The grey decision model and its application based on generalized greyness of interval grey number

Abstract

Purpose

In order to solve the decision-making problem that the attributive weight and attributive value are both interval grey numbers, this paper tries to construct a multi-attribute grey decision-making model based on generalized greyness of interval grey number.

Design/methodology/approach

Firstly, according to the nature of the generalized gresness of interval grey number, the generalized weighted greyness distance between interval grey numbers is given, and the transformation relationship between greyness distance and real number distance is analyzed. Then according to the objective function that the square sum of generalized weighted greyness distances from the decision scheme to the best scheme and the worst scheme is the minimum, a multi-attribute grey decision-making model is constructed, and the simplified form of the model is given. Finally, the grey decision-making model proposed in this paper is applied to the evaluation of technological innovation capability of 6 provinces in China to verify the effectiveness of the model.

Findings

The results show that the grey decision-making model proposed in this paper has a strict mathematical foundation, clear physical meaning, simple calculation and easy programming. The application example shows that the grey decision model in this paper is feasible and effective. The research results not only enrich the grey system theory, but also provide a new way for the decision-making problem that the attributive weights and attributive values are interval grey numbers.

Practical implications

The decision-making model proposed in this paper does not need to seek the optimal solution of the attributive weight and the attributive value, and can save the decision-making labor and capital investment. The model in this paper is also suitable for the decision-making problem that deals with the coexistence of interval grey numbers and real numbers.

Originality/value

The paper succeeds in realizing the multi-attribute grey decision-making model based on generalized gresness and its simplified forms, which provide a new method for grey decision analysis.

Citation

Li, L. and Li, X. (2024), "The grey decision model and its application based on generalized greyness of interval grey number", Grey Systems: Theory and Application, Vol. 14 No. 4, pp. 641-670. https://doi.org/10.1108/GS-01-2024-0003

1. Introduction

Multi-attribute decision-making is to synthesize the multi-attributive eigenvalues into a decision value, and then rank the decision schemes and select the optimal decision scheme. The research content of multi-attribute decision-making mainly includes the preprocessing of attributive value, the determination of attributive weight, the construction of decision model and its optimization method (Xu, 2004). Many scholars have conducted in-depth discussions on the decision-making problem with the attributive value of real numbers and interval numbers, and obtained rich results of theoretical research and practical application (Liu et al., 2014; Sun, 2020).

In practical decision-making problems, due to the complexity of the decision-making environment and influencing factors, it is often difficult to give the exact value of the attribute but only the range of its value. This kind of number that only knows the value range but not its exact value is called a grey number (Liu, 2021). When the attribute values are grey numbers, the decision analysis needs to adopt the grey decision-making method. The grey decision-making is one of the research contents of grey system theory, and the main methods include grey relational decision-making, grey target decision-making, grey clustering decision-making and grey situation decision-making and so on. By searching the literature, it is found that the first three methods of grey decision-making have more research results, while the research results of grey situation decision-making are relatively few. Because of the lack of scientific expression of grey number, the early research of grey decision-making mainly used the theory and method of grey system, such as grey relational analysis and grey clustering, to solve the decision-making problem of attribute value being real number. With the deepening of research, scholars gradually attach importance to the decision-making problem of attribute values being interval grey numbers (Luo and Liu, 2005), and put forward some new decision-making models one after another, for example, the grey decision model based on the degree of separation (Zhong et al., 2008), the grey decision model based on the comprehensive degree of proximity (Xi and Wei, 2019), the multi-attribute decision-making method based on the Choquet integral (Wang and Dang, 2015), the grey association decision-making model based on the area (Zeng et al., 2013; Jiang et al., 2015) and so on. However, these studies treat interval grey numbers as interval numbers and fail to make use of all the information of interval grey numbers. In 2010, Professor Liu Sifeng proposed using “Kernel and greyness” to express interval grey number and using “Kernel and greyness” to carry out interval grey number operation (Liu et al., 2010), which has solved the problem of interval grey number operation to some extent. Subsequently, some scholars use the “Kernel and greyness” of interval grey number in the research of grey decision-making, and put forward the method of multi-attribute decision-making based on the kernel and greyness of interval grey number (Liu et al., 2017; Li et al., 2017), etc., but the decision also needs to compare the size of the interval grey number. In order to improve the reliability of decision-making, some scholars introduce regret theory and prospect theory into grey decision-making, such as the grey target decision model based on regret theory (Guo et al., 2018), the multi-attribute decision-making model based on prospect theory and evidential reasoning (Xiong and Wang, 2019), the grey multi-attribute group decision-making method considering the psychological behavior of decision-makers (Luo et al., 2021), the grey off-target deviation degree method for group decision-making with regret aversion (Qian et al., 2021) and so on. At present, some new grey decision models are emerging, such as the dynamic grey target decision model based on interval grey number (Zhou et al., 2020), the decision model based on clustering comprehensive evaluation value (Zhao et al., 2022), the multi-objective grey hierarchical group consensus model (Gu et al., 2023), three-parameter interval grey number dynamic TOPSIS model (Li et al., 2023), and the increasing information and taking large method (Li and Li, 2023a) and so on. The grey decision theory and method have been widely used to solve practical engineering problems, for example, impact analysis and classification of aircraft functional failures (Zhang et al., 2021), drought grade assessment (Luo et al., 2022), evaluation of rural tourism development level (Yan et al., 2023), evaluation of regional green innovation vitality (Zhou et al., 2023) and so on. Although these results enrich the theory and method of grey decision-making, they are not based on the scientific expression of interval grey numbers.

