Constructing multistage linear fuzzy ellipsoid number to express multi-channel uncertain digital information and application

Abstract

In this paper, the problems of expressing and fusing multi-channel uncertain digital information is studied. The concept of a special high-dimensional fuzzy number called multi-level linear fuzzy ellipsoid number is given, and a method of constructing such high dimensional fuzzy number to express multi-channel uncertain digital information is established. Then a calculation formula of the centroid of multistage linear fuzzy ellipsoid number is deduced. And then, as an application example of multi-channel uncertain digital information fusion, a specific example is given to show ranking some objects which are characterized by multi-channel uncertain digital information by using the obtained results and the concept of fuzzy order on high dimensional fuzzy number space.

Keywords

Fuzzy numbers fuzzy ellipsoid numbers multistage linear fuzzy ellipsoid number constructing fuzzy numbers expressing multi-channel uncertain digital information multi-channel uncertain digital information fusion

1 Introduction

The representation of information is an important research topic in information fusion and intelligent computing. Fuzzy number is a good tool for representing uncertain or imprecise numerical information. In [29, 31], Wang and Shi et al. suggested using one-dimensional or high-dimensional fuzzy cell number and fuzzy ellipsoid number to express the uncertain or imprecise information, respectively.

The concept of fuzzy numbers was introduced by Chang and Zadeh [3] in 1972. Since then, problems related to the theory and application of fuzzy numbers have been widely studied and rich results have been obtained (see for example [2, 5, 8, 12, 20, 39]). Recently, there has still been a lot of work on the theory and application of fuzzy numbers., see for example, Wang, Lv and Zhu et al. studied the problems of approximation of 1-dimensional multi-knots piecewise linear fuzzy numbers and multi-stage step type fuzzy numbers in [17, 27] and [41], respectively; Wang and Wang proposed the methods of expressing and ranking of uncertain digital information by using multi-knots piecewise linear fuzzy numbers in [33] Pavlačka and Pavlačková studied the properties of the fuzzy weighted average of fuzzy numbers with normalized fuzzy weights in [24]; Mashchenko analyzed the operation of minimization with a fuzzy set of the indices of operands for fuzzy numbers in [19]; Kumar and Khepar et al. gave a broad overview of current techniques in fuzzy numbers and its applications in [15]; Hai and Lv et al. studied the problems of approximation of 2-cell and n-cell numbers in [10] and [18], respectively.

Many scholars have been engaged in or are working on using fuzzy numbers to express uncertain digital information, and have also achieved some good results. For example, Muzzioli and Torricelli set stock price as fuzzy number, presented a new one period model in [21]; Wu used fuzzy numbers to generate pricing boundaries of European options based on Black Scholes formula in [37]; Muzzioli, Reynaerts and Torricelli regarded the stock’s rising factor and falling factor as fuzzy numbers in [22, 23], respectively; Thavaneswaran, Appadoo and Andrĺȩ-Sĺćnchez et al. studied the pricing of European options by fuzzing the maturity price of stocks into trapezoid fuzzy numbers or triangular fuzzy numbers in [1, 25], respectively; in [16], Li and Zeng et al. proposed a new fuzzy regression model based on trapezoidal fuzzy number and least absolute deviation method; following the framework of Klein in [14], Xu and Liu derived a fuzzy binomial tree pricing model by modelling the firm value and default recovery rate as fuzzy numbers in [38]; Chen and Hu et al. considered pricing European call options under a fuzzy environment in [4]; Wang and Shi et al. gave a specific method to solve the fuzzy number coefficients of the random fuzzy number linear regression equation in [30]; Wang and Shi et al. proposed methods of constructing fuzzy cell numbers and fuzzy ellipsoid numbers to express multi-channel uncertain digital information in [29, 31], respectively; Wang and Wang et al. established a method of option pricing in the form of fuzzy number based on fuzzy number binary tree model in [34].

Of course, using fuzzy numbers to express uncertain digital information is not the ultimate purpose. By constructing fuzzy numbers to express uncertain digital information, the space of objects characterized by uncertain digital information can be transformed into a fuzzy number space, and then appropriate aggregation operators can be established in the fuzzy number space to achieve the ultimate purpose of processing the target, such as ranking (see for example [6, 7, 9, 11, 13, 35]), identifying (see for example [26, 29]), classifying (see for example [32, 40]), etc.

In the existing work of using fuzzy numbers and fuzzy number vectors to express single channel and multichannel uncertain or imprecise digital information (quantities), for the convenience of calculations, the fuzzy numbers and fuzzy number vectors used are all basically rectangular fuzzy numbers (interval numbers), triangular fuzzy numbers and trapezoidal fuzzy numbers and high dimensional fuzzy numbers with the components of rectangular fuzzy numbers (interval numbers), triangular fuzzy numbers and trapezoidal. In fact, this can really bring great convenience to the application. However, it is also because the structure of such fuzzy numbers is too simple, using them to express uncertain information will inevitably lose some information, which reduces the rationality and accuracy of the applications.

In this paper, in order to overcome or reduce the defect of losing too much information and increase the rationality and accuracy of expressing uncertain information, we establish a method of constructing multistage Linear fuzzy ellipsoid numbers to to express multi-channel uncertain digital information. In addition, for the convenience of aggregation operations of high-dimensional fuzzy numbers and applications, we also derive the calculation formula of the centroid of multistage Linear fuzzy ellipsoid numbers.

The specific structure of this paper is as follows: In Section 2, we briefly review some basic notions, definitions and results about fuzzy numbers, which will be used in this paper. In Section 3, by a theorem we obtain, we give the definition of r-stage linear fuzzy n-ellipsoid numbers. In Section 4, we establish a method and the specific steps of constructing r-stage linear fuzzy n-ellipsoid numbers to represent multi-channel uncertain digital information, and give an example to show how to use the method to construct such numbers. In Section 5, we derive a calculation formula of the centroid of multistage Linear fuzzy ellipsoid number, and give an example to show how to use the calculation formula to work out the centroid of a multistage Linear fuzzy ellipsoid number. In Section 6, as an example of application in multi-channel uncertain digital information fusion, we give a specific practical example to show how to use the results obtained by us and a fuzzy order on high dimensional fuzzy number space introduced in [36] to rank some objects which are characterized by multi-channel uncertain digital information. in Section 7, we make a conclusion to this paper.

2 Basic definitions and notations

Let n be a natural number, R be the real number set, R⁺ be the positive 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 fuzzy set u and r ∈ (0, 1], the r-level set {x ∈ Rⁿ : u (x) ≥ r} of u is denoted by [u] ^r, and the closure of support of u is denoted by [u] ⁰, 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.

If u ∈ Eⁿ, and for each r ∈ [0, 1] , [u] ^r is a (closed) 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 [31]), then u is called a fuzzy n-ellipsoid number [31]. And the collection of all (closed) n-ellipsoids is denoted by E (Rⁿ), and the collection of all fuzzy n-ellipsoid numbers is denoted by E (Eⁿ).

In this paper, we still use the notation in [31]: $\begin{matrix} (\begin{matrix} a_{1} & b_{1} \\ a_{2} & b_{2} \\ ⋮ & ⋮ \\ a_{n} & b_{n} \end{matrix}) (in short, {(a_{i}, b_{i})}_{i = 1}^{n}) \\ = {(x_{1}, x_{2}, \dots, x_{n}) \in R^{n} : \sum_{i = 1}^{n} \frac{(x_{i} - \frac{b_{i} + a_{i}}{2})^{2}}{(\frac{b_{i} - a_{i}}{2})^{2}} \leq 1} . \end{matrix}$ So, for u ∈ E (Eⁿ) and r ∈ [0, 1], [u] ^r can be denoted as ${(\underline{u_{i}} (r), \bar{u_{i}} (r))}_{i = 1}^{n}$ .

Theorem 1. [31] If u ∈ E (Eⁿ), then for i = 1, 2, ⋯ , n, $\underline{u_{i}} (r)$ , $\bar{u_{i}} (r)$ are real-valued functions on [0, 1], and satisfy

(1) $\underline{u_{i}} (r)$ are non-decreasing and left continuous;

(2) $\bar{u_{i}} (r)$ are non-increasing and left continuous;

(3) $\underline{u_{i}} (r) \leq \bar{u_{i}} (r)$ (it is equivalent to $\underline{u_{i}} (1) \leq \bar{u_{i}} (1)$ );

(4) $\underline{u_{i}} (r)$ , $\bar{u_{i}} (r)$ are right continuous at r = 0.

Conversely if a_i (r), b_i (r), i = 1, 2, ⋯ , n are real-valued functions on [0, 1] which satisfy the (1)-(4) above, then there exists a unique u ∈ E (Eⁿ) such that $[u]^{r} = {(x_{1}, \dots, x_{n}) \in R^{n} : \sum_{i = 1}^{n} \frac{(x_{i} - \frac{b_{i} (r) + a_{i} (r)}{2})^{2}}{(\frac{b_{i} (r) - a_{i} (r)}{2})^{2}} \leq 1}$ for any r ∈ [0, 1].

Let u_i ∈ E (= E¹), i = 1, 2, ⋯ , n. The ordered class u₁, u₂, ⋯ , u_n is called a n-dimensional fuzzy vector, and denoted as (u₁, u₂, ⋯ , u_n). The collection of all n-dimensional fuzzy vectors (i.e. the Cartesian product ${\overset{︷}{E \times E \times \dots \times E}}^{n}$ ) is called the n-dimensional fuzzy vector space, and denoted as (E) ⁿ.

By Theorem 3.1 in [28], we see fuzzy n-ellipsoid number and n-dimensional fuzzy vector can represent each other, and the representation is unique, so E (Eⁿ) and (E) ⁿ may be regarded as identity. If (u₁, u₂, ⋯ , u_n) ∈ (E) ⁿ is the unique representation of u ∈ E (Eⁿ), then it can be denoted as u = F_e (u₁, u₂, ⋯ , u_n).

If u ∈ E and is defined as $u (x) = {\begin{matrix} \frac{x - a}{b - a}, & x \in [a, b), \\ 1, & x \in [b, c], \\ \frac{d - x}{d - c}, & x \in (c, d], \\ 0, & x \notin [a, d], \end{matrix}$ then u is called a trapezoid fuzzy number, and denoted as u = F (a, b, c, d), where a, b, c, d ∈ R with a ≤ b ≤ c ≤ d.

3 Multistage Linear fuzzy ellipsoid numbers

In order to introduce the concept of multistage Linear fuzzy ellipsoid numbers, we first give the following result:

Theorem 2. Let 0 = r₀ < r₁ < ⋯ < r_m-1 < r_m = 1, a_ij, b_ij ∈ R with a_i0 ≤ a_i1 ≤ ⋯ ≤ a_i(m-1) ≤ a_im ≤ b_im ≤ b_i(m-1) ≤ ⋯ ≤ b_i1 ≤ b_i0 for any i = 1, 2, ⋯ , n and j = 0, 1, ⋯ , m, and $\begin{matrix} a_{i} (r) \\ = {\begin{matrix} a_{i 0} + (a_{i 1} - a_{i 0}) \frac{r - r_{0}}{r_{1} - r_{0}}, & r \in [0, r_{1}] \\ a_{i 1} + (a_{i 2} - a_{i 1}) \frac{r - r_{1}}{r_{2} - r_{1}}, & r \in (r_{1}, r_{2}] \\ ⋮ \\ a_{i (m - 1)} + (a_{im} - a_{i (m - 1)}) \frac{r - r_{m - 1}}{r_{m} - r_{m - 1}}, & r \in (r_{m - 1}, 1] \end{matrix} \end{matrix}$ (1) and $\begin{matrix} b_{i} (r) \\ = {\begin{matrix} b_{i 0} + (b_{i 1} - b_{i 0}) \frac{r - r_{0}}{r_{1} - r_{0}}, & r \in [0, r_{1}] \\ b_{i 1} + (b_{i 2} - b_{i 1}) \frac{r - r_{1}}{r_{2} - r_{1}}, & r \in (r_{1}, r_{2}] \\ ⋮ \\ b_{i (m - 1)} + (b_{im} - b_{i (m - 1)}) \frac{r - r_{m - 1}}{r_{m} - r_{m - 1}}, & r \in (r_{m - 1}, 1] \end{matrix} \end{matrix}$ (2) Then there exists a unique u ∈ E (Eⁿ) such that $[u]^{r} = {(\underline{u_{i}} (r), \bar{u_{i}} (r))}_{i = 1}^{n} = {(a_{i} (r), b_{i} (r))}_{i = 1}^{n}$ for any r ∈ [0, 1].

