The result greyness problem of the grey relational analysis and its solution

Abstract

This paper discusses how to deal with the greyness problem in the system from the perspective of “result”. Aiming at the greyness problem of the traditional grey relational analysis result, an information fusion grey relational analysis method based on D-S evidence theory and multi solution information fusion is proposed, which mends the traditional grey relational analysis method. The results show that the method proposed in the study has better effect than the traditional grey relational analysis method, and has higher accuracy in the wear particle identification, which indicates that it can further expand the application scope of the grey relational analysis method.

Keywords

Grey relational analysis greyness of grey relational analysis D-S evidence theory information fusion

1 Introduction

A theory and method of grey system to deal with grey uncertainty [1]. At present, the grey system theory has formed a relatively complete system, and has formed a basic method with its own characteristics and strong practicality, including grey relational analysis, grey dynamic model, grey prediction method, grey situation decision-making, multi-dimensional grey assessment, multi-dimensional grey planning, grey five step modelling and grey redundancy control. In the grey system, we can understand the connotation of “grey” from two perspectives. On the one hand, from the perspective of “cause”, grey refers to the lack of sufficient information to identify the truth of the object, that is, the lack of information; on the other hand, from the perspective of “result”, grey refers to the multiple solutions (greyness) of the cognitive result. As to current literature on the grey theory academic field, it mainly focuses on dealing with the “grey” in the system from the perspective of “cause”, while there is little research on dealing with the “grey” in the system from the perspective of “result”. The purpose of this paper is to discuss how to handle the greyness problem in the system from the latter. There are many methods in the system of grey system theory, but the greyness problem of the result of the grey relational analysis is the most typical. Therefore, this paper focuses on the result greyness problem of relational analysis.

Since the grey relational analysis model was put forward, it has been a hot and active branch of grey system theory [2]. The existing research on grey relational analysis is mainly divided into two fields: theoretical research and applied research. In the field of theoretical research, many scholars have improved the grey relational analysis model proposed by Professor Deng, and have made many valuable achievements. Yuan C et al. [3] defined the approximate real starting point and calculated the generalized grey relation. The results show that the generalized grey relation has greater advantages. Balamurugan S [4] studied the multi-attribute decision-making of transesterification process by using grey relational analysis. For more research on the improvement of the grey relational analysis model, please refer to the review literature [5 –7]. In the field of applied research, many scholars have successfully applied grey relational analysis to different problems such as decision-making evaluation, identification of influencing factors, economic prediction, and so on [8]. Suvvari et al. [9] applied the grey relational analysis to evaluate the accounting performance of the Indian life insurance industry and found that the return rate with negative value played a key role in determining the financial performance of Indian life insurance companies. Song et al. [10] used the improved grey relational analysis model to study the eight pilot carbon trading markets in China and explored the relational degree of each factor from the two dimensions of space and time. Nurcan and koksl [11] developed a financial failure prediction model based on grey relational analysis, logistic regression analysis, and data envelopment analysis, analyzed the company’s financial reports on Istanbul stock exchange from 2014 to 2016, and determined the indicators affecting the company’s financial status. Xie et al. [12] applied grey relational analysis to identify the time lag between input and output variables, improving the accuracy of traffic and related emission prediction models. However, there is no special research on the result greyness problem of grey relational analysis. The above-mentioned studies mostly improved and applied the grey relational analysis method from the perspective of “causes”, without considering the greyness of the result of grey relational analysis. The greyness of the grey relational analysis result mainly comes from three aspects: (1) different grey relational operators may cause relational order drift; (2) different resolution coefficients may cause relational order drift; (3) different measurement equations may cause relational order drift. In view of this, this research will deal with the “grey” research in the system from the perspective of “results”, and study the grey problem of the results of relationship analysis.

In this paper, we discuss the result greyness problem of grey relational analysis caused by different grey relational operators. Different grey relational operators can be regarded as different dimensions of the observation object, or as a necessary mean to supplement the poor information; the result drift is naturally the result of different observation angles. In this paper, the D-S evidence theory is adopted to fuse the grey relational analysis result from different grey relational operators to obtain a systematic recognition result.

The rest of this paper is arranged as follows. In section 2, the grey relational analysis, the greyness of grey relational analysis result, and the D-S evidence theory are introduced as the preliminaries. In section 3, the information fusion method for the grey relational analysis is proposed. In section 4, the proposed method is applied to the real economic problem scenario to provide decision makers with a more systematic analysis result. Section 5 concluded the paper.