Because the interval grey number is different from the interval number, if the interval grey number is treated as the interval number, the information may be lost or the decision result may not be consistent with the reality (Li and Li, 2022a; Jiang et al., 2017). Therefore, the scientific expression of interval grey number is an important basis of constructing grey decision model. Greyness is an index to measure the uncertainty of interval grey number. The greater the greness, the greater the uncertainty of interval grey number is; otherwise, the opposite. Dialectically speaking, the grey uncertainty of interval grey number is absolute from the micro (range of values) and relative from the macro (background domain). Therefore, according to the principles of syllogism and dialectics, the authors in this paper give the definition of the generalized greyness of interval grey number, and propose to use a three-dimensional greyness vector to express interval grey number and establish the greyness relational degree model for the attributive index with interval grey number (Li and Li, 2022b). Then, the properties of the generalized greyness of interval grey number are further analyzed, and the conservation principle of the generalized greyness is given (Li and Li, 2023b). The generalized greyness of interval grey numbers can not only express interval grey numbers scientifically but also provide a theoretical basis for ranking interval grey number (Li and Li, 2023c), grey evaluation, grey decision and grey prediction (Ren et al., 2022; Yu et al., 2023) and so on.

Compared to decisions with exact values, the uncertain decision-making problem is complicated when the attributive values are interval grey numbers. If the attributive weight and attributive value are both interval grey numbers, the uncertain decision-making analysis will be more complicated. By searching the literature, this research has not been reported so far. The generalized greyness provides a new expression way for interval grey number scientifically, but how to use generalized greyness to solve the decision-making problem with grey uncertainty needs further study. Therefore, according to the generalized greyness of the interval grey number, this paper firstly gives the greyness distance between interval grey numbers and analyzes its properties. Then, according to the objective function that the square sum of the generalized weighted greyness distances from the decision scheme to the best decision and the worst decision is the minimum, the grey decision-making model with interval grey numbers for the attributive value and weight is constructed. Finally, the evaluation example of technical innovation capability of 6 provinces in China and two compared cases are used to verify the effectiveness of the model proposed in this paper.

The main contributions of this article are as follows. (1) The definition of greyness distance of the multi-attribute with interval grey number is put forward based on the generalized greyness, and the relationship between the greyness distance and real distance is revealed. (2) According to the greyness distance of interval grey number, the multi-attribute grey decision-making model is proposed, which provides a new way to deal with the decision-making problem that attributive weight and attributive value are both interval grey numbers and the coexistence of interval grey numbers and real numbers.