Proof. For each i = 1, 2, ⋯ , n, it can be directly seen that a_i (r) and b_i (r) are all left continuous on (0, 1]. Let r′, r″ ∈ [0, 1] with r′ ≤ r″. By 0 = r₀ < r₁ < ⋯ < r_m-1 < r_m = 1, it is known that there exist j₁, j₂ ∈ {r_i : j = 1, 2, ⋯ , m} with j₁ ≤ j₂ such that r_j₁-1 ≤ r′ ≤ r_{j
₁} ≤ j₂ ≤ r″ ≤ r_j₂+1. For each i = 1, 2, ⋯ , n, by the definition of a_i (r) and a_i0 ≤ a_i1 ≤ ⋯ ≤ a_i(m-1) ≤ a_im, it can be derived that $\begin{matrix} a_{i} (r^{'}) = a_{i (j_{1} - 1)} + (a_{{ij}_{1}} - a_{i (j_{1} - 1)}) \frac{r^{'} - r_{j_{1} - 1}}{r_{j_{1}} - r_{j_{1} - 1}} \\ \leq a_{i (j_{1} - 1)} + (a_{{ij}_{1}} - a_{i (j_{1} - 1)}) \frac{r_{j_{1}} - r_{j_{1} - 1}}{r_{j_{1}} - r_{j_{1} - 1}} \\ = a_{{ij}_{1}} \\ \leq a_{{ij}_{2}} \\ \leq a_{{ij}_{2}} + (a_{i (j_{2} + 1)} - a_{{ij}_{2}}) \frac{r^{″} - r_{j_{2}}}{r_{j_{2} + 1} - r_{j_{2}}} \\ = a_{i} (r^{″}), \end{matrix}$ so a_i (r) is non-decreasing on [0, 1]. Similarly, it can be also proved that b_i (r) is non-increasing on [0, 1] for i = 1, 2, ⋯ , n. Therefore, a_i (r), b_i (r), i = 1, 2, ⋯ , n satisfy the conditions (1) and (2) of Theorem 1.

For each i = 1, 2, ⋯ , n, by the definitions of a_i (r) and b_i (r), it can be get that $a_{i} (1) = a_{i (m - 1)} + (a_{im} - a_{i (m - 1)}) \frac{1 - r_{m - 1}}{1 - r_{m - 1}} = a_{im} \leq b_{im} = b_{i (m - 1)} + (b_{im} - b_{i (m - 1)}) \frac{1 - r_{m - 1}}{1 - r_{m - 1}} = b_{i} (1)$ . So, the condition (3) of Theorem 1 also holds for a_i (r), b_i (r), i = 1, 2, ⋯ , n.

For each i = 1, 2, ⋯ , n, by the definitions of a_i (r) and b_i (r), it is obvious that a_i (r), b_i (r), i = 1, 2, ⋯ , n satisfy the condition (4) of Theorem 1.

Thus, by Theorem 1, the correctness of the conclusion of the Theorem 2 can be seen. The proof of Theorem 2 is completed.

For a m × n matrix A = (a_ij) _m×n, its transpose matrix is denoted as $A^{T} = (a_{ij})_{m \times n}^{T}$ .

Definition 1. Let r = (r₀, r₁, ⋯ , r_m) ^T with 0 = r₀ < r₁ < ⋯ < r_m-1 < r_m = 1. For A = (a₁, a₂, ⋯ , a_n) and B = (b₁, b₂, ⋯ , b_n), where a_i = (a_i0, a_i1, ⋯ , a_im) ^T, b_i = (b_i0, b_i1, ⋯ , b_im) ^T with a_i0 ≤ a_i1 ≤ ⋯ ≤ a_i(m-1) ≤ a_im ≤ b_im ≤ b_i(m-1) ≤ ⋯ ≤ b_i1 ≤ b_i0 (i = 1, 2, ⋯ , n), if a_i (r) and b_i (r), i = 1, 2, ⋯ , n are respectively defined by Expressions (1) and (2) in Theorem 2, then the unique fuzzy n-ellipsoid number u defined by a_i (r), b_i (r), i = 1, 2, ⋯ , n (see Theorem 2) is called a r-stage linear fuzzy n-ellipsoid number, and denoted as $u = F_{S L E} (\begin{matrix} r_{0} \\ r_{1} \\ ⋮ \\ r_{m} \end{matrix} : \begin{matrix} a_{10} & \dots & a_{n 0} \\ a_{11} & \dots & a_{n 1} \\ ⋮ & ⋮ \\ a_{1 m} & \dots & a_{nm} \end{matrix}; \begin{matrix} b_{10} & \dots & b_{n 0} \\ b_{11} & \dots & b_{n 1} \\ ⋮ & ⋮ \\ b_{1 m} & \dots & b_{nm} \end{matrix})$ (3) in short, $u = F_{SLE} (r : A; B)$ (4) and the collection of all r-stage linear fuzzy n-ellipsoid numbers is denoted by E_r-SL (Eⁿ). And for each such r, a r-stage linear fuzzy n-ellipsoid number is said to be a multistage Linear fuzzy n-ellipsoid number (MSLF-n-ellipsoid number) as m ≥ 2, and the collection of all multistage Linear fuzzy n-ellipsoid numbers is denoted by E_MSL (Eⁿ).

Remark 1. Let u ∈ E (Eⁿ) with u = (u₁, u₂, ⋯ , u_n) (u_i ∈ E, i = 1, 2, ⋯ , n). If for i = 1, 2, ⋯ , n, u_i is a trapezoid fuzzy number u_i = F ( a_i, b_i, c_i, d_i), then by the definitions of r-stage linear fuzzy n-ellipsoid number and trapezoid fuzzy number, we easily see that u is a r-stage linear fuzzy n-ellipsoid number $F_{S L E} (\begin{matrix} 0 \\ 1 \end{matrix} : \begin{matrix} a_{1} & a_{2} & \dots & a_{n} \\ b_{1} & b_{2} & \dots & b_{n} \end{matrix}; \begin{matrix} d_{1} & d_{2} & \dots & d_{n} \\ c_{1} & c_{2} & \dots & c_{n} \end{matrix})$ with $r = (\begin{matrix} 0 \\ 1 \end{matrix})$ .

Example 1. Let r = (0, 0.5, 0.8, 1) ^T, $A = (a_{1}, a_{2}) = (\begin{matrix} 0 & 1 \\ 0.4 & 1.4 \\ 1 & 2.2 \\ 1.8 & 3 \end{matrix})$ , $B = (b_{1}, b_{2}) = (\begin{matrix} 3 & 5 \\ 2.8 & 4.4 \\ 2.5 & 4 \\ 2 & 3.4 \end{matrix})$ . Then u = F_S L E ( r : A ; B ) is a r-stage linear fuzzy 2-ellipsoid number with r = (0, 0.5, 0.8, 1) ^T (see Fig. 1).

Fig. 1

u = F_SLE (r : A ; B).

4 Construction of multistage linear fuzzy ellipsoid numbers

As mentioned in Introduction, in order to be easy to use, the existing work basically uses rectangular, triangular and trapezoidal fuzzy numbers with very simple structure to express uncertain or imprecise information. In [29, 31], authors set up methods of constructing high dimensional fuzzy cell numbers and fuzzy ellipsoid number with the components of rectangular or triangular or trapezoidal numbers to express multichannel uncertain or imprecise information. When dealing with some multichannel uncertain or imprecise information, such as identification, ranking and classification, if the required accuracy is high, such a construction method will be insufficient in accuracy. In this Section, a construction method of multistage Linear fuzzy ellipsoid numbers will be given, so that more reasonable and objective results as processing multichannel uncertain or imprecise information can be get.

Suppose that a certain object (denoted by O) to be processed is characterized by multiple (denoted by n) uncertain digital quantity features (denoted by O₁, O₂, ⋯ , O_n in turn), and the following set of data are the statistical values of the K₁, K₂, ⋯ , K_n samples for respectively uncertain digital quantity feature O₁, O₂, ⋯ , O_n that are arbitrarily taken from the object O:

$\begin{matrix} O_{1} & O_{2} & \dots & O_{n} \\ Sample 1 : & o_{11} & o_{21} & \dots & o_{n 1} \\ Sample 2 : & o_{12} & o_{22} & \dots & o_{n 2} \\ ⋮ & ⋮ & ⋮ & ⋮ \\ Sample K_{i} : & o_{1 K_{1}} & o_{2 K_{2}} & \dots & o_{{nK}_{n}} \end{matrix}$ (5)

For any given r = (r₀, r₁, ⋯ , r_m) ^T with 0 = r₀ < r₁ < ⋯ < r_m-1 < r_m = 1 (where m < min {K_i : i = 1, 2, ⋯ , n}), in the following, a method of constructing a r-stage linear fuzzy ellipsoid number to express the object O to be processed is about to be given.

Remark 2. In the process of obtaining the above Data Set (4), the statistical data of corresponding characteristics can come from different samples, and the number of samples to be taken can also be different, that is, it is unnecessary to require K₁ = K₂ = ⋯ = K_n.

For the convenience of writing, first introduce the following notations.

For each i = 1, 2, ⋯ n, {o_ij : j = 1, 2, ⋯ , K_i} is denoted by σ (O_i). For any x ∈ R with x > 0, [x] is the integer part of the positive real number x, i.e., the maximum integer not exceeding x. For x ∈ R and l ∈ (0, + ∞), a closed interval with length l is denoted by I (l), and the closed interval with center x and length l is denoted by I (x, l), i.e., $I (x, l) = [x - \frac{l}{2}, x + \frac{l}{2}]$ . For a finite set A (where, it can be stipulated that the set A can have the same elements), and |A| is used to represent the number of all elements in set A (the same elements are recorded according to the number of repetitions. For example, | {a, b, a} |=3).

Method and Steps

Step 1: For each i = 1, 2, …, n, calculate $μ_{i} = \frac{1}{K_{i}} \sum_{j = 1}^{K_{i}} o_{ij}$ (6)

Step 2: For each i = 1, 2, ⋯ n, let $N_{i} = [\frac{K_{i}}{m}]$ , $\underline{o_{i}} = min {o_{ij} : j = 1, 2, \dots, K_{i}}$ and $\bar{o_{i}} = max {o_{ij} : j = 1, 2, \dots, K_{i}}$ , and find l_im = min{ l ∈ R⁺ : |I (l) ∩ σ (O_i) | = N_i }. Let $\begin{matrix} \underline{x_{im}} = inf {x \in [\underline{o_{i}}, \bar{o_{i}}] : | I (x, l_{im}) \cap σ (O_{i}) | = N_{i}}, \\ \bar{x_{im}} = sup {x \in [\underline{o_{i}}, \bar{o_{i}}] : | I (x, l_{im}) \cap σ (O_{i}) | = N_{i}}, \end{matrix}$ and take $a_{im} = min {μ_{i}, \underline{x_{im}}, \bar{x_{im}}} - \frac{l_{im}}{2}$ and $b_{im} = max {μ_{i}, \underline{x_{im}}, \bar{x_{im}}} + \frac{l_{im}}{2}$ .