2 Preliminaries

2.1 Grey relational analysis

2.1.1 Grey relational operator

In the system analysis using the grey relational analysis model, if the meaning and dimension of the system behavior variables involved are identical, the relational degree can be directly calculated. If the meaning and dimension of the system behavior variables involved are different, before calculating the relational degree, it is necessary to use the grey relational operator to transform the system behavior variables into dimensionless data with very similar quantities. The function of the grey relational operator is to transform the original sequence data into dimensionless data with similar order of magnitude, which is a necessary basic step for grey relational analysis.

Different grey relational operators constitute the grey relational operator set. There are three kinds of grey relational operators commonly used in the previous literature, which are the averaging operator, the initialling operator, and the interval operator. The definitions of the three operators are introduced below.

Definition 1. Let sequence X_i = (x_i (1) , x_i (2) , … , x_i (q)), D₁ is the averaging operator, and X_iD₁ = (x_i (1) d₁, x_i (2) d₁, …, x_i (q) d₁), where $x_{i} (k) d_{1} = \frac{x_{i} (k)}{{\bar{X}}_{i}}, {\bar{X}}_{i} = \frac{1}{n} \sum_{k = 1}^{n} x_{i} (k), k = 1, 2, . . ., q$ (1)

Then X_iD₁ is called the image of the sequence under the averaging operator D₁, which can be called the average image for short.

Definition 2. Let sequence X_i = (x_i (1) , x_i (2) , … , x_i (q)), D₂ is the initialing operator, and X_iD₂ = (x_i (1) d₂, x_i (2) d₂, …, x_i (q) d₂), where $x_{i} (k) d_{2} = \frac{x_{i} (k)}{x_{i} (1)}, x_{i} (1) \neq 0, k = 1, 2, . . ., q$ (2)

Then X_iD₂ is called the image of the sequence under the initialling operator D₂, which can be called the initial image for short.

Definition 3. Let sequence X_i = (x_i (1) , x_i (2) , … , x_i (q)), D₃ is the interval operator, and X_iD₃ = (x_i (1) d₃, x_i (2) d₃, …, x_i (q) d₃), where $x_{i} (k) d_{3} = \frac{x_{i} (k) - min_{k} x_{i} (k)}{max_{k} x_{i} (k) - min_{k} x_{i} (k)}, k = 1, 2, . . ., q$ (3)

Then X_iD₃ is called the image of the sequence under the interval operator D₃, which can be called the interval image for short.

2.1.2 Grey relational degree

The grey relational degree generally measures the correlation degree between factors by the degree of similarity or difference of the development trend between factors. Definition 4. Let the system behaviour data sequence is X_i = (x_i (1) , x_i (2) , … , x_i (q)), where i = 0, 1, 2, . . . , p. Select X₀ = (x₀ (1) , x₀ (2) , …, x₀ (q)) as the reference sequence and select X_i = (x_i (1) , x_i (2) , …, x_i (q)) as the comparison sequence, where i = 1, 2, . . . , p. Then

$\begin{matrix} r (x_{0} (k), x_{i} (k)) \\ = \frac{min_{i} min_{k} | x_{0} (k) -_{i} (k) | + ζ max_{i} max_{k} | x_{0} (k) -_{i} (k) |}{| x_{0} (k) -_{i} (k) | + ζ max_{i} max_{k} | x_{0} (k) -_{i} (k) |} . \end{matrix}$ (4) $r (X_{0}, X_{i}) = \frac{1}{q} \sum_{k = 1}^{n} γ (x_{0} (k), x_{i} (k)), k = 1, 2, . . ., q$ (5)

Where ζ is the distinguishing coefficient and ζ ∈ [0, 1], usually ζ = 0.5 [13]. r (X₀, X_i) is called the grey relational degree between X₀ and X_i, abbreviated as r_0i; r (x₀ (k) , x_i (k)) is called the grey relational coefficient of point k, abbreviated as r_0i (k).

2.1.3 Greyness of grey relational analysis result

Grey system includes three basic concepts: grey number, greyness and grey element. Grey number refers to the number that only knows the approximate range but does not know the exact value. It is generally divided into information type, conceptual type and hierarchical type; greyness represents the measure of grey level, and the grey coefficient is generally called grey element. Among them, the greyness of the grey number can reflect the uncertainty of people’s understanding of the grey system. The greyness of grey relational analysis result means that the order of grey relational degree calculated based on different grey relational operators may have more than one result, that is, multiple solutions. It can be seen from section 2.1.1 that the grey relational images of the original sequences processed by different grey relational operators are different, so the grey relational degrees calculated according to Equation 4 and Equation 5 are not equal, which may lead to different grey relational orders based on grey relational degree ranking. This is the drift of the grey relational order caused by the different grey relational operators, which results in the result greyness problem of the relational analysis. Besides, the difference between the distinguishing coefficient and the measurement formula may also lead to the drift of the grey relational order, resulting in the result greyness problem of the grey relational analysis.