The remainder of this research is organized as follows. In section 2, the definition and properties of greyness distance of interval grey numbers are put forward, and the relationship between the greyness distance and real distance is analyzed. In section 3, the multi-attribute grey decision-making model based on the greyness distance is proposed, and its simplified forms are given. In section 4, the application and comparison examples are provided to illustrate the advantages and contributions of this study. Finally, the conclusions are summarized in section 5.

2. The greyness distance and its property of interval grey number

This section first introduces the concept of generalized greyess of interval grey number, then gives the greyness distance between interval grey numbers and analyzes its properties, which provides a basis for establishing the multi-attribute grey decision making model.

Grey number is the basic element of grey mathematics, but it is different from interval number. Grey number is divided into discrete grey number and interval grey number, and interval grey number is divided into the continuous interval grey number and discontinuous interval grey number (Liu, 2021). Without explanation, the grey number described below refers to the continuous interval grey number without value information distribution.

2.1 The generalized greyness of interval grey number

Definition 1.

Li and Li (2022b) suppose the bounded background domain of interval grey number is Ω=[p,q], for ∀⊗∈[a,b]⊂Ω, a,b,p,q are real numbers, and p≤a<b≤q, if

(1)g∘(⊗1)=μ(⊗1)μ(Ω),g∘(⊗2)=μ(⊗2)μ(Ω),g∘(⊗3)=μ(⊗3)μ(Ω)

They are called the greyness of the lower bound domain ⊗1, value domain ⊗2 and upper bound domain ⊗3 of interval grey number ⊗ respectively. They are collectively called the generalized greyness of interval grey number ⊗, also called the three-dimensional greyness of interval grey number. ⊗1, ⊗2 and ⊗3 are called the lower bound domain, value domain and upper bound domain of interval grey number ⊗ respectively, and they are collectively called the three domains of interval grey number, and ⊗1=[p,a], ⊗2=[a,b], ⊗3=[b,q]. Among, μ(⊗1), μ(⊗2) and μ(⊗3) are called the measure of the lower bound domain, value domain and upper bound domain of interval grey number ⊗ respectively, and they are collectively called the generalized measure of interval grey number, and μ(⊗1)=a−p, μ(⊗2)=b−a, μ(⊗3)=q−b. μ(Ω) is the measure of the background domain Ω, and μ(Ω)=q−p.

Definition 1 shows that although the exact value of interval grey number ⊗ is uncertain, the lower bound domain, value domain and upper bound domain of interval grey number ⊗ are definite interval numbers. Generally speaking, the three domains of different interval grey numbers are different and varied, so this paper also uses the grey number symbol ⊗ to represent the three domains of interval grey number.

As we all know, the occurrence of random events is uncertain, but the probability can be used to accurately express the likelihood of their occurrence, and provide a basis for the analysis of decision-making and prediction of random events. Similarly, using generalized greyness to express interval grey number has similar characteristics. It can not only make full use of all information of interval grey number, but also transform uncertainty analysis into certainty analysis.

The greyness of three domains of interval grey numbers is not only interlinked and relative, but also opposite, and is both static and dynamic. This is the dialectical philosophical basis of interval grey number analysis. The generalized greyness of interval grey numbers not only embodies the relationship between the whole (background domain) and the local (three domains), but also embodies the relationship between certainty and uncertainty.

Theorem 1.

Li and Li (2023b) stated that for bounded background domain, infinite background domain, infinitesimal background domain, and dynamically varying interval grey number and background domain, the sum of greyness of three domains for any interval grey number ⊗ is equal to 1, namely

(2)g∘(⊗1)+g∘(⊗2)+g∘(⊗3)=1

Theorem 1 shows that the generalized greyness of interval grey numbers is not only consistent with the grey immortal axiom (Liu, 2021), but also conserved. Therefore, Theorem 1 is called the conservation principle of generalized greyness of the interval grey number.