Step 3: For each i = 1, 2, ⋯ n, find $\begin{matrix} l_{i (m - 1)} = min {l \in R^{+} : I (l) \supset [a_{im}, b_{im}], | I (l) \cap σ (O_{i}) | \\ \geq r_{m - 1} N_{i} + (1 - r_{m - 1}) K_{i}} . \end{matrix}$ Let $\begin{matrix} \underline{x_{i (m - 1)}} = inf {x \in [\underline{o_{i}}, \bar{o_{i}}] : I (x, l_{i (m - 1)}) \supset [a_{im}, b_{im}], \\ | I (x, l_{i (m - 1)}) \cap σ (O_{i}) | \geq r_{m - 1} N_{i} + (1 - r_{m - 1}) K_{i}}, \\ \bar{x_{i (m - 1)}} = sup {x \in [\underline{o_{i}}, \bar{o_{i}}] : I (x, l_{i (m - 1)}) \supset [a_{im}, b_{im}], \\ | I (x, l_{i (m - 1)}) \cap σ (O_{i}) | \geq r_{m - 1} N_{i} + (1 - r_{m - 1}) K_{i}} . \end{matrix}$ Take $a_{i (m - 1)} = \underline{x_{i (m - 1)}} - \frac{l_{i (m - 1)}}{2}$ and $b_{i (m - 1)} = \bar{x_{i (m - 1)}} + \frac{l_{i (m - 1)}}{2}$ .

Step 4: For each i = 1, 2, ⋯ n, find $\begin{matrix} l_{i (m - 2)} = min {l \in R^{+} : I (l) \supset [a_{i (m - 1)}, b_{i (m - 1)}], | I (l) \cap \\ σ (O_{i}) | \geq r_{m - 2} N + (1 - r_{m - 2}) K_{i}} . \end{matrix}$ Let $\begin{matrix} \underline{x_{i (m - 2)}} \\ = inf {x \in [\underline{o_{i}}, \bar{o_{i}}] : I (x, l_{i (m - 2)}) \supset [a_{i (m - 1)}, b_{i (m - 1)}], \\ | I (x, l_{i (m - 2)}) \cap σ (O_{i}) | \geq r_{m - 2} N_{i} + (1 - r_{m - 2}) K_{i}}, \\ \bar{x_{i (m - 2)}} \\ = sup {x \in [\underline{o_{i}}, \bar{o_{i}}] : I (x, l_{i (m - 2)}) \supset [a_{i (m - 1)}, b_{i (m - 1)}], \\ | I (x, l_{i (m - 2)}) \cap σ (O_{i}) | \geq r_{m - 2} N_{i} + (1 - r_{m - 2}) K_{i}} . \end{matrix}$ Take $a_{i (m - 2)} = \underline{x_{i (m - 2)}} - \frac{l_{i (m - 2)}}{2}$ and $b_{i (m - 2)} = \bar{x_{i (m - 2)}} + \frac{l_{i (m - 2)}}{2}$ .

⋮

Step k: For each i = 1, 2, ⋯ n, find $\begin{matrix} l_{i (m - k + 2)} = min {l \in R^{+} : I (l) \supset [a_{i (m - k + 3)}, b_{i (m - k + 3)}], \\ | I (l) \cap σ (O_{i}) | \geq r_{m - k + 2} N_{i} + (1 - r_{m - k + 2}) K_{i}} . \end{matrix}$

Let $\begin{matrix} \underline{x_{i (m - k + 2)}} \\ = inf {x \in [\underline{o_{i}}, \bar{o_{i}}] : I (x, l_{i (m - k + 2)}) \supset [a_{i (m - k + 3)}, b_{i (m - k + 3)}], \\ | I (x, l_{i (m - k + 2)}) \cap σ (O_{i}) | \geq r_{m - k + 2} N_{i} + (1 - r_{m - k + 2}) K_{i}}, \\ \bar{x_{i (m - k + 2)}} \\ = sup {x \in [\underline{o_{i}}, \bar{o_{i}}] : I (x, l_{i (m - k + 2)}) \supset [a_{i (m - k + 3)}, b_{i (m - k + 3)}], \\ | I (x, l_{i (m - k + 2)}) \cap σ (O_{i}) | \geq r_{m - k + 2} N_{i} + (1 - r_{m - k + 2}) K_{i}} . \end{matrix}$ Take $a_{i (m - k + 2)} = \underline{x_{i (m - k + 2)}} - \frac{l_{i (m - k + 2)}}{2}$ and $b_{i (m - k + 2)} = \bar{x_{i (m - k + 2)}} + \frac{l_{i (m - k + 2)}}{2}$ .

⋮

Step m+1: For each i = 1, 2, ⋯ n, find $\begin{matrix} l_{i 1} = min {l \in R^{+} : I (l) \supset [a_{i 2}, b_{i 2}], | I (l) \cap σ (O_{i}) | \\ \geq r_{1} N_{i} + (1 - r_{1}) K_{i}} . \end{matrix}$ Let $\begin{matrix} \underline{x_{i 1}} = inf {x \in [\underline{o_{i}}, \bar{o_{i}}] : I (x, l_{i 1}) \supset [a_{i 2}, b_{i 2}], \\ | I (x, l_{i 1}) \cap σ (O_{i}) | \geq r_{1} N_{i} + (1 - r_{1}) K_{i}}, \\ \bar{x_{i 1}} = sup {x \in [\underline{o_{i}}, \bar{o_{i}}] : I (x, l_{i 1}) \supset [a_{i 2}, b_{i 2}], \\ | I (x, l_{i 1}) \cap σ (O_{i}) | \geq r_{1} N_{i} + (1 - r_{1}) K_{i}} . \end{matrix}$ Take $a_{i 1} = \underline{x_{i 1}} - \frac{l_{i 1}}{2}$ and $b_{i 1} = \bar{x_{i 1}} + \frac{l_{i 1}}{2}$ .

Step m+2: For each i = 1, 2, ⋯ n, find $\begin{matrix} l_{i 0} = min {l \in R^{+} : I (l) \supset [a_{i 1}, b_{i 1}], | I (l) \cap σ (O_{i}) | \\ \geq \frac{r_{1}}{2} N_{i} + (1 - \frac{r_{1}}{2}) K_{i}} . \end{matrix}$ Let $\begin{matrix} \underline{x_{i 0}} = inf {x \in [\underline{o_{i}}, \bar{o_{i}}] : I (x, l_{i 0}) \supset [a_{i 1}, b_{i 1}], \\ | I (x, l_{i 0}) \cap σ (O_{i}) | \geq \frac{r_{1}}{2} N_{i} + (1 - \frac{r_{1}}{2}) K_{i}}, \\ \bar{x_{i 0}} = sup {x \in [\underline{o_{i}}, \bar{o_{i}}] : I (x, l_{i 0}) \supset [a_{i 1}, b_{i 1}], \\ | I (x, l_{i 0}) \cap σ (O_{i}) | \geq \frac{r_{1}}{2} N_{i} + (1 - \frac{r_{1}}{2}) K_{i}} . \end{matrix}$ Take $a_{i 0} = 2 \underline{x_{i 0}} - a_{i 1} - l_{i 0}$ and $b_{i 0} = 2 \bar{x_{i 0}} - b_{i 1} + l_{i 0}$ (i.e., a_i0 is the coordinate (on first axis) of the intersection of the straight line passing through point (a_i1, r₁) and point $(\underline{x_{i 0}} - \frac{l_{i 0}}{2}, \frac{r_{1}}{2})$ and the first axis, and b_i0 is the coordinate (on first axis) of the intersection of the straight line passing through point (b_i1, r₁) and point $(\bar{x_{i 0}} + \frac{l_{i 0}}{2}, \frac{r_{1}}{2})$ and the first axis).

Step m+3: From the constructing of a_ij and b_ij (i = 1, 2, ⋯ , n, j = 0, 1, ⋯ , m), it is easy to see that a_i0 ≤ a_i1 ≤ ⋯ ≤ a_i(m-1) ≤ a_im ≤ b_im ≤ b_i(m-1) ≤ ⋯ ≤ b_i1 ≤ b_i0 for any i = 1, 2, ⋯ , n, so u = F_SLE (r : A ; B) is a r-stage linear fuzzy ellipsoid number, and it can be used to express the object O to be processed, where A = (a₁, a₂, ⋯ , a_n), B = (b₁, b₂, ⋯ , b_n), a_i = (a_i0, a_i1, ⋯ , a_im) ^T, b_i = (b_i0, b_i1, ⋯ , b_im) ^T.

Example 2. Suppose that the object O to be processed is characterized by 3 uncertain digital quantity features O₁, O₂, O₃, and the following set of data are the statistical values of the K₁ = 25, K₂ = 20, K₃ = 30 samples for respectively uncertain digital quantity feature O₁, O₂, O₃ that are arbitrarily taken from the object O: $\begin{matrix} Sample & O_{1} & O_{2} & O_{3} \\ 1 & 9 & 6.92 & 76 \\ 2 & 16 & 5.60 & 91 \\ 3 & 13 & 6.99 & 76 \\ 4 & 20 & 3.89 & 87 \\ 5 & 14 & 2.77 & 78 \\ 6 & 22 & 4.10 & 81 \\ 7 & 17 & 5.75 & 89 \\ 8 & 16 & 5.85 & 93 \\ 9 & 14 & 3.14 & 88 \\ 10 & 20 & 3.95 & 85 \\ 11 & 13 & 3.82 & 96 \\ 12 & 14 & 5.74 & 81 \\ 13 & 21 & 6.49 & 82 \\ 14 & 12 & 4.34 & 68 \\ 15 & 9 & 6.22 & 65 \end{matrix} \begin{matrix} Sample & O_{1} & O_{2} & O_{3} \\ 16 & 17 & 5.77 & 88 \\ 17 & 16 & 2.87 & 80 \\ 18 & 14 & 5.45 & 91 \\ 19 & 18 & 4.52 & 71 \\ 20 & 9 & 6.81 & 84 \\ 21 & 9 & 85 \\ 22 & 22 & 95 \\ 23 & 9 & 81 \\ 24 & 18 & 81 \\ 25 & 10 & 93 \\ 26 & 91 \\ 27 & 68 \\ 28 & 63 \\ 29 & 69 \\ 30 & 64 \end{matrix}$ (7)

In the following, the thing to do is to construct a r-stage linear fuzzy ellipsoid number with r = (r₀, r₁, r₂, r₃, r₄) ^T = (0, 0.4, 0.7, 0.9, 1) ^T to express the object O.

Step 1: By $μ_{1} = \frac{1}{25} \sum_{j = 1}^{25} o_{1 j}$ , $μ_{2} = \frac{1}{20} \sum_{j = 1}^{20} o_{2 j}$ and $μ_{3} = \frac{1}{30} \sum_{j = 1}^{30} o_{3 j}$ , it can be get that μ₁ = 15.16, μ₂ = 5.05 and μ₃ = 81.33, respectively.