2.2 D-S evidence theory

D-S evidence theory was first proposed by Dempster in 1967 and further developed by Shafer in 1976. Therefore, it is called the Dempster-Shater evidence theory, and D-S evidence theory for short. This method has the advantages of processing subjective judgments and integrating uncertain information [14]. As an imprecise reasoning theory, the upper and lower bounds of probability are obtained by multivalued mapping. D-S evidence theory has been widely used in robotics, medical diagnosis, decision analysis and other fields to solve the problem of information uncertainty because it can better express important concepts such as “uncertainty” and “unknown”.

In D-S evidence theory, the set of all possible outcomes of a certain event that is mutually exclusive is called a frame of discernment, denoted by Θ, and a subset of the frame of discernment is called a proposition. All the subsets of Θ form the power set of Θ, denoted as 2^Θ. Assuming that Θ contains n mutually exclusive possible outcomes, the frame of discernment and its power set can be expressed as follows: ${\begin{matrix} Θ = {θ_{1}, θ_{2}, \dots, θ_{n}} \\ 2^{Θ} = {A | A \subseteq Θ} \end{matrix}$ (6)

Definition 5. Let Θ be a frame of discernment. A basic probability assignment function (also called mass function) on Θ is defined as a mapping m : 2^Θ → [0, 1], m satisfies $m (\emptyset) = 0, \sum_{A \subseteq Θ} m (A) = 1 .$ (7)

Definition 6. Let Θ be a frame of discernment, the Dempster’s combination rule is as follows:

$\begin{matrix} (m_{1} \oplus m_{2} \oplus . . . \oplus m_{n}) (A) \\ = \frac{1}{1 - K} \sum_{A_{1} \cap A_{2} . . . \cap A_{n} = A} m_{1} (A_{1}) m_{2} (A_{2}) . . . m_{n} (A_{n}) \end{matrix}$ (8) where A ⊆ Θ, A≠ ∅, and K = ∑_{A₁∩A₂...∩A_n=∅}m₁ (A₁) m₂ (A₂) . . . m_n (A_n).

3 Information fusion algorithm for grey relational analysis based D-S evidence theory

3.1 Algorithm idea

If the grey relational operators are regarded as the pieces of evidence and the comparison sequence is regarded as the frame of discernment, the D-S evidence theory can be adopted to deal with the result greyness problem of the grey relational analysis.

Suppose the comparison sequence of grey relational analysis is X ={ X₁, X₂, …, X_p }, then the comparison sequence is mapped to the frame of discernment Θ of D-S evidence theory, namely Θ ={ X₁, X₂, …, X_p }. Suppose the grey relational operator set is D = {D₁, D₂, . . . , D_k}, the corresponding grey relational degree sequence of D_k is $R^{k} = {r_{0 i}^{k} | i = 1, 2, . . ., p}$ , Rk is normalized and denoted as ${\bar{R}}^{k}$ , then there are $φ : R^{k} \to {\bar{R}}^{k}$ , $r_{0 i}^{k} \in R^{k}$ , ${\bar{r}}_{0 i}^{k} \in {\bar{R}}^{k}$ , i = 1, 2, . . . , p. $\forall r_{0 i}^{k} \in R^{k}$ : ${\begin{matrix} {\bar{r}}_{0 i}^{k} = φ (r_{0 i}^{k}) \\ φ (r_{0 i}^{k}) = {\frac{r_{0 i}^{k}}{\sum_{i = 1}^{p} r_{0 i}^{k}} | i = 1, 2, . . ., p} \end{matrix}$ (9)

Map the grey relational operator set _D to the evidence set E = {e₁, e₂, . . . , e_k} of D-S evidence theory, that is D_k → e_k, and map the sequence ${\bar{R}}^{k}$ to the mass function of evidence ek: $mass (e_{k}) = {{\bar{r}}_{0 i}^{k} |, i = 1, 2, . . ., p}$ (10)

According to equation 8, calculate the information fused grey relational degree sequence R ={R_0i|i = 1, 2, . . . , p}.