Definition 2.

Li and Li (2023c) suppose any interval grey number ⊗∈Ω∈R, R represents the real number domain, g∘(⊗1),g∘(⊗2) and g∘(⊗3) are the greyness of three domains of interval grey number ⊗ respectively. If let x=g∘(⊗1), y=g∘(⊗2), z=g∘(⊗3), then the plane x+y+z=1 is called the greyness space plane of interval grey number.

From definition 2, the essence of the unity of the generalized greyness is to map the interval grey numbers to the points on the same greyness space plane, realizing the transformation and dimension expansion of the interval grey number (from 1 dimension to 3 dimensions), and realizing the relative quantitative and precise expression of the grey property of interval grey number.

If the bounded background domain Ω is given, then any interval grey number ⊗ or real number (whitening value of grey number) can be represented by the generalized greyness vector with real number form, that is

(3)g∘(⊗)=(g∘(⊗1),g∘(⊗2),g∘(⊗3))

Therefore, the generalized greyness is a scientific expression of interval grey number, and it provides a theoretical basis for the comprehensive analysis of interval grey number. An important property of generalized greyness of interval grey number is given as follows.

Theorem 2.

In the bounded background domain Ω, the generalized greyness of interval grey numbers is a linear transformation, which can automatically convert any interval grey number ⊗[a,b] to the interval grey number ⊗[a′,b′] in interval [0, 1], and a′=g∘(⊗1), b′=g∘(⊗1)+g∘(⊗2).

Proof It can be certified by using definition 1 (omitted).

From theorem 2 and definition 1, it can be seen that the generalized greyness of interval grey number can not only convert any interval grey number on the bounded background domain to the interval grey number on the interval [0, 1] automatically, but also realize the dimensionless processing of attribute value automatically.

2.2 The greyness distance and its properties

The distance is a commonly used measurement form to measure the degree of difference between decision schemes. The generalized greyness maps the interval grey numbers to the points on the same greyness space plane, and the distance between two points on the greyness space plane can naturally reflect the difference between the two interval grey numbers. Therefore, based on the generalized greyness of interval grey number, the greyness distance of interval grey numbers is given, and the relationship between the greyness distance and real number distance is analyzed as follows.

Definition 3.

Li and Li (2022b) suppose there are two interval grey numbers ⊗xj,⊗xk∈Ω, and their generalized greyness vectors are as follows respectively.

g∘(⊗xj)=(g∘(⊗r1j),g∘(⊗r2j),g∘(⊗r3j))

g∘(⊗xk)=(g∘(⊗r1k),g∘(⊗r2k),g∘(⊗r3k))

and 0≤g∘(⊗rlj)≤1, 0≤g∘(⊗rlk)≤1, ∑l=13g∘(⊗rlj)=1, ∑l=13g∘(⊗rlk)=1, l=1,2,3. If

(4)djk=‖g∘(⊗rj)−g∘(⊗rk)‖={λ∑l=13|g∘(⊗rlj)−g∘(⊗rlk)|α}1/α

Then it is called the greyness distance between the interval grey numbers ⊗xj and ⊗xk. ‖x‖ represents the norm, and α is the distance parameter, usually taken as α=1 or α=2. When α=1, equation (4) is called the Hamming greyness distance; when α=2, equation (4) is called the Euclidean greyness distance. λ is the conversion coefficient between the greyness distance and real number distance, usually taken as λ=0.5.

From definition 3, the greyness distance is essentially the distance between two points on the greyness space plane x+y+z=1. Compared with the ordinary distance in Euclidean space, the greyness distance can be called the sub-Euclidean space distance. Therefore, the conversion coefficient λ=0.5 is a bridge between the greyness distance and real number distance.

Theorem 3.

If the greyness distance of interval grey numbers ⊗j,⊗k satisfies definition 3, then

1. dkk=0.

2. djk=dkj.

3. 0≤djk≤1.

Proof From equation (4), the properties (1) and (2) hold. It is easy to prove that the property (3) holds (omitted).