Step 2: By $N_{i} = [\frac{K_{i}}{4}]$ (i = 1, 2, 3), it is easy see that N₁ = 6, N₂ = 5 and N₃ = 7. Find $\begin{matrix} \underline{o_{1}} = min {o_{1 j} : j = 1, 2, \dots, 25} = 9, \\ \bar{o_{1}} = max {o_{1 j} : j = 1, 2, \dots, 25} = 22, \\ \underline{o_{2}} = min {o_{1 j} : j = 1, 2, \dots, 20} = 2.77, \\ \bar{o_{2}} = max {o_{1 j} : j = 1, 2, \dots, 20} = 6.99, \\ \underline{o_{3}} = min {o_{1 j} : j = 1, 2, \dots, 30} = 63, \\ \bar{o_{3}} = max {o_{1 j} : j = 1, 2, \dots, 30} = 96 . \end{matrix}$ By l_i4 = min{ l ∈ R⁺ : |I (l) ∩ σ (O_i) | = N_i } (i = 1, 2, 3), it can be found that l₁₄ = 1.00, l₂₄ = 0.38, l₃₄ = 4.00.

By $\underline{x_{i 4}} = inf {x \in [\underline{o_{i}}, \bar{o_{i}}] : | I (x, l_{i 4}) \cap σ (O_{i}) | = N_{i}}$ and $\bar{x_{i 4}} = sup {x \in [\underline{o_{i}}, \bar{o_{i}}] : | I (x, l_{i 4}) \cap σ (O_{i}) |$ = N_i} (i = 1, 2, 3), it can be calculated that $\underline{x_{14}} = 13.50$ , $\bar{x_{14}} = 13.50$ , $\underline{x_{24}} = 5.70$ , $\bar{x_{24}} = 5.70$ , $\underline{x_{34}} = 80.00$ and $\bar{x_{34}} = 82.00$ .

By $a_{i 4} = min {μ_{i}, \underline{x_{i 4}}, \bar{x_{i 4}}} - \frac{l_{i 4}}{2}$ and $b_{i 4} = max {μ_{i}, \underline{x_{i 4}}, \bar{x_{i 4}}} + \frac{l_{i 4}}{2}$ (i = 1, 2, 3), it can be get that a₁₄ = 13.00, b₁₄ = 15.66, a₂₄ = 4.86, b₂₄ = 5.89, a₃₄ = 78.00, b₃₄ = 84.00.

Step 3: By l_i3 = min { l ∈ R⁺ : I (l) ⊃ [a_i4, a_i4] , |I (l) ∩σ (O_i) |≥0.9N_i + (1 - 0.9) K_i } (i = 1, 2, 3), it can be found that l₁₃ = 3.20, l₂₃ = 1.56, l₃₃ = 7.40.

By $\underline{x_{i 3}} = inf {x \in [\underline{o_{i}}, \bar{o_{i}}] : I (x, l) \supset [a_{i 4}, b_{i 4}], | I (x, l_{i_{3}})$ ∩σ (O_i) | ≥ 0.9N_i + 0.1K_i } and $\bar{x_{i 3}} = sup {x \in [\underline{o_{i}}, \bar{o_{i}}] :$ I (x, l) ⊃ [a_i4, b_i4] , |I (x, l_{i
₃}) ∩ σ (O_i) | ≥ 0.9N_i + 0.1K_i } (i = 1, 2, 3), it can be get that $\underline{x_{13}} = 14.44$ , $\bar{x_{13}} = 14.44$ , $\underline{x_{23}} = 5.46$ , $\bar{x_{23}} = 5.46$ , $\underline{x_{33}} = 81.40$ , $\bar{x_{33}} = 81.40$ .

By $a_{i 3} = \underline{x_{i 3}} - \frac{l_{i 3}}{2}$ and $b_{i 3} = \bar{x_{i 3}} + \frac{l_{i 3}}{2}$ (i = 1, 2, 3), it can be get that a₁₃ = 12.80, b₁₃ = 16.00, a₂₃ = 4.68, b₂₃ = 6.24, a₃₃ = 77.70, b₃₃ = 85.10.

Step 4: By l_i2 = min { l ∈ R⁺ : I (l) ⊃ [a_i3, a_i3] , |I (l) ∩σ (O_i) |≥0.7N_i + (1 - 0.7) K_i } (i = 1, 2, 3), it can be found that l₁₂ = 4.86, l₂₂ = 2.50, l₃₂ = 12.60.

By $\underline{x_{i 2}} = inf {x \in [\underline{o_{i}}, \bar{o_{i}}] : I (x, l_{i_{2}}) \supset [a_{i 3}, b_{i 3}], | I (x, l_{i_{2}})$ ∩σ (O_i) | ≥ 0.7N_i + 0.3K_i } and $\bar{x_{i 2}} = sup {x \in [\underline{o_{i}}, \bar{o_{i}}] :$ I (x, l_{i
₂}) ⊃ [a_i3, b_i3] , |I (x, l_{i
₂}) ∩ σ (O_i) | ≥ 0.7N_i + 0.3K_i } (i = 1, 2, 3), it can be get that $\underline{x_{12}} = 14.58$ , $\bar{x_{12}} = 14.58$ , $\underline{x_{22}} = 5.00$ , $\bar{x_{22}} = 5.00$ , $\underline{x_{32}} = 81.90$ , $\bar{x_{32}} = 81.90$ .

By $a_{i 2} = \underline{x_{i 2}} - \frac{l_{i 2}}{2}$ and $b_{i 2} = \bar{x_{i 2}} + \frac{l_{i 2}}{2}$ (i = 1, 2, 3), it can be obtained that a₁₂ = 12.15, b₁₂ = 17.01,a₂₂ = 3.75, b₂₂ = 6.25, a₃₂ = 75.60,b₃₂ = 88.20.

Step 5: By l_i1 = min { l ∈ R⁺ : I (l) ⊃ [a_i2, a_i2] , |I (l) ∩σ (O_i) |≥0.4N_i + (1 - 0.4) K_i } (i = 1, 2, 3), it can be seen that l₁₁ = 10.00, l₂₁ = 3.26, l₃₁ = 23.00.

By $\underline{x_{i 1}} = inf {x \in [\underline{o_{i}}, \bar{o_{i}}] : I (x, l_{i_{1}}) \supset [a_{i 2}, b_{i 2}], | I (x, l_{i_{1}})$ ∩σ (O_i) | ≥ 0.4N_i + 0.6K_i } and $\bar{x_{i 1}} = sup {x \in [\underline{o_{i}}, \bar{o_{i}}] :$ I (x, l_{i
₁}) ⊃ [a_i2, b_i2] , |I (x, l_{i
₁}) ∩ σ (O_i) | ≥ 0.4N_i + 0.6K_i } (i = 1, 2, 3), it can be get that $\underline{x_{11}} = 15.00$ , $\bar{x_{11}} = 15.00$ , $\underline{x_{21}} = 4.89$ , $\bar{x_{21}} = 4.89$ , $\underline{x_{31}} = 80.50$ , $\bar{x_{31}} = 80.50$ .

By $a_{i 1} = \underline{x_{i 1}} - \frac{l_{i 1}}{2}$ and $b_{i 1} = \bar{x_{i 1}} + \frac{l_{i 1}}{2}$ (i = 1, 2, 3), it can be seen that a₁₁ = 10.00, b₁₁ = 20.00, a₂₁ = 3.26, b₂₁ = 6.52, a₃₁ = 69.00, b₃₁ = 92.00.

Step 6: By l_i0 = min { l ∈ R⁺ : I (l) ⊃ [a_i1, a_i1] , |I (l) $\cap σ (O_{i}) | \geq \frac{0.4}{2} N_{i} + (1 - \frac{0.4}{2}) K_{i}}$ (i = 1, 2, 3), it can be found that l₁₀ = 13.34, l₂₀ = 4.36, l₃₀ = 30.70.

By $\underline{x_{i 0}} = inf {x \in [\underline{o_{i}}, \bar{o_{i}}] : I (x, l_{i_{0}}) \supset [a_{i 1}, b_{i 1}], | I (x, l_{i_{0}})$ ∩σ (O_i) |≥0.2N_i + 0.8K_i } and $\bar{x_{i 0}} = sup$ { $x \in [\underline{o_{i}}, \bar{o_{i}}] :$ I (x, l_{i
₀}) ⊃ [a_i1, b_i1] , |I (x , l_{i
₀}) ∩ σ (O_i) | ≥ 0.2N_i + 0.8K_i } (i = 1, 2, 3), it can be get that $\underline{x_{10}} = 13.34$ , $\bar{x_{10}} = 13.34$ , $\underline{x_{20}} = 4.19$ , $\bar{x_{20}} = 4.19$ , $\underline{x_{30}} = 76.75$ , $\bar{x_{30}} = 76.75$ .

By $a_{i 0} = 2 \underline{x_{i 0}} - a_{i 1} - l_{i 0}$ and $b_{i 0} = 2 \bar{x_{i 0}} - b_{i 1} + l_{i 0}$ (i = 1, 2, 3), it is easy get that a₁₀ = 3.34, b₁₀ = 20.02, a₂₀ = 1.10, b₂₀ = 6.56, a₃₀ = 53.80, b₃₀ = 92.20.

Step 7: Let $\begin{matrix} A = (\begin{matrix} a_{10} & a_{20} & a_{30} \\ a_{11} & a_{21} & a_{31} \\ a_{12} & a_{22} & a_{32} \\ a_{13} & a_{23} & a_{33} \\ a_{14} & a_{24} & a_{34} \end{matrix}) = (\begin{matrix} 3.34 & 1.10 & 53.80 \\ 10.00 & 3.26 & 69.00 \\ 12.15 & 3.75 & 75.60 \\ 12.80 & 4.68 & 77.70 \\ 12.00 & 4.86 & 78.00 \end{matrix}), \\ B = (\begin{matrix} b_{10} & b_{20} & b_{30} \\ b_{11} & b_{21} & b_{31} \\ b_{12} & b_{22} & b_{32} \\ b_{13} & b_{23} & b_{33} \\ b_{14} & b_{24} & b_{34} \end{matrix}) = (\begin{matrix} 20.02 & 6.56 & 92.20 \\ 20.00 & 6.52 & 92.00 \\ 17.01 & 6.25 & 88.20 \\ 16.00 & 6.24 & 85.10 \\ 15.66 & 5.89 & 84.00 \end{matrix}) . \end{matrix}$ Then the r-stage linear fuzzy ellipsoid numbers u = F_SLE (r : A ; B) with r = (0, 0.4, 0.7, 0.9, 1) ^T can be used to express the object O to be processed.

5 Calculation of centroid of MSLF-n-ellipsoid numbers

In [36], for general multidimensional fuzzy numbers, authors introduced the centroid concept, which can characterize the fuzzy number more reasonably than the mean for special multi dimensional fuzzy numbers such as fuzzy cell numbers or fuzzy ellipsoid numbers, so that the processing method of multi channel uncertain or imprecise information such as ranking, classifying and recognizing based on the concept of the centroid will be more reasonable than that based on the concept of the mean.

Definition 2. [36] Let u ∈ Eⁿ. For any i = 1, 2, ⋯ , n, the n-dimensional real number vector (C₁ (u) , C₂ (u) , ⋯, C_n (u)) is called the centroid of multi dimensional fuzzy number u, and denoted as C (u), i.e., C (u) = (C₁ (u) , C₂ (u) , ⋯ , C_n (u)), where, $C_{i} (u) = \frac{{\overset{︷}{\int \dots \int}}_{[u]^{0}}^{n} x_{i} u (x_{1}, \dots, x_{n}) {dx}_{1} \dots {dx}_{n}}{{\overset{︷}{\int \dots \int}}_{[u]^{0}}^{n} u (x_{1}, \dots, x_{n}) {dx}_{1} \dots {dx}_{n}}$ (8)

For the convenience of calculation in application, in the following, the calculation formula of centroid for a multistage Linear fuzzy ellipsoid number u is given.