Rank the fused grey relational degree R_0i to get the information fused grey relational order.

There are two cases in the final result of information fused grey relational order:

It is consistent with the grey relational order obtained based on one or more operators in the grey relational operator set $D$ . This result indicates that the perspective of comprehensive analysis is more inclined to the ranking result of one or more grey relational operators.

It is not consistent with the grey relational order obtained based on any operators in the grey relational operator set $D$ . This result indicates that a new grey relational order is generated from the perspective of comprehensive analysis.

The purpose of the information fusion algorithm for grey relational analysis in this paper is not to obtain a standard and more accurate solution, but to provide a more systematic and comprehensive analysis perspective for the greyness problem existing in the grey relational analysis method, to provide more comprehensive and better decision support for relevant decision makers.

3.2 Algorithm steps

Step 1: Determine the reference sequence and the comparison sequences. Select X₀ = (x₀ (1) , x₀ (2) , …, x₀ (q)) as the reference sequence and X_i = (x_i (1) , x_i (2) , …, x_i (q)) as the comparison sequence, where i = 1, 2, … p.

Step 2: The grey relational operators D₁, D₂, . . . D_k are selected to perform dimensionless processing on the original reference sequence and the comparison sequence to obtain the corresponding image sequence. The image sequence obtained based on the D_k operator is: X_iD_k = (x_i (1) d_k, x_i (2) d_k, …, x_i (q) d_k), where i = 0, 1, 2, … p.

Step 3: Calculate the grey relational degree. Using Equation 4 and Equation 5 in section 2.1.2 to calculate the grey relational degree $r_{0 i}^{k}$ based on image sequences of different grey relational operators, the results are shown in Table 1.

Table 1
Value of the grey relational degree

X _i D ₁ D ₂ . . . D _k

X ₁ $r_{01}^{1}$ $r_{01}^{2}$ . . . $r_{01}^{k}$

X ₂ $r_{02}^{1}$ $r_{02}^{2}$ . . . $r_{02}^{k}$

. . . . . . . . . . . . . . .

X _p $r_{0 p}^{1}$ $r_{0 p}^{2}$ . . . $r_{0 p}^{k}$

Step 4: Normalize $r_{0 i}^{k}$ by Equations 10, and map it to the mass function of D-S evidence theory. The results are shown in Table 2.

Table 2

Value of the mass function

X _i	m (D₁)	m (D₂)	. . .	m (D_k)
X ₁	$φ (r_{01}^{1})$	$φ (r_{01}^{2})$	. . .	$φ (r_{01}^{k})$
X ₂	$φ (r_{02}^{1})$	$φ (r_{02}^{2})$	. . .	$φ (r_{02}^{k})$
. . .	. . .	. . .	. . .	. . .
X _p	$φ (r_{0 p}^{1})$	$φ (r_{0 p}^{2})$	. . .	$φ (r_{0 p}^{k})$

Step 5: Calculate the fused grey relational degree R_0i according to Equation 8, and the results are shown in Table 3.

Table 3

Value of the fused grey relational degree

X _i	m (D₁) ⊕ m (D₂) ⊕ . . . ⊕ m (D_k)
X ₁	R ₀₁
X ₂	R ₀₂
. . .	. . .
X _p	R _0p

Step 6: Rank the fused grey relational degree R_0i to obtain the final information fused grey relational order.

By synthesizing the above steps, the grey relational analysis algorithm fused with D-S evidence theory (Algorithm 1) is obtained and the pseudo-code is as follows:

Algorithm 1: The information fusion algorithm for grey relational analysis Input: Reference sequence X₀ and comparison sequence X_i

Output: Information fused grey relational order

1. For each operator D_k ∈ D

2. Calculate the image sequence X₀D_k and X_iD_k;

3. Calculate the grey relational degree sequence R^k;

4. End

5. For each grey relational degree

r_{0 i}^{k} \in R^{k}

6. Calculate

{\bar{r}}_{0 i}^{k}

;

7. Map

{\bar{r}}_{0 i}^{k}

to mass (e_k)

8. End

9. Calculate the fused grey relational degree R_0i

10. Rank R_0i to obtain the information fused grey relational order

3.3 Numerical example

Here we designed two numerical examples and compared the result of the information fusion method proposed in this paper with the results of the traditional grey relational analysis method to show the result greyness problem of the grey relational analysis mentioned in section 2.1.3. These two numerical examples correspond to the two cases of the final result of information fused grey relational order mentioned in section 3.1.