Definition 3 provides the greyness distance between two interval grey numbers without considering weights, but in practice, decision-making is often based on multiple attributes, and the attributive weight needs to be considered. Due to the complexity of influencing factors and the limitation of people’s cognitive level, it is often difficult to provide an accurate value of the weight, and only the approximate range of the value can be obtained. For the case where the attributive weight and attributive value are both interval grey numbers, the definition of grey weight is put forward, and then the generalized weighted greyness distance is given and its properties are analyzed as follows.

Definition 4.

Suppose the weight vector of m attributes is as follows.

(5)⊗W=(⊗w1,⊗w2,⋯,⊗wm)

⊗wi∈[ui,vi]⊂[0,1], i=1,2,⋯,m, and 0≤ui<vi≤1. If the whitening value ⊗ˆwi∈[ui,vi] of the weight satisfies the following normalization condition, namely

(6)∑i=1m⊗ˆwi=1

Then the weight is called the weight with interval grey number, or the grey weight for short.

Definition 5.

Suppose the generalized greyness vector of grey weight of the i th attribute is as follows.

(7)g∘(⊗wi)=(g∘(⊗w1i),g∘(⊗w2i),g∘(⊗w3i))T

and 0≤g∘(⊗wli)≤1, ∑l=13g∘(⊗wli)=1, i=1,2,⋯,m, l=1,2,3. If

(8)g∘(⊗W)=[g∘(⊗w11) g∘(⊗w12)⋯g∘(⊗w1m)g∘(⊗w21) g∘(⊗w22)⋯g∘(⊗w2m)g∘(⊗w31) g∘(⊗w32)⋯g∘(⊗w3m)]

It is called the generalized greyness matrix of grey weight, or the greyness matrix of grey weight for short.

From definition 5, the grey weight can be represented by greyness matrix, which provides a basis for fully utilizing the information of grey weight.

Theorem 4.

Suppose Ω=[0,1], if the grey weight ⊗wi∈[ui,vi]⊂Ω, i=1,2,⋯,m, and 0≤ui<vi≤1, then

1. 0≤∑i=1mui<1<∑i=1mvi≤m.

2. If g∘(⊗w2i)=g∘(⊗w2s), i≠s, i,s=1,2,⋯,m, then g∘(⊗w2i)≤1m, i=1,2,⋯,m.

Proof (1) Because the grey weight ⊗wi∈[ui,vi]⊂[0,1], i=1,2,⋯,m, 0≤ui<vi≤1, and the whitening value of the weight is ⊗ˆwi∈[ui,vi], that is 0≤ui≤⊗ˆwi≤vi≤1, then

(9)0≤∑i=1mui≤∑i=1m⊗ˆwi≤∑i=1mvi≤m

Due to ∑i=1m⊗ˆwi=1, so 0≤∑i=1mui≤1≤∑i=1mvi≤m.

If ∑i=1mui=1=∑i=1mvi, then ui=vi, that is, the weight must be a real weight, which contradicts the assumption that the weight is a grey weight.

Therefore, when the weight is grey weight, 0≤∑i=1mui<1<∑i=1mvi≤m must hold.

• (2) Because the grey weight ⊗wi∈[ui,vi]⊂Ω, i=1,2,⋯,m, the greyness of its value domain ⊗w2i=[ui,vi] is as follows.

(10)g∘(⊗w2i)=μ(⊗w2i)μ(Ω)=vi−ui1−0=vi−ui

Because the grey weight ⊗wi has whitening weight ⊗ˆwi∈[ui,vi], and meet ∑i=1m⊗ˆwi=1, if we assume that grey weights are dynamic, then the possible maximum change range of the value domain of grey weight ⊗wi is ⊗w2i=⊗[0,vi]. From equation (10), the greyness of the maximum value domain ⊗w2i=[0,vi] of grey weight ⊗wi is as follows.