For x_i ∈ R, i = 1, 2, ⋯ , n, let $\prod_{j = 1}^{n} x_{j} = x_{1} x_{2} \dots x_{n}$ .

For x = (x₁, x₂, ⋯ , x_n) ∈ Rⁿ, Let Σ_Π₀ (x) =1, and $\begin{matrix} Σ_{Π_{l}} (x) = \sum_{1 \leq i_{1} < \dots < i_{l} \leq n} x_{i_{1}} \dots x_{i_{j}} \end{matrix}$ i.e., the sum of all the products of l non repeating numbers in the n numbers x₁, ⋯ , x_n for each l = 1, ⋯ , n.

Theorem 3. let r = (r₀, r₁, ⋯ , r_m) ^T with 0 = r₀ <r₁ < ⋯ < r_m-1 < r_m = 1, u = F_SLE (r : A ; B) ∈ E_r-SL (Eⁿ), where, A = (a₁, a₂, ⋯ , a_n), B = (b₁, b₂, ⋯ , b_n), a_i = (a_i0, a_i1, ⋯ , a_im) ^T, b_i = (b_i0, b_i1, ⋯ , b_im) ^T with a_i0 ≤ a_i1 ≤ ⋯ ≤ a_i(m-1) ≤ a_im ≤ b_im ≤ b_i(m-1) ≤ ⋯ ≤ b_i1 ≤ b_i0 (i = 1, 2, ⋯ , n). Then for any i = 1, 2, ⋯ , n, the ith component C_i (u) of centroid C (u) = (C₁ (u) , C₂ (u) , ⋯ , C_n (u)) is $C_{i} (u) = \frac{D_{i 1}^{*} (D_{i 2}^{*} + D_{i 3}^{*})}{D_{*}}$ (9)where, $\begin{matrix} D_{i 1}^{*} = \sum_{k = 1}^{m} (α_{ik}^{+} \prod_{j = 1}^{n} α_{jk}^{-}), \\ D_{i 2}^{*} = \frac{r_{k}^{n + 2} - r_{k - 1}^{n + 2}}{n + 2} + β_{ik}^{+} Σ_{Π_{n}} (β_{k}^{-}) (r_{k} - r_{k - 1}), \\ D_{i 3}^{*} = \sum_{l = 1}^{n} \frac{(β_{ik}^{+} Σ_{Π_{l - 1}} (β_{k}^{-}) + Σ_{Π_{l}} (β_{k}^{-})) (r_{k}^{n - l + 2} - r_{k - 1}^{n - l + 2})}{n - l + 2}, \\ D_{*} = \sum_{k = 1}^{m} [(\prod_{j = 1}^{n} α_{jk}^{-}) \sum_{l = 0}^{n} \frac{Σ_{Π_{l}} (β_{k}^{-}) (r_{k}^{n - l + 1} - r_{k - 1}^{n - l + 1})}{n - l + 1}] \\ α_{ik}^{+} = \frac{a_{ik} + b_{ik} - (a_{i (k - 1)} + b_{i (k - 1)})}{r_{k} - r_{k - 1}}, \\ β_{ik}^{+} = \frac{(a_{i (k - 1)} + b_{i (k - 1)}) r_{k} - (b_{ik} + a_{ik}) r_{k - 1}}{a_{ik} + b_{ik} - (a_{i (k - 1)} + b_{i (k - 1)})}, \\ α_{ik}^{-} = \frac{b_{ik} - a_{ik} - (b_{i (k - 1)} - a_{i (k - 1)})}{r_{k} - r_{k - 1}}, \\ β_{ik}^{-} = \frac{(b_{i (k - 1)} - a_{i (k - 1)}) r_{k} - (b_{ik} - a_{ik}) r_{k - 1}}{b_{ik} - a_{ik} - (b_{i (k - 1)} - a_{i (k - 1)})}, \end{matrix}$ and $β_{k}^{-} = (β_{1 k}^{-}, β_{2 k}^{-}, \dots, β_{nk}^{-})$ (k = 1, 2, ⋯ , m).

Proof. For each i = 1, 2, ⋯ , n, by Theorem 2, it is known that as r ∈ [r_k-1, r_k], $\begin{matrix} \underline{u_{i}} (r) + \bar{u_{i}} (r) \\ = a_{i (k - 1)} + (a_{ik} - a_{i (k - 1)}) \frac{r - r_{k - 1}}{r_{k} - r_{k - 1}} + b_{i (k - 1)} \\ + (b_{ik} - b_{i (k - 1)}) \frac{r - r_{k - 1}}{r_{k} - r_{k - 1}} \\ = \frac{a_{ik} + b_{ik} - (a_{i (k - 1)} + b_{i (k - 1)})}{r_{k} - r_{k - 1}} (r + \frac{(a_{i (k - 1)} + b_{i (k - 1)}) r_{k} - (b_{ik} + a_{ik}) r_{k - 1}}{a_{ik} + b_{ik} - (a_{i (k - 1)} + b_{i (k - 1)})}), \end{matrix}$

$\begin{matrix} \bar{u_{i}} (r) - \underline{u_{i}} (r)) \\ = b_{i (k - 1)} + (b_{ik} - b_{i (k - 1)}) \frac{r - r_{k - 1}}{r_{k} - r_{k - 1}} \\ - (a_{i (k - 1)} + (a_{ik} - a_{i (k - 1)}) \frac{r - r_{k - 1}}{r_{k} - r_{k - 1}}) \\ = \frac{b_{ik} - a_{ik} - (b_{i (k - 1)} - a_{i (k - 1)})}{r_{k} - r_{k - 1}} (r + \frac{(b_{i (k - 1)} - a_{i (k - 1)}) r_{k} - (b_{ik} - a_{ik}) r_{k - 1}}{b_{ik} - a_{ik} - (b_{i (k - 1)} - a_{i (k - 1)})}) . \end{matrix}$

For each i = 1, 2, ⋯ , n and k = 1, 2, ⋯ , m, by the definitions of $α_{ik}^{+}, β_{ik}^{+}, α_{ik}^{-}, β_{ik}^{-}$ , it can be obtained that for i = 1, 2, ⋯ , n, $\underline{u_{i}} (r) + \bar{u_{i}} (r) = α_{ik}^{+} (r + β_{ik}^{+})$ , as r ∈ [r_k-1, r_k], so $\begin{matrix} \prod_{j = 1}^{n} (\bar{u_{j}} (r) - \underline{u_{j}} (r)) \\ = (\prod_{j = 1}^{n} α_{jk}^{-}) \prod_{j = 1}^{n} (r + β_{jk}^{-}) \\ = (\prod_{j = 1}^{n} α_{jk}^{-}) [r^{n} + (\sum_{1 \leq j_{1} \leq n} β_{j_{1} k}^{-}) r^{n - 1} \\ + (\sum_{1 \leq j_{1} < j_{2} \leq n} β_{j_{1} k}^{-} β_{j_{2} k}^{-}) r^{n - 2} + \dots \\ + (\sum_{1 \leq j_{1} < j_{2} < \dots < j_{l} \leq n} β_{j_{1} k}^{-} β_{j_{2} k}^{-} \dots β_{j_{l} k}^{-}) r^{n - l} + \dots \\ + β_{1 k}^{-} β_{2 k}^{-} \dots β_{nk}^{-}] . \end{matrix}$ Further, it can be obtained that $\begin{matrix} (\underline{u_{i}} (r) + \bar{u_{i}} (r)) \prod_{j = 1}^{n} (\bar{u_{j}} (r) - \underline{u_{j}} (r)) \\ = (α_{ik}^{+} \prod_{j = 1}^{n} α_{jk}^{-}) [r^{n + 1} + (β_{ik}^{+} + \sum_{1 \leq j_{1} \leq n} β_{j_{1} k}^{-}) r^{n} \\ + (β_{ik}^{+} \sum_{1 \leq j_{1} \leq n} β_{j_{1} k}^{-} + \sum_{1 \leq j_{1} < j_{2} \leq n} β_{j_{1} k}^{-} β_{j_{2} k}^{-}) r^{n - 1} \\ + \dots + (β_{ik}^{+} \sum_{1 \leq j_{1} < j_{2} < \dots < j_{l} \leq n} β_{j_{1} k}^{-} β_{j_{2} k}^{-} \dots β_{j_{l} k} \\ + \sum_{1 \leq j_{1} < j_{2} < \dots < j_{l + 1} \leq n} β_{j_{1} k}^{-} β_{j_{2} k}^{-} \dots β_{j_{l + 1} k}^{-}) r^{n - l} \\ + \dots + (β_{ik}^{+} \sum_{1 \leq j_{1} < j_{2} < \dots < j_{n - 1} \leq n} β_{j_{1} k}^{-} β_{j_{2} k}^{-} \dots β_{j_{n - 1} k}^{-} \\ + β_{1 k}^{-} β_{2 k}^{-} \dots β_{nk}^{-}) r + β_{ik}^{+} β_{1 k}^{-} β_{2 k}^{-} \dots β_{nk}^{-}] \end{matrix}$ for i = 1, 2, ⋯ , n and k = 1, 2, ⋯ , m. Then for each k = 1, 2, ⋯ , m, it can be otained (by the definitions of $β_{k}^{-}$ and $Σ_{Π_{l}} (β_{k}^{-})$ ) that as r ∈ [r_k-1, r_k], $\begin{matrix} \prod_{j = 1}^{n} (\bar{u_{j}} (r) - \underline{u_{j}} (r)) \\ = (\prod_{j = 1}^{n} α_{jk}^{-}) [\sum_{l = 0}^{n} Σ_{Π_{l}} (β_{k}^{-}) r^{n - l}], \end{matrix}$ and for i = 1, 2, ⋯ , n, k = 1, 2, ⋯ , m, $\begin{matrix} (\underline{u_{i}} (r) + \bar{u_{i}} (r)) \prod_{j = 1}^{n} (\bar{u_{j}} (r) - \underline{u_{j}} (r)) \\ = (α_{ik}^{+} \prod_{j = 1}^{n} α_{jk}^{-}) [r^{n + 1} + \sum_{l = 1}^{n} (β_{ik}^{+} Σ_{Π_{l - 1}} (β_{k}^{-}) \\ + Σ_{Π_{l}} (β_{k}^{-})) r^{n - l + 1} + β_{ik}^{+} Σ_{Π_{n}} (β_{k}^{-})], \end{matrix}$ where, $β_{k}^{-} = (β_{1 k}^{-}, β_{2 k}^{-}, \dots, β_{nk}^{-})$ . Therefore, it can be get that $\begin{array}{l} \int_{0}^{1} \prod_{j = 1}^{n} (\bar{u_{j}} (r) - \underline{u_{j}} (r)) d r \\ = \sum_{k = 1}^{m} \int_{r_{k - 1}}^{r_{k}} (\prod_{j = 1}^{n} α_{j k}^{-}) [\sum_{l = 0}^{n} Σ_{Π_{l}} (β_{k}^{-}) r^{n - l}] d r \\ = \sum_{k = 1}^{m} [(\prod_{j = 1}^{n} α_{j k}^{-}) \sum_{l = 0}^{n} \frac{Σ_{Π_{l}} (β_{k}^{-}) (r_{k}^{n - l + 1} - r_{k - 1}^{n - l + 1})}{n - l + 1}] , \\ \int_{0}^{1} (\underline{u_{i}} (r) + \bar{u_{i}} (r)) \prod_{j = 1}^{n} (\bar{u_{j}} (r) - \underline{u_{j}} (r)) d r \\ = \sum_{k = 1}^{m} \int_{r _{k - 1}}^{r _{k}} ( α _{i k}^{+} \prod _{j = 1}^{n} α _{j k}^{-} ) [r^{n + 1} + \sum_{l = 1}^{n} (β _{i k}^{+} Σ _{Π _{l - 1}} (β_{k}^{-}) \\ + Σ_{Π_{l}} (β_{k}^{-}) ) r^{n - l + 1} + β_{i k}^{+} Σ_{Π_{n}} (β_{k}^{-})] d r \\ = \sum _{k = 1}^{m} ( α _{i k}^{+} \prod _{j = 1}^{n} α ^{-} _{j k} ) [\frac{r_{k} ^{n + 2} - r^{n + 2} _{k - 1}}{n + 2} + β_{i k}^{+} Σ_{Π_{n}} (β^{-} _{k}) (r _{k} - r _{k - 1}) \\ + \sum_{l = 1}^{n} \frac{(β_{i k}^{+} Σ_{Π_{l - 1}} (β_{k}^{-}) + Σ_{Π_{l}} (β_{k}^{-}) ) (r _{k}^{n - l + 2} - r _{k - 1}^{n - l + 2})}{n - l + 2} ] \end{array}$ for each i = 1, 2, ⋯ , n. By the definitions of $D_{i 1}^{*}, D_{i 2}^{*}, D_{i 3}^{*}$ (i = 1, 2, 3) and D_*, it is easy seen that $\begin{matrix} \int_{0}^{1} \prod_{j = 1}^{n} (\bar{u_{j}} (r) - \underline{u_{j}} (r)) dr = D_{*}, \\ \int_{0}^{1} (\underline{u_{i}} (r) + \bar{u_{i}} (r)) \prod_{j = 1}^{n} (\bar{u_{j}} (r) - \underline{u_{j}} (r)) dr = D_{i 1}^{*} (D_{i 2}^{*} + D_{i 3}^{*}) . \end{matrix}$ Thus, by Theorem 5.1 in [36], it can be get that $\begin{matrix} C_{i} (u) = \frac{\int_{0}^{1} (\underline{u_{i}} (r) + \bar{u_{i}} (r)) \prod_{j = 1}^{n} (\bar{u_{j}} (r) - \underline{u_{j}} (r)) dr}{2 \int_{0}^{1} \prod_{j = 1}^{n} (\bar{u_{j}} (r) - \underline{u_{j}} (r)) dr} \\ = \frac{D_{i 1}^{*} (D_{i 2}^{*} + D_{i 3}^{*})}{2 D_{*}} \end{matrix}$ for i = 1, 2, ⋯ , n. The proof of theorem is completed.