3.3.1 Numerical example 1

The numerical example 1 as shown in Table 4. X₀ is the reference sequence, and X₁ X₂ X₃ X₄ are the comparison sequences.

Table 4
Numerical example 1

X _i x_i(1) x_i(2) x_i(3) x_i(4)

X ₀ 8.532 9.071 8.962 11.053

X ₁ 2.181 2.462 2.785 3.121

X ₂ 0.440 0.493 0.474 0.496

X ₃ 0.767 0.747 0.912 1.034

X ₄ 0.247 0.254 0.286 0.313

X _i	x_i(1)	x_i(2)	x_i(3)	x_i(4)
X ₀	8.532	9.071	8.962	11.053
X ₁	2.181	2.462	2.785	3.121
X ₂	0.440	0.493	0.474	0.496
X ₃	0.767	0.747	0.912	1.034
X ₄	0.247	0.254	0.286	0.313

The calculation results are shown in Table 5. Firstly, it can be observed from Table 5 that in this example, the orders of grey relational ranking based on the averaging operator (D₁), initializing operator (D₂), and interval operator (D₃) are different, which indicates the existence of greyness of grey relational analysis result. Secondly, the order of fused grey relational degree is consistent with that obtained by the averaging operator (D₁), indicating that the result of grey relational analysis in the comprehensive perspective tends to be more inclined to the result based on the averaging operator.

Table 5

Comparison of ranking results

Comparison sequence	Averaging operator (D₁)		Initializing operator (D₂)		Interval operator (D₃)		Fused grey relational degree
	$r_{0 i}^{1}$	Order	$r_{0 i}^{2}$	Order	$r_{0 i}^{3}$	Order	R _0i	Order
X ₁	0.678	2	0.605	4	0.812	1	0.235	2
X ₂	0.620	4	0.719	2	0.697	4	0.219	4
X ₃	0.652	3	0.673	3	0.737	3	0.228	3
X ₄	0.722	1	0.770	1	0.810	2	0.317	1

3.3.2 Numerical example 2

Fine tune the data in Table 4 to get numerical example 2 (see Table 6).

Table 6
Numerical example 2

X _i x_i (1) x_i (2) x_i (3) x_i (4)

X ₀ 8.532 9.071 8.962 11.053

X ₁ 2.281 2.462 2.785 3.121

X ₂ 0.440 0.493 0.474 0.496

X ₃ 0.767 0.847 0.912 1.034

X ₄ 0.217 0.254 0.286 0.313

X _i	x_i (1)	x_i (2)	x_i (3)	x_i (4)
X ₀	8.532	9.071	8.962	11.053
X ₁	2.281	2.462	2.785	3.121
X ₂	0.440	0.493	0.474	0.496
X ₃	0.767	0.847	0.912	1.034
X ₄	0.217	0.254	0.286	0.313

Based on the averaging operator (D₁), initializing operator (D₂), interval operator (D₃), and the proposed method in this paper, the results of grey relational order is shown in Table 7.

Table 7

Comparison of ranking results

Comparison sequence	Averaging operator (D₁)		Initializing operator (D₂)		Interval operator (D₃)		Fused grey relational degree
	$r_{0 i}^{1}$	Order	$r_{0 i}^{2}$	Order	$r_{0 i}^{3}$	Order	R _0i	Order
X ₁	0.670	2	0.745	1	0.864	1	0.296	2
X ₂	0.577	4	0.744	2	0.697	4	0.206	3
X ₃	0.716	1	0.743	3	0.827	2	0.303	1
X ₄	0.618	3	0.615	4	0.749	3	0.195	4

From Table 7 it can be seen that the ranking result obtained by the grey relational analysis method fused with DS evidence theory is different from the ranking results obtained based on the grey relational operator D₁, D₂, D₃ respectively, which indicates that a new grey relational order is generated from the perspective of the comprehensive averaging operator, initializing operator and interval operator, and the recommended grey relational order from the perspective of comprehensive analysis is X₃ ≻ X₁ ≻ X₂ ≻ X₄.

To sum up, under the two numerical examples, it can be seen that the ranking result of the grey relational analysis method based on DS evidence theory is better than that based on the grey relational operator, which shows that the grey relational analysis method is more effective after DS evidence theory is integrated.