(11)g∘(⊗w2i)=vi

if the whitening weight of grey weight ⊗wi takes the upper limit value of the interval [0,vi], that is, ⊗ˆwi=vi, i=1,2,⋯,m, then

(12)∑i=1m⊗ˆwi=∑i=1mvi=∑i=1mg∘(⊗w2i)=1

if g∘(⊗w2i)=g∘(⊗w2s), i≠s, i,s=1,2,⋯,m, then

∑i=1mg∘(⊗w2i)=1⇒mg∘(⊗w2i)=1⇒g∘(⊗w2i)=1m

in general, due to the fact that the whitening weights of the grey weight do not all take the upper limit value of the interval [0,vi] at the same time, then g∘(⊗w2i)≤1m, i=1,2,⋯,m. Therefore, theorem 4 holds.

From theorem 4, when determining the grey weight, the sum of the lower limit values of the value domain of grey weight should be less than 1, and the change range of the value domain of grey weight should generally not greater than 1m at the same time, and m is the attribute number of the decision. Therefore, theorem 4 provides a basic principle for determining grey weight.

For the multi-attribute decision, the weighted greyness distance is given as follows.

Definition 6.

Suppose m attributive values and weights are all interval grey numbers, and the background domains of each attributive value are Ωi=[pi,qi]∈R respectively, and pi<qi, i=1,2,⋯,m . The i th attributive values of the decision schemes j and k are ⊗xij∈[eij,fij]⊂Ωi, ⊗xik∈[eik,fik]⊂Ωi respectively, and the grey weight of the i th attribute is ⊗wi∈[ui,vi]⊂[0,1], and 0≤ui<vi≤1, and their generalized greyness vectors are as follows.

(13)g∘(⊗xij)=(g∘(⊗r1ij),g∘(⊗r2ij),g∘(⊗r3ij))

(14)g∘(⊗xik)=(g∘(⊗r1ik),g∘(⊗r2ik),g∘(⊗r3ik))

(15)g∘(⊗wi)=(g∘(⊗w1i),g∘(⊗w2i),g∘(⊗w3i))

0≤g∘(⊗xlij)≤1, 0≤g∘(⊗xlik)≤1, 0≤g∘(⊗wli)≤1, ∑l=13g∘(⊗xlij)=1, ∑l=13g∘(⊗xlik)=1, ∑l=13g∘(⊗wli)=1, i=1,2,⋯,m, l=1,2,3. If

(16)djk={∑i=1mλiα∑l=13[g∘(⊗wli)|g∘(⊗rlij)−g∘(⊗rlik)|]α}1/α

It is called the generalized weighted greyness distance between decision schemes j and k, abbreviated as greyness distance. α is the distance parameter, usually taken as α=1 or α=2. λiα is the conversion coefficient between the weighted greyness distance and the weighted real number distance of the i th attribute, namely

(17)λiα=(g∘(⊗w1i))α(g∘(⊗w1i))α+(g∘(⊗w3i))α

(g∘(⊗w1i))α+(g∘(⊗w3i))α≠0.

It should be noted that equation (16) is the weighted greyness distance when the attributive weight and attributive value are both interval grey numbers, which is an extension of the classical distance. This model can fully utilize the certainty and uncertainty information of grey weight and grey attribute.

Theorem 5.

For any interval grey numbers ⊗j,⊗k,⊗s, the generalized weighted greyness distance satisfying definition 6 has the following properties, namely

1. dkk=0.

2. djk=dkj.

3. djk≤djs+dsk.

Proof From definition 6, the above properties hold (omitted).

From theorem 5, the generalized weighted greyness distance and real number distance have similar properties.

Theorem 6.

If the interval grey numbers ⊗xij,⊗xik,⊗wi all have whitening values ⊗ˆxij,⊗ˆxik,⊗ˆwi respectively, then the weighted greyness distance equation (16) can be simplified as follows.

(18)djk={∑i=1m[g∘(⊗w1i)|g∘(⊗r1ij)−g∘(⊗r1ik)|]α}1/α

Proof If the interval grey numbers ⊗xij,⊗xik,⊗wi all have whitening values ⊗ˆxij,⊗ˆxik,⊗ˆwi respectively, then their generalized greyness vectors are as follows.