Example 3. In Example 1, it can be seen that r = (r₀, r₁, r₂, r₃) = (0, 0.5, 0.8, 1), and $\begin{matrix} A = (a_{1}, a_{2}) = (\begin{matrix} a_{10} & a_{20} \\ a_{11} & a_{21} \\ a_{12} & a_{22} \\ a_{13} & a_{23} \end{matrix}) = (\begin{matrix} 0 & 1 \\ 0.4 & 1.4 \\ 1 & 2.2 \\ 1.8 & 3 \end{matrix}), \\ B = (b_{1}, b_{2}) = (\begin{matrix} b_{10} & b_{20} \\ b_{11} & b_{21} \\ b_{12} & b_{22} \\ b_{13} & b_{23} \end{matrix}) = (\begin{matrix} 3 & 5 \\ 2.8 & 4.4 \\ 2.5 & 4 \\ 2 & 3.4 \end{matrix}) . \end{matrix}$ For each i = 1, 2 and k = 1, 2, 3, by $\begin{matrix} α_{ik}^{+} = \frac{a_{ik} + b_{ik} - (a_{i (k - 1)} + b_{i (k - 1)})}{r_{k} - r_{k - 1}}, \\ β_{ik}^{+} = \frac{(a_{i (k - 1)} + b_{i (k - 1)}) r_{k} - (b_{ik} + a_{ik}) r_{k - 1}}{a_{ik} + b_{ik} - (a_{i (k - 1)} + b_{i (k - 1)})} \\ α_{ik}^{-} = \frac{b_{ik} - a_{ik} - (b_{i (k - 1)} - a_{i (k - 1)})}{r_{k} - r_{k - 1}}, \\ β_{ik}^{-} = \frac{(b_{i (k - 1)} - a_{i (k - 1)}) r_{k} - (b_{ik} - a_{ik}) r_{k - 1}}{b_{ik} - a_{ik} - (b_{i (k - 1)} - a_{i (k - 1)})} \end{matrix}$ it can be worked out that $\begin{matrix} α_{11}^{+} = 0.4, & β_{11}^{+} = 7.5, & α_{11}^{-} = - 1.2, & β_{11}^{-} = - 2.5, \\ α_{12}^{+} = 1, & β_{12}^{+} = 2.7, & α_{12}^{-} = - 3, & β_{12}^{-} = - 1.3, \\ α_{13}^{+} = 1.5, & β_{13}^{+} = 1.533, & α_{13}^{-} = - 6.5, & β_{13}^{-} = - 1.031, \\ α_{21}^{+} = - 0.4, & β_{21}^{+} = - 15, & α_{21}^{-} = - 2, & β_{21}^{-} = - 2, \\ α_{22}^{+} = 1.333, & β_{22}^{+} = 3.85, & α_{22}^{-} = - 4, & β_{22}^{-} = - 1.25, \\ α_{23}^{+} = 1, & β_{23}^{+} = 5.4, & α_{23}^{-} = - 7, & β_{23}^{-} = - 1.057 \end{matrix}$ and $β_{1}^{-} = (β_{11}^{-}, β_{21}^{-}) = (- 2.5, - 2), β_{2}^{-} = (β_{12}^{-}, β_{22}^{-}) = (- 1.3, - 1.25), β_{3}^{-} = (β_{13}^{-}, β_{23}^{-}) = (- 1.031, - 1.057)$ and $\begin{matrix} Σ_{Π_{1}} (β_{1}^{-}) = β_{11}^{-} + β_{21}^{-} = (- 2.5) + (- 2) = - 4.5 \\ Σ_{Π_{2}} (β_{1}^{-}) = β_{11}^{-} β_{21}^{-} = (- 2.5) \times (- 2) = 5 \\ Σ_{Π_{1}} (β_{2}^{-}) = β_{12}^{-} + β_{22}^{-} = (- 1.3) + (- 1.25) = - 2.55 \\ Σ_{Π_{2}} (β_{2}^{-}) = β_{12}^{-} β_{22}^{-} = (- 1.3) \times (- 1.25) = 1.625 \\ Σ_{Π_{1}} (β_{3}^{-}) = β_{13}^{-} + β_{23}^{-} = (- 1.031) + (- 1.057) = - 2.088 \\ Σ_{Π_{2}} (β_{3}^{-}) = β_{13}^{-} β_{23}^{-} = (- 1.031) \times (- 1.057) = 1.090 \end{matrix}$ Further, it can be worked out that $\begin{array}{l} D_{11}^{*} (D_{13}^{*} + D_{12}^{*}) \\ = \sum _{k = 1}^{3} ( α _{1 k}^{+} \prod _{j = 1}^{2} α ^{-} _{j k} ) [\frac{r_{k} ^{2 + 2} - r^{2 + 2} _{k - 1}}{2 + 2} + β_{1 k}^{+} Σ_{Π_{2}} (β^{-} _{k}) (r _{k} - r _{k - 1}) \\ + \sum_{l = 1}^{2} \frac{(β_{1 k}^{+} Σ_{Π_{l - 1}} (β_{k}^{-}) + Σ_{Π_{l}} (β_{k}^{-}) ) (r _{k}^{2 - l + 2} - r _{k - 1}^{2 - l + 2})}{2 - l + 2} ] \\ =20 .2335, \\ D_{21}^{*} (D_{23}^{*} + D_{22}^{*}) \\ = \sum _{k = 1}^{3} ( α _{2 k}^{+} \prod _{j = 1}^{2} α ^{-} _{j k} ) [\frac{r_{k} ^{2 + 2} - r^{2 + 2} _{k - 1}}{2 + 2} + β_{2 k}^{+} Σ_{Π_{2}} (β^{-} _{k}) (r _{k} - r _{k - 1}) \\ + \sum_{l = 1}^{2} \frac{(β_{2 k}^{+} Σ_{Π_{l - 1}} (β_{k}^{-}) + Σ_{Π_{l}} (β_{k}^{-}) ) (r _{k}^{2 - l + 2} - r _{k - 1}^{2 - l + 2})}{2 - l + 2} ] \\ =37 .9787, \\ D_{*} \\ = \sum_{k = 1}^{3} [ (\prod_{j = 1}^{2} α_{j k}^{-}) \sum_{l = 0}^{2} \frac{Σ_{Π_{l}} (β_{k}^{-}) (r_{k}^{2 - l + 1} - r_{k - 1}^{2 - l + 1})}{2 - l + 1}] \\ =6 .4006 . \end{array}$ Therefore, by Calculation Formula (9), it is get that $\begin{matrix} C_{1} (u) = \frac{D_{11}^{*} (D_{13}^{*} + D_{12}^{*})}{2 D_{*}} = \frac{20.2335}{2 \times 6.4006} = 1.5806, \\ C_{1} (u) = \frac{D_{21}^{*} (D_{23}^{*} + D_{22}^{*})}{2 D_{*}} = \frac{37.9787}{2 \times 6.4006} = 2.9668 . \end{matrix}$ Thus, the centroid of the r-stage linear fuzzy 2-ellipsoid number $\begin{matrix} u = F_{SLE} (\begin{matrix} 0 \\ 0.5 \\ 0.8 \\ 1 \end{matrix} : \begin{matrix} 0 & 1 \\ 0.4 & 1.4 \\ 1 & 2.2 \\ 1.8 & 3 \end{matrix}; \begin{matrix} 3 & 5 \\ 2.8 & 4.8 \\ 2.5 & 4 \\ 2 & 3.4 \end{matrix}) \end{matrix}$ is C (u) = (C₁ (u) , C₂ (u)) = (1.5806, 2.9668).

6 Applications

In the above two sections, a method to construct multistage Linear fuzzy ellipsoid numbers is established, and the calculation formula of the centroid of multistage Linear fuzzy ellipsoid numbers is given. In this way, the obtained results can be applied to the fusion of multi-channel uncertain digital information, such as ranking, recognition and classification, by constructing aggregation operators in High-dimensional fuzzy number space.

In this section, the problem of ranking uncertain digital information in the specific example given in [36] is taken as an example to show the application of the results obtained in this paper.

Example 4. In Example 6.3 in [36], it is required ranking the four Manufacturing enterprises $A, B, C, D$ according to the following four evaluating evidences:

Evidence 1 (denoted by E₁): the score of Asset Operating Status of the enterprise;

Evidence 2 (denoted by E₂): the score of Product Quality of the enterprise;

Evidence 3 (denoted by E₃): the score of Production Capacity of the enterprise;

Evidence 4 (denoted by E₄): the score of Environmental Protection Evaluation of the enterprise.