4 Two application cases

Case 1: The economic development of a region largely depends on the region’s independent innovation capabilities. Many scholars in economics-related fields have demonstrated the fact that there is a relationship between innovation input and regional economic growth, but there are few studies on the impact of different innovation input indicators on regional economic growth. At present, the international indicators for measuring the ability of independent innovation mainly include the number of people engaged in R&D activities, the number of R&D expenditures, the proportion of R&D expenditures to the gross regional product (GDP), and the number of invention patents granted. Therefore, the method proposed in this article can be used to evaluate the impact of these four indicators on the regional economy. This paper selects the statistics related to GDP and innovation input and output of each city in Jiangsu Province of China in 2012 as the data for application case [15].

Step 1: Determine the reference sequence and the comparison sequences. As shown in Table 8, the GDP of each city in Jiangsu Province in 2012 is selected as the reference sequence X₀, and the comparison sequences are the number of people engaged in R&D activities X₁, the amount of R&D expenditures X₂, the proportion of R&D expenditures to the gross regional product (GDP) X₃, and the number of invention patents granted X₄.

Table 8
GDP and innovation input output data of Jiangsu Province in 2012

City X₀/billion RMB X₁/person X₂/billion RMB X₃/% X₄/patents

Nanjing 720.157 90905.00 206.08 2.92 3010.00

Suzhou 121.165 127467.00 281.02 2.50 391.00

Wuxi 756.815 73378.00 197.78 2.65 423.00

Changzhou 396.987 47472.00 97.88 2.50 262.00

Zhenjiang 263.042 28728.00 55.50 2.31 497.00

Yangzhou 293.320 24063.00 62.02 2.05 104.00

Taizhou 270.167 17439.00 57.76 1.94 3.00

Nantong 455.867 39688.00 99.38 2.27 141.00

Xuzhou 401.658 28772.00 67.12 1.61 174.00

Yancheng 312.000 21621.00 42.34 1.50 30.00

Huaian 192.091 9093.00 22.73 1.27 69.00

Lianyungang 160.342 10381.00 22.44 1.40 60.00

Suqian 152.203 7455.00 15.07 1.02 3.00

City	X₀/billion RMB	X₁/person	X₂/billion RMB	X₃/%	X₄/patents
Nanjing	720.157	90905.00	206.08	2.92	3010.00
Suzhou	121.165	127467.00	281.02	2.50	391.00
Wuxi	756.815	73378.00	197.78	2.65	423.00
Changzhou	396.987	47472.00	97.88	2.50	262.00
Zhenjiang	263.042	28728.00	55.50	2.31	497.00
Yangzhou	293.320	24063.00	62.02	2.05	104.00
Taizhou	270.167	17439.00	57.76	1.94	3.00
Nantong	455.867	39688.00	99.38	2.27	141.00
Xuzhou	401.658	28772.00	67.12	1.61	174.00
Yancheng	312.000	21621.00	42.34	1.50	30.00
Huaian	192.091	9093.00	22.73	1.27	69.00
Lianyungang	160.342	10381.00	22.44	1.40	60.00
Suqian	152.203	7455.00	15.07	1.02	3.00

Step 2: Three most commonly used grey relational operators: the averaging operator (D₁), the initializing operator (D₂), and the interval operator (D₃), are applied to perform dimensionless processing on the reference sequence and the comparison sequence. The image sequences obtained are shown in Tables 9 –11 respectively.

Table 9

The average image

City	X ₀	X ₁	X ₂	X ₃	X ₄
Nanjing	2.082	2.245	2.183	1.463	7.573
Suzhou	0.350	3.148	2.977	1.253	0.984
Wuxi	2.188	1.812	2.095	1.328	1.064
Changzhou	1.148	1.172	1.037	1.253	0.659
Zhenjiang	0.761	0.709	0.588	1.158	1.250
Yangzhou	0.848	0.594	0.657	1.027	0.262
Taizhou	0.781	0.431	0.612	0.972	0.008
Nantong	1.318	0.980	1.053	1.138	0.355
Xuzhou	1.161	0.710	0.711	0.807	0.438
Yancheng	0.902	0.534	0.449	0.752	0.075
Huaian	0.555	0.225	0.241	0.636	0.174
Lianyungang	0.464	0.256	0.238	0.702	0.151
Suqian	0.440	0.184	0.160	0.511	0.008