(19)g∘(⊗xij)=(g∘(⊗r1ij),g∘(⊗r2ij),g∘(⊗r3ij))=(g∘(⊗r1ij),0,g∘(⊗r3ij))

(20)g∘(⊗xik)=(g∘(⊗r1ik),g∘(⊗r2ik),g∘(⊗r3ik))=(g∘(⊗r1ik),0,g∘(⊗r3ik))

(21)g∘(⊗wi)=(g∘(⊗w1i),g∘(⊗w2i),g∘(⊗w3i))=(g∘(⊗w1i),0,g∘(⊗w3i))

and they meet g∘(⊗r1ij)+g∘(⊗r3ij)=1, g∘(⊗r1ik)+g∘(⊗r3ik)=1, g∘(⊗w1i)+g∘(⊗w3i)=1.

Therefore, |g∘(⊗r1ij)− g∘(⊗r1ik)|=|g∘(⊗r3ij)− g∘(⊗r3ik)|, |g∘(⊗r2ij)− g∘(⊗r2ik)|=0.

Substitute equations (19), (20) and (21) into equation (16), then

∑l=13[g∘(⊗wli)|g∘(⊗rlij)−g∘(⊗rlik)|]α=[g∘(⊗w1i)|g∘(⊗r1ij)−g∘(⊗r1ik)|]α+0+[g∘(⊗w3i)|g∘(⊗r3ij)−g∘(⊗r3ik)|]α=[(g∘(⊗w1i))α+(g∘(⊗w3i))α]|g∘(⊗r1ij)−g∘(⊗r1ik)|α

djk={∑i=1m(g°(⨂w1i))α(g°(⨂w1i))α+(g°(⨂w3i))α∑l=13[g°(⨂wli)|g°(⨂rlij)−g°(⨂rlik)|]α}1α={∑i=1m(g°(⨂w1i))α(g°(⨂w1i))α+(g°(⨂w3i))α[(g°(⨂w1i))α+(g°(⨂w3i))α]|g°(⨂r1ij)−g°(⨂rlik)|α}1α={∑i=1m[(g°(⨂w1i)|g°(⨂r1ij)−g°(⨂rlik)|]α}1α

Therefore, theorem 6 holds.

Theorem 6 shows that when the attributive weight and attributive value of the interval grey numbers are whitened and converted into real numbers, the greyness distance is equal to the greyness distance of the lower bound domain of interval grey number. Obviously, the generalized weighted greyness distance of interval grey number is an extension of the greyness distance of real number.

Inference 1.

If the whitening values ⊗ˆxij,⊗ˆxik,⊗ˆwi of interval grey numbers ⊗xij,⊗xik,⊗wi are the numbers within the interval [0, 1], and they are rewritten as rij,rik,wi, then the generalized weighted greyness distance of equation (18) can be rewritten as follows.

(22)djk={∑i=1m[wi|rij−rik|]α}}1/α

Inference 1 shows that if the attributive value takes a real number within the interval [0, 1], the generalized weighted greyness distance of the real number is equal to the weighted distance of the real number. Obviously, the generalized weighted greyness distance of interval grey number is an extension of the weighted real number distance within the interval [0, 1].

3. Grey decision-making model based on greyness distance

The purpose of decision-making is to rank the schemes and select the best scheme. In order to achieve this goal, we usually choose positive ideal scheme and negative ideal scheme as the referenced scheme. For the decision-making problem that the attributive weight and attributive value are both interval grey numbers, according to the generalized greyness of interval grey number, the definitions of the best scheme and the worst scheme are given first, then the grey decision-making model is established based on the greyness distance.

3.1 Best scheme and worst scheme

Definition 7.