Here, it is assumed that 20 experts are invited to give the corresponding scores (the percentage system is adopted) to each Manufacturing enterprise of the four Manufacturing enterprises for the four evaluating evidences. For example, the following data set is the scores of Manufacturing enterprise $A$ :

$\begin{matrix} E_{1} & E_{2} & E_{3} & E_{4} \\ Expert 1 : & 88 & 85 & 66 & 86 \\ Expert 2 : & 73 & 64 & 68 & 75 \\ Expert 3 : & 80 & 63 & 49 & 69 \\ Expert 4 : & 91 & 60 & 85 & 52 \\ Expert 5 : & 89 & 77 & 53 & 66 \\ Expert 6 : & 61 & 45 & 62 & 65 \\ Expert 7 : & 81 & 85 & 77 & 67 \\ Expert 8 : & 70 & 64 & 55 & 65 \\ Expert 9 : & 70 & 72 & 61 & 67 \\ Expert 10 : & 72 & 69 & 58 & 67 \end{matrix} \begin{matrix} E_{1} & E_{2} & E_{3} & E_{4} \\ Expert 11 : & 86 & 87 & 66 & 86 \\ Expert 12 : & 76 & 61 & 71 & 72 \\ Expert 13 : & 77 & 66 & 46 & 72 \\ Expert 14 : & 93 & 58 & 85 & 52 \\ Expert 15 : & 88 & 78 & 52 & 67 \\ Expert 16 : & 60 & 46 & 61 & 66 \\ Expert 17 : & 79 & 87 & 75 & 69 \\ Expert 18 : & 66 & 64 & 55 & 65 \\ Expert 19 : & 68 & 73 & 59 & 69 \\ Expert 20 : & 76 & 65 & 62 & 63 \end{matrix}$

First: Let r = (r₀, r₁, r₂, r₃, r₄) ^T = (0, 0.4, 0.7, 0.9, 1) ^T and N₁ = N₂ = N₃ = N₄ = 5. By the method of constructing MSLF-ellipsoid numbers proposed in Section 4 (See also Example 2), the r-stage linear fuzzy 4-ellipsoid number $u_{A} = F_{SLE} (r : \underline{A}; \bar{A})$ with r = (0, 0.4, 0.7, 0.9, 1) ^T can be constructed to express Manufacturing enterprise $A$ , where $\begin{matrix} \underline{A} = (\begin{matrix} {\underline{a}}_{10} & {\underline{a}}_{20} & {\underline{a}}_{30} & {\underline{a}}_{40} \\ {\underline{a}}_{11} & {\underline{a}}_{21} & {\underline{a}}_{31} & {\underline{a}}_{41} \\ {\underline{a}}_{12} & {\underline{a}}_{22} & {\underline{a}}_{32} & {\underline{a}}_{42} \\ {\underline{a}}_{13} & {\underline{a}}_{23} & {\underline{a}}_{33} & {\underline{a}}_{43} \\ {\underline{a}}_{14} & {\underline{a}}_{24} & {\underline{a}}_{34} & {\underline{a}}_{44} \end{matrix}) = (\begin{matrix} 51.36 & 48.40 & 28.52 & 53.78 \\ 66.00 & 56.00 & 46.48 & 60.94 \\ 66.67 & 57.24 & 53.24 & 62.79 \\ 70.00 & 59.67 & 57.36 & 64.12 \\ 75.20 & 63.00 & 58.60 & 65.00 \end{matrix}), \\ \bar{A} = (\begin{matrix} {\bar{a}}_{10} & {\bar{a}}_{20} & {\bar{a}}_{30} & {\bar{a}}_{40} \\ {\bar{a}}_{11} & {\bar{a}}_{21} & {\bar{a}}_{31} & {\bar{a}}_{41} \\ {\bar{a}}_{12} & {\bar{a}}_{22} & {\bar{a}}_{32} & {\bar{a}}_{42} \\ {\bar{a}}_{13} & {\bar{a}}_{23} & {\bar{a}}_{33} & {\bar{a}}_{43} \\ {\bar{a}}_{14} & {\bar{a}}_{24} & {\bar{a}}_{34} & {\bar{a}}_{44} \end{matrix}) = (\begin{matrix} 88.04 & 96.00 & 80.25 & 100.00 \\ 88.00 & 78.00 & 69.27 & 72.02 \\ 80.01 & 69.48 & 69.27 & 68.77 \\ 80.00 & 69.45 & 69.27 & 68.70 \\ 80.00 & 69.45 & 69.70 & 68.50 \end{matrix}) . \end{matrix}$

With the same method, u_B, u_C, u_D can be worked out to respectively express Manufacturing enterprises $B$ , $C$ and $D$ . Suppose that $u_{B} = F_{SLE} (r : \underline{B}; \bar{B})$ , $u_{C} = F_{SLE} (r : \underline{C}; \bar{C})$ and $u_{A} = F_{SLE} (r : \underline{D}; \bar{D})$ with r = (0, 0.4, 0.7, 0.9, 1) ^T and $\begin{matrix} \underline{B} = (\begin{matrix} {\underline{b}}_{10} & {\underline{b}}_{20} & {\underline{b}}_{30} & {\underline{b}}_{40} \\ {\underline{b}}_{11} & {\underline{b}}_{21} & {\underline{b}}_{31} & {\underline{b}}_{41} \\ {\underline{b}}_{12} & {\underline{b}}_{22} & {\underline{b}}_{32} & {\underline{b}}_{42} \\ {\underline{b}}_{13} & {\underline{b}}_{23} & {\underline{b}}_{33} & {\underline{b}}_{43} \\ {\underline{b}}_{14} & {\underline{b}}_{24} & {\underline{b}}_{34} & {\underline{b}}_{44} \end{matrix}) = (\begin{matrix} 55.00 & 54.50 & 53.20 & 63.10 \\ 58.00 & 57.00 & 59.00 & 65.90 \\ 67.80 & 60.30 & 62.80 & 71.20 \\ 73.50 & 65.30 & 68.00 & 75.30 \\ 75.00 & 69.50 & 70.30 & 78.60 \end{matrix}), \end{matrix}$

$\begin{matrix} \bar{B} = (\begin{matrix} {\bar{b}}_{10} & {\bar{b}}_{20} & {\bar{b}}_{30} & {\bar{b}}_{40} \\ {\bar{b}}_{11} & {\bar{b}}_{21} & {\bar{b}}_{31} & {\bar{b}}_{41} \\ {\bar{b}}_{12} & {\bar{b}}_{22} & {\bar{b}}_{32} & {\bar{b}}_{42} \\ {\bar{b}}_{13} & {\bar{b}}_{23} & {\bar{b}}_{33} & {\bar{b}}_{43} \\ {\bar{b}}_{14} & {\bar{b}}_{24} & {\bar{b}}_{34} & {\bar{b}}_{44} \end{matrix}) = (\begin{matrix} 98.45 & 95.00 & 97.37 & 99.83 \\ 95.00 & 91.20 & 93.20 & 95.20 \\ 88.30 & 84.00 & 87.18 & 93.70 \\ 85.00 & 79.50 & 84.50 & 88.60 \\ 80.00 & 73.20 & 78.90 & 85.20 \end{matrix}), \\ \underline{C} = (\begin{matrix} {\underline{c}}_{10} & {\underline{c}}_{20} & {\underline{c}}_{30} & {\underline{c}}_{40} \\ {\underline{c}}_{11} & {\underline{c}}_{21} & {\underline{c}}_{31} & {\underline{c}}_{41} \\ {\underline{c}}_{12} & {\underline{c}}_{22} & {\underline{c}}_{32} & {\underline{c}}_{42} \\ {\underline{c}}_{13} & {\underline{c}}_{23} & {\underline{c}}_{33} & {\underline{c}}_{43} \\ {\underline{c}}_{14} & {\underline{c}}_{24} & {\underline{c}}_{34} & {\underline{c}}_{44} \end{matrix}) = (\begin{matrix} 43.00 & 51.20 & 52.00 & 59.00 \\ 55.80 & 58.00 & 58.40 & 63.50 \\ 62.60 & 65.30 & 60.50 & 66.30 \\ 68.90 & 69.00 & 61.00 & 68.70 \\ 70.00 & 72.70 & 63.20 & 69.03 \end{matrix}), \\ \bar{C} = (\begin{matrix} {\bar{c}}_{10} & {\bar{c}}_{20} & {\bar{c}}_{30} & {\bar{c}}_{40} \\ {\bar{c}}_{11} & {\bar{c}}_{21} & {\bar{c}}_{31} & {\bar{c}}_{41} \\ {\bar{c}}_{12} & {\bar{c}}_{22} & {\bar{c}}_{32} & {\bar{c}}_{42} \\ {\bar{c}}_{13} & {\bar{c}}_{23} & {\bar{c}}_{33} & {\bar{c}}_{43} \\ {\bar{c}}_{14} & {\bar{c}}_{24} & {\bar{c}}_{34} & {\bar{c}}_{44} \end{matrix}) = (\begin{matrix} 98.20 & 97.30 & 91.00 & 90.60 \\ 90.00 & 95.00 & 83.50 & 82.20 \\ 86.30 & 89.70 & 78.30 & 80.20 \\ 79.40 & 81.50 & 74.00 & 78.00 \\ 75.00 & 78.30 & 69.70 & 74.20 \end{matrix}), \\ \underline{D} = (\begin{matrix} {\underline{d}}_{10} & {\underline{d}}_{20} & {\underline{d}}_{30} & {\underline{d}}_{40} \\ {\underline{d}}_{11} & {\underline{d}}_{21} & {\underline{d}}_{31} & {\underline{d}}_{41} \\ {\underline{d}}_{12} & {\underline{d}}_{22} & {\underline{d}}_{32} & {\underline{d}}_{42} \\ {\underline{d}}_{13} & {\underline{d}}_{23} & {\underline{d}}_{33} & {\underline{d}}_{43} \\ {\underline{d}}_{14} & {\underline{d}}_{24} & {\underline{d}}_{34} & {\underline{d}}_{44} \end{matrix}) = (\begin{matrix} 58.50 & 52.00 & 65.40 & 63.30 \\ 66.30 & 64.30 & 69.20 & 65.30 \\ 71.40 & 70.50 & 73.00 & 69.00 \\ 75.00 & 73.00 & 75.30 & 70.02 \\ 80.00 & 77.80 & 79.25 & 73.12 \end{matrix}), \\ \bar{D} = (\begin{matrix} {\bar{d}}_{10} & {\bar{d}}_{20} & {\bar{d}}_{30} & {\bar{d}}_{40} \\ {\bar{d}}_{11} & {\bar{d}}_{21} & {\bar{d}}_{31} & {\bar{d}}_{41} \\ {\bar{d}}_{12} & {\bar{d}}_{22} & {\bar{d}}_{32} & {\bar{d}}_{42} \\ {\bar{d}}_{13} & {\bar{d}}_{23} & {\bar{d}}_{33} & {\bar{d}}_{43} \\ {\bar{d}}_{14} & {\bar{d}}_{24} & {\bar{d}}_{34} & {\bar{d}}_{44} \end{matrix}) = (\begin{matrix} 98.12 & 99.50 & 97.50 & 91.50 \\ 96.00 & 97.00 & 96.80 & 89.20 \\ 92.30 & 93.40 & 96.80 & 87.60 \\ 86.30 & 89.80 & 93.50 & 83.00 \\ 85.00 & 88.30 & 89.50 & 79.70 \end{matrix}) . \end{matrix}$

Second: From Formula (9) (see Example 3), $\begin{matrix} C (u_{A}) = (72.0756, 70.6259, 55.5268, 73.7550), \\ C (u_{B}) = (75.3556, 81.7420, 73.1929, 79.1975), \\ C (u_{C}) = (73.4867, 75.7358, 71.1381, 85.3252), \\ C (u_{D}) = (72.5416, 81.5938, 84.0865, 82.1875) . \end{matrix}$

Third: Taking weight $p = (\frac{1}{4}, \frac{1}{4}, \frac{1}{4}, \frac{1}{4})$ , we can work out the fuzzy order $≺_{p}^{F}$ proposed in [36] (Definition 6.1 in [36]) as follows: $\begin{matrix} ≺_{p}^{F} (u_{A}, u_{B}) = 1.0000 (i . e ., ≺_{p}^{F} (u_{B}, u_{A}) = 0.0000), \\ ≺_{p}^{F} (u_{A}, u_{C}) = 1.0000 (i . e ., ≺_{p}^{F} (u_{C}, u_{A}) = 0.0000), \\ ≺_{p}^{F} (u_{A}, u_{D}) = 1.0000 : (i . e ., ≺_{p}^{F} (u_{D}, u_{A}) = 0.0000), \\ ≺_{p}^{F} (u_{B}, u_{C}) = 0.3816 (i . e ., ≺_{p}^{F} (u_{C}, u_{B}) = 0.6184), \\ ≺_{p}^{F} (u_{B}, u_{D}) = 0.8242 (i . e ., ≺_{p}^{F} (u_{D}, u_{B}) = 0.1758), \\ ≺_{p}^{F} (u_{C}, u_{D}) = 0.8216 (i . e ., ≺_{p}^{F} (u_{D}, u_{C}) = 0.1784), \end{matrix}$ so $u_{A} ≺_{p}^{1} u_{C} ≺_{p}^{0.6184} u_{B} ≺_{p}^{0.8242} u_{D} .$

Thus, a ranking result (from excellent to bad) of the four Manufacturing enterprises can be get: $D$ , $B$ , $C$ , $A$ . And it is also seen that the degrees of which $D$ is superior to $B$ , $B$ is superior to $C$ and $C$ is superior to $A$ are 0.8242, 0.6184 and 1, respectively.