Table 10

The initial image

City	X ₀	X ₁	X ₂	X ₃	X ₄
Nanjing	1.000	1.000	1.000	1.000	1.000
Suzhou	0.168	1.402	1.364	0.856	0.130
Wuxi	1.051	0.807	0.960	0.908	0.141
Changzhou	0.551	0.522	0.475	0.856	0.087
Zhenjiang	0.365	0.316	0.269	0.791	0.165
Yangzhou	0.407	0.265	0.301	0.702	0.035
Taizhou	0.375	0.192	0.280	0.664	0.001
Nantong	0.633	0.437	0.482	0.777	0.047
Xuzhou	0.558	0.317	0.326	0.551	0.058
Yancheng	0.433	0.238	0.205	0.514	0.010
Huaian	0.267	0.100	0.110	0.435	0.023
Lianyungang	0.223	0.114	0.109	0.479	0.020
Suqian	0.211	0.082	0.073	0.349	0.001

Table 11

The interval image

City	X ₀	X ₁	X ₂	X ₃	X ₄
Nanjing	0.942	0.695	0.718	1.000	1.000
Suzhou	0.000	1.000	1.000	0.779	0.129
Wuxi	1.000	0.549	0.687	0.858	0.140
Changzhou	0.434	0.333	0.311	0.779	0.086
Zhenjiang	0.223	0.177	0.152	0.679	0.164
Yangzhou	0.271	0.138	0.177	0.542	0.034
Taizhou	0.234	0.083	0.161	0.484	0.000
Nantong	0.527	0.269	0.317	0.658	0.046
Xuzhou	0.441	0.178	0.196	0.311	0.057
Yancheng	0.300	0.118	0.103	0.253	0.009
Huaian	0.112	0.014	0.029	0.132	0.022
Lianyungang	0.062	0.024	0.028	0.200	0.019
Suqian	0.049	0.000	0.000	0.000	0.000

Step 3: Calculate the grey relational degree $r_{0 i}^{k}$ based on the three image sequences respectively. The calculation results are shown in Table 12.

Table 12

Value of the grey relational degree

X _i	D ₁	D ₂	D ₃
X ₁	0.889	0.786	0.765
X ₂	0.899	0.800	0.786
X ₃	0.906	0.761	0.772
X ₄	0.784	0.676	0.743

Step 4: Normalize $r_{0 i}^{k}$ by Equations 10, and map it to the mass function of D-S evidence theory. The results are shown in Table 13.

Table 13

Value of the mass function

X _i	m (D₁)	m (D₂)	m (D₃)
X ₁	0.256	0.260	0.249
X ₂	0.258	0.265	0.256
X ₃	0.261	0.252	0.252
X ₄	0.225	0.224	0.242

Step 5: Calculate the fused grey relational degree R_0i according to Equation 8, and the results are shown in Table 14.

Table 14

Value of the fused grey relational degree

X _i	Fused grey relational degree
X ₁	0.264
X ₂	0.279
X ₃	0.263
X ₄	0.194

Step 6: Rank and compare the above calculation results. The results are shown in Table 15.

Table 15

Comparison of ranking results

Comparison sequence	Averaging operator (D₁)		Initializing operator (D₂)		Interval operator (D₃)		Fused grey relational degree
	$r_{0 i}^{1}$	Order	$r_{0 i}^{2}$	Order	$r_{0 i}^{3}$	Order	R _0i	Order
$r_{0 i}^{1}$	Order	$r_{0 i}^{2}$	Order	$r_{0 i}^{3}$	Order	R _0i	Order
X ₁	0.890	3	0.786	2	0.765	3	0.264	2
X ₂	0.899	2	0.801	1	0.786	1	0.279	1
X ₃	0.906	1	0.761	3	0.772	2	0.263	3
X ₄	0.784	4	0.676	4	0.743	4	0.194	4

It can be seen from Table 15 that in this application case, the result of the grey relational analysis method fused with the D-S evidence theory is more inclined to the result based on the initializing operator, which means that the amount of R&D expenditure has the greatest impact on GDP growth, followed by the number of people engaged in R&D activities on GDP growth, and the proportion of R&D expenditures in GDP has the third-largest impact on GDP growth, and the number of patents granted for inventions has the least impact on GDP growth.

Case 2: Wear particles can directly reflect the wear process and wear state of the engine. Accurate identification of wear particles is a key link to achieve engine fault diagnosis and state detection. As a single intelligent method has certain limitations in wear particle identification, the grey relational analysis method integrating D-S evidence theory is used to intelligently identify wear particles, and the identification effect is compared with the traditional grey relational analysis method.

