = 0.55; (2) First: City $C$ , Second: City $A$ , City $E$ , Third: City $B$ , City $D$ in the level r = 0.59; (3) First: City $C$ , Second: City $A$ , City $E$ , City $B$ , City $D$ in the level r = 0.62.

Remark 5. In Example 1, from $u_{D} {\tilde{≺}}_{p}^{0.55} u_{B} {\tilde{≺}}_{p}^{0.59} u_{E} {\tilde{≺}}_{p}^{0.55} u_{A} {\tilde{≺}}_{p}^{0.62} u_{C}$ we not only can get the ranking (from good to bad) of the five cities: City $C$ , City $A$ , City $E$ , City $B$ , City $D$ , but also see the degrees of good and bad between cities. And we can also get dynamic rankings which change with the change of the level r. Compared with the ranking methods proposed in [18, 22] by using weak orders or metrics, these just reflect the advantages of the ranking method set up by us in this paper.

5. Conclusion

In this paper, we introduced the concepts of left mean, right mean and mean for a fuzzy ellipsoid number and left absolute difference, right absolute difference and absolute difference for two fuzzy ellipsoid numbers, and proved that the absolute difference is a metric on the fuzzy ellipsoid number space. Then based on the concepts of the mean and the absolute difference, we defined a fuzzy binary relation on the fuzzy ellipsoid number space, and showed that it has some properties, such as fuzzy transitivity, etc, which can embody that the fuzzy binary relation has the attributes of order (so we call it a fuzzy order). And then we gave a practical example to show how to set up a dynamic ranking method which changes with the change of the level r for some objects which can be characterized by some multichannel uncertain digital information by using our proposed fuzzy order. Finally we concluded a remark to illustrate the advantage of the dynamic ranking method. In the future, we will use our method to rank possible faults according to the probability of occurrence in industrial alarm systems.

Footnotes

Acknowledgments

This work is partially supported by the Nature Science Foundation of China (Nos. 61771174 and 61433001) and Zhejiang Provincial Nature Science Foundation of China under Grant No. LY19F030018.

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