In addition, if the level r = 0.6184, r = 0.8242 and r = 1 are taken, the following three dynamic rankings (from excellent to bad) are can obtained which changes with level r for the four Manufacturing enterprises: (1) First: $D$ , Second: $B$ , Third: $C$ , Fourth: $A$ as r = 0.6184; (2) First: $D$ , Second: $B$ and $C$ , Third: $A$ as r = 0.8242; (3) First: $D$ , $B$ and $C$ , Second: $A$ as r = 1.

Remark 3. 1. From the structural point of view, it is obvious that using the fuzzy number constructed here to express multi-channel uncertain or inaccurate information is more objective and reasonable than its components are triangular or trapezoidal fuzzy numbers. 2. Example 4 is just an example. The method of constructing fuzzy numbers given in this paper and other types of aggregation operators can be also used to realize the fusion of multichannel uncertain or inaccurate digital information, such as recognition, classification and clustering.

7 Conclusion

In this paper, first, a theorem (Theorem 2) was given, by which the concept of r-stage linear fuzzy n-ellipsoid numbers (Definition 1) is defined. Second, a method of constructing r-stage linear fuzzy n-ellipsoid numbers was established to express multi-channel uncertain digital information, and an example (Example 2) was given to show how to use the method to construct such numbers. Third, for the convenience of aggregation operations of high-dimensional fuzzy numbers and applications, the calculation formula (Theorem 3) of the centroid of multistage Linear fuzzy ellipsoid numbers was derived, and an example (Example 3) was given to show how to use the calculation formula to work out the centroid of a multistage Linear fuzzy ellipsoid number. At last, a practical application example (Example 4) was given to show how to use the results obtained and a fuzzy order on high dimensional fuzzy number space introduced in [36] to rank some objects which are characterized by multi-channel uncertain digital information.

It should be noted that our results and existing aggregation operators such as recognition type aggregation operators (see for example, [29, 26]) and classification or cluster type aggregation operators (see for example, [32]) can be also used to identify and classify or cluster multi-channel uncertain or inaccurate digital information.

In addition, compared with fuzzy ellipsoid numbers whose components are trapezoidal or triangular fuzzy numbers, although our method of constructing MSLF-ellipsoid numbers to express multi-channel uncertain digital information has more reasonable and accurate advantages, it also clearly has more complex computational shortcomings. In order to reduce computational complexity, the problem of constructing a multistage platform type ellipsoid numbers to express multi-channel uncertain digital information can be explored in the future.

References

Andrĺȩ-Sĺénchez

, Pricing European options with triangular fuzzy parameters: Assessing alternative triangular approximations in the Spanish stock option market, International Journal of Fuzzy Systems 20 (2018), 1624–1643.

Arotaritei

and Ionescu

, Fuzzy Voronoi diagram for dis-joint fuzzy numbers of dimension two, Journal of Intelligent and Fuzzy Systems 26 (2014), 1253–1262.

Chang

S.S.L.

and Zadeh

L.A.

, On fuzzy mappings and control, IEEE Trans Syst Man Cybernet 2 (1972), 1–34.

Chen

H.M.

, Hu

C.F.

and Yeh

W.C.

, Considers developing the fuzzy pattern of European call option under the assumption of the stock return being a Gaussian fuzzy number, Soft Computing 23(2019), 13351–13374.

Coroianu

, Gagolewski

and Grzegorzewski

, Nearest piecewise linear approximation of fuzzy numbers, Fuzzy Sets and Systems 233 (2014), 26–51.

Deng

, Comparing and ranking fuzzy numbers using ideal solutions, Applied Mathematical Modelling 38 (2014), 1638–1646.

Eslamipoor

, Haji

M.J.

and Sepehriar

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

Gong

and Hao

, Fuzzy Laplace transform based on the Henstock integral and its applications in discontinuous fuzzy systems, Fuzzy Sets and Systems 358 (2019), 1–28.

and Xuan

, A new approach for ranking fuzzy numbers based on possibility theory, Journal of Computational and Applied Mathematics 309 (2017), 674–682.

10.

Hai

, Gong

and Chen

, Weighted pseudometric approximation of 2-dimensional fuzzy numbers by fuzzy 2-cell prismoid numbers preserving the centroid, Fuzzy Sets and Systems 387(2020), 158–178.

11.

Haji

M.J.

, Zare

H.K.

, Eslamipoor

and Sepehriar

, A developed distance method for ranking generalized fuzzy numbers, Neural Comput and Applic 25 (2014), 727–731.

12.

Hong

D.H.

, The law of large numbers and renewal process forT-related weighted fuzzy numbers on R^p, Information Sciences 228 (2013), 45–60.

13.

Kim

J.D.

, Moon

E.L.

, Jeong

and Hong

D.H.

, Ranking methods for fuzzy numbers: The solution to Brunelli and Mezei’s conjecture, Fuzzy Sets and Systems 315 (2017), 109–113.

14.

Klein

, Pricing Black-Scholes options with correlated credit risk, Journal of Banking & Finance 20 (1996), 1211–1229.

15.

Kumar

, Khepar

and Yadav

, et al, A systematic review on generalized fuzzy numbers and its applications: past, present and future, Archives of Computational Methods in Engineering 29 (2022), 5213–5236.

16.

, Zeng

, Xie

and Yin

, A new fuzzy regression model based on least absoluted eviation, Engineering Applications of Artificial Intelligence 52 (2016), 54–64.

17.

and Wang

, Approximations of fuzzy numbers by usingr - s-piecewise linear fuzzy numbers based on weighted metric, Mathematics 145 (2022), 1–10.

18.

, Wang

and Li

, Approximations of n-dimensional fuzzy numbers by using α - β-knots piecewise linear fuzzy n-cell numbers, Fuzzy Sets and Systems. https://doi.org/10.1016/j.fss.2023.02.015.

19.

Mashchenko

S.O.

, Minimum of fuzzy numbers with a fuzzy set of operands, Cybernetics and Systems Analysis 58 (2022), 210–219.

20.

Moreno-Garcia

, Linares

L.J.

, Rodriguez-Benitez

and Castillo

E.D.

, Fuzzy numbers from raw discrete data using linear regression, Information Sciences 233 (2013), 1–14.

21.

Muzzioli

and Torricelli

, A multiperiod binomial model forpricing options in a vague world, Journal of Economic Dynamicsand Control 28 (2004), 861–887.

22.

Muzzioli

and Reynaerts

, American option pricing with imprecise risk-neutral probabilities, International Journal of Approximate Reasoning 49 (2008), 140–147.

23.

Muzzioli

and Torricelli

, A model for pricing an option with afuzzy payoff, Fuzzy Economic Review 5 (2009), 49–61.

24.

Pavlaçka

and Pavlačková

, On the properties of thefuzzy weighted average of fuzzy numbers with normalized fuzzy weights, Iranian Journal of Fuzzy Systems 18 (2021), 1–17.

25.

Thavaneswaran

, Appadoo

S.S.

and Frank

, Binary option pricing using fuzzy numbers, Applied Mathematics Letters 28(2013), 65–72.

26.

Wang

and Li

, The ranking of multi-dimensional uncertain information based on metrics on the fuzzy ellipsoid number space, IEEE Transactions on Fuzzy Systems 25 (2017), 614–626.

27.

Wang

, Shen

and Wang

, Approximation of fuzzy numbers byusing multi-knots piecewise linear fuzzy numbers, Journal ofIntelligent & Fuzzy Systems 39 (2020), 3597–3615.

28.

Wang

, Shi

, Agarwal

R.K.

and Shi

, On fuzzy ellipsoid numbersand membership functions, Journal of Intelligent & FuzzySystems 31 (2016), 391–403.

29.

Wang

, Shi

and Messenger

, Representation of uncertain multi-channel digital signal spaces and study of pattern recognition based on metrics and diferrence values on fuzzy n-ell number spaces, IEEE Trans on Fuzzy Systems 17 (2009), 421–439.

30.

Wang

, Shi

and Wang

, Solving fuzzy regression equation and its approximation for random fuzzy variable and their application, Soft Computing 24 (2020), 919–933.

31.

Wang

, Shi

, Wang

and Zhang

, Fuzzy n-ellipsoid numbersand representations of uncertain multi channel digital information, IEEE Trans on Fuzzy Systems 22 (2014), 1113–1126.

32.

Wang

, Shi

and Wen

, Fuzzy approximation relations on fuzzyn-cell number space and their applications in classification, Information Sciences 181 (2011), 3846–3860.

33.

Wang

and Wang

, Expressing and ranking of uncertain digital information by using multi-knots piecewise linear fuzzy numbers, International Journal of Fuzzy Systems 24 (2022), 3067–3081.

34.

Wang

, Wang

and Tang

, Fuzzy option pricing based on fuzzynumber binary tree model, IEEE Trans on Fuzzy Systems 30(2022), 3548–3558.

35.

Wang

, wang

, Qin

and Zhang

, Dynamic ranking of multi-dimensional uncertain information based on fuzzy order on the fuzzy ellipsoid number space, Journal of Intelligent & Fuzzy Systems 36 (2019), 6555–6566.

36.

Wang

and Xu

, Calculation of centroid of high dimensional fuzzy number and application, Iranian Journal of Fuzzy Systems 18 (2021), 29–44.

37.

H.C.

, Using fuzzy sets theory and Black Scholes formula to generate pricing boundaries of European options, Applied Mathematics and Computation 185 (2007), 136–146.

38.

and Liu

, A binomial tree approach to pricing vulnerable option in a vague world, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 26 (2018), 143–162.

39.

Yang

and Gong

, Ill-posedness for fuzzy Fredholm integral equations of the first kind and regularization methods, Fuzzy Sets and Systems 358 (2019), 132–149.

40.

Yazdi

H.S.

, GhasemiGol

and Effati

, Hierarchical tree clustering of fuzzy number, Journal of Intelligent & Fuzzy Systems 26 (2014), 541–550.

41.

Zhu

, Wang

and Li

, Weighted approximation of fuzzy numbersby using m - n-Step Type Fuzzy Number, (Accepted), Iranian Journal of Fuzzy Systems.