Table 16

Identification Results and Accuracy of Different Methods

Abrasive grain	Grey Relational Analysis Method Based on D-S Evidence Theory	Traditional Grey Relational Analysis Method
Normal abrasive particles	0.1266	0.1198
Strict slippery	0.1412	0.1641
Cutting abrasive	0.1228	0.1148
Fatigue abrasive	0.1251	0.1359
Flaky abrasive	0.1267	0.1203
Spherical abrasive	0.1218	0.1197
Oxide	0.1203	0.1120
Recognition accuracy	96.89%	90.64%

The results show that, compared with the traditional grey relational analysis method, the grey relational analysis method based on D-S evidence theory can improve the discrimination and accuracy of wear particle identification, and has good universality, adaptability and fault tolerance, which can provide effective reference for engine wear identification.

5 Discussion and conclusion

Grey relational analysis is a very active branch of grey system theory. It mainly judges the connection tightness between different sequences according to the similarity of the geometric shape of sequence curves, which is more suitable for dynamic history analysis. For the grey coefficients with clear information and unclear information, the grey relational analysis can be used for analysis and discussion, and the correlation order of various information can be described by using the relational degree. D-S evidence theory has the advantages of processing subjective judgment and integrating uncertain information, and can better express important concepts such as “uncertainty” and “unknown”. It has been widely used in robotics, medical diagnosis, decision analysis and other fields to solve the problem of information uncertainty.

Aiming at the grey problem of results caused by different grey relational operators, an information fusion method for grey relational analysis is proposed in this study. This method takes grey relation operator as evidence, comparison sequence as identification framework, and then uses D-S evidence theory to deal with grey problems. Two numerical examples and a practical application example show that the information fusion method of grey relational analysis can conduct more systematic and comprehensive analysis and provide better decision support for relevant decision-makers. In terms of accurate identification of wear particles, the identification accuracy of the grey relational analysis method combined with D-S evidence theory and the traditional grey relational analysis method is 96.89% and 90.64% respectively. It can be explained that the information fusion method of grey relational analysis proposed in this paper solves the grey problem of the results of grey relational analysis to a certain extent, and expands the application scope of grey relational analysis method. In the future, research will be carried out on the accuracy of sequences and equations.

Conflicts of interest

The authors declare that there is no conflict of interest regarding the publication of this paper.

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The result greyness problem of the grey relational analysis and its solution

Abstract

Keywords

1 Introduction

2 Preliminaries

2.1 Grey relational analysis

2.1.1 Grey relational operator

2.2 D-S evidence theory

3.1 Algorithm idea

Table 1 Value of the grey relational degree X i D 1 D 2 . . . D k X 1 r 01 1 r 01 2 . . . r 01 k X 2 r 02 1 r 02 2 . . . r 02 k . . . . . . . . . . . . . . . X p r 0 p 1 r 0 p 2 . . . r 0 p k

3.3.1 Numerical example 1

Table 4 Numerical example 1 X i x i (1) x i (2) x i (3) x i (4) X 0 8.532 9.071 8.962 11.053 X 1 2.181 2.462 2.785 3.121 X 2 0.440 0.493 0.474 0.496 X 3 0.767 0.747 0.912 1.034 X 4 0.247 0.254 0.286 0.313

Table 6 Numerical example 2 X i x i (1) x i (2) x i (3) x i (4) X 0 8.532 9.071 8.962 11.053 X 1 2.281 2.462 2.785 3.121 X 2 0.440 0.493 0.474 0.496 X 3 0.767 0.847 0.912 1.034 X 4 0.217 0.254 0.286 0.313

Conflicts of interest

References

Table 1
Value of the grey relational degree

X _i D ₁ D ₂ . . . D _k

X ₁ $r_{01}^{1}$ $r_{01}^{2}$ . . . $r_{01}^{k}$

X ₂ $r_{02}^{1}$ $r_{02}^{2}$ . . . $r_{02}^{k}$

. . . . . . . . . . . . . . .

X _p $r_{0 p}^{1}$ $r_{0 p}^{2}$ . . . $r_{0 p}^{k}$

Table 4
Numerical example 1

X _i x_i(1) x_i(2) x_i(3) x_i(4)

X ₀ 8.532 9.071 8.962 11.053

X ₁ 2.181 2.462 2.785 3.121

X ₂ 0.440 0.493 0.474 0.496

X ₃ 0.767 0.747 0.912 1.034

X ₄ 0.247 0.254 0.286 0.313

Table 6
Numerical example 2

X _i x_i (1) x_i (2) x_i (3) x_i (4)

X ₀ 8.532 9.071 8.962 11.053

X ₁ 2.281 2.462 2.785 3.121

X ₂ 0.440 0.493 0.474 0.496

X ₃ 0.767 0.847 0.912 1.034

X ₄ 0.217 0.254 0.286 0.313