Suppose the decision scheme set is U ={u1,u2,⋯,un}, j=1,2,⋯,n, and the attribute index set is V={v1,v2,⋯,vm}, i=1,2,⋯,m. The interval grey number of the i th attributive value of the j th scheme is ⊗rij∈[eij,fij]⊂Ωi, Ωi=[pi,qi], and pi≤eij<fij≤qi. The generalized greyness vector of ⊗rij is as follows.

g∘(⊗rij)=(g∘(⊗r1ij),g∘(⊗r2ij),g∘(⊗r3ij))

Suppose there are decision schemes ⊗A and ⊗B, i∈{1,2,⋯,m}, if the generalized greyness vectors of the benefit-type attributive value are as follows, namely

(23)g∘(⊗ai)=(g∘(⊗a1i),g∘(⊗a2i),g∘(⊗a3i))=(1,0,0)

(24)g∘(⊗bi)=(g∘(⊗b1i),g∘(⊗b2i),g∘(⊗b3i))=(0,0,1)

If the generalized greyness vectors of the cost-type attributive value are as follows, namely

(25)g∘(⊗ai)=(g∘(⊗a1i),g∘(⊗a2i),g∘(⊗a3i))=(0,0,1)

(26)g∘(⊗bi)=(g∘(⊗b1i),g∘(⊗b2i),g∘(⊗b3i))=(1,0,0)

Then ⊗A and ⊗B are called the best scheme and worst scheme respectively, or positive ideal scheme and negative ideal scheme.

From definition 7, the attributive values of the best scheme and the worst scheme exist objectively, and they are automatically generated by the generalized greyness of interval grey number, and they are all whitening values on the background domain. Its physical meaning is as follows: for the benefit-type (Bigger is better) attributes, the attributive value of the best scheme takes the upper limit value (maximum value) of its background domain interval, and the attribute value of the worst scheme takes the lower limit value (minimum value) of its background domain interval. The opposite is true for cost-type (smaller is better) attributes.

For computational convenience, for cost-type (smaller is better) attributes, before calculating the generalized greyness, it can be converted to the benefit-type attributes, or after calculating the generalized greyness, the greyness g∘(⊗r1ij) of lower bound domain and the greyness g∘(⊗r3ij) of upper bound domain can be swapped. For ease of programming, the generalized greyness vector after swapping is still written as g∘(⊗rij)=(g∘(⊗r1ij),g∘(⊗r2ij),g∘(⊗r3ij)).

3.2 The multi-attribute grey decision model

Definition 8.

If

(27)dja={∑i=1mλiα∑l=13[g∘(⊗wli)|g∘(⊗rlij)−g∘(⊗ali)|]α}1/α

then it is called the generalized weighted greyness distance from the scheme j to the best scheme ⊗A, or greyness distance-best for short. α is the distance parameter, generally taken as α=1 or α=2. λiα is the conversion coefficient (see equation (17)). If

(28)djb={∑i=1mλiα∑l=13[g∘(⊗wli)|g∘(⊗rlij)−g∘(⊗bli)|]α}1/α

Then it is called the generalized weighted greyness distance from the scheme j to the worst scheme ⊗B, or greyness distance-worst for short.

The grey system theory focuses on the analysis and utilization of concept’s connotative information of things. From the perspective of information, in order to describe the degree to which the scheme j approaches the best scheme and the worst scheme, the definition of proximity is given below.

Definition 9.

Suppose σj and σjc represent the comprehensive measure index of the scheme j relative to the best scheme and the worst scheme, and 0≤σj≤1, 0≤σjc≤1, if

(29)σj+σjc=1

Then σj and σjc are called the proximity of the scheme j to the best scheme and the worst scheme, respectively.

Obviously, the proximity satisfying equation (29) has infinite solutions. In order to establish the objective function for solving proximity, the following takes σj and σjc as the weight to further expand the definition of greyness distance.

Definition 10.

Suppose σj and σjc represent the proximity of the scheme j to the best scheme and the worst scheme respectively, dja and djb represent the generalized weighted greyness distance from the scheme j to the best scheme and the worst scheme respectively, then Dja=σjdja and Djb=σjcdjb are called the weighted greyness distance-best and weighted greyness distance-worst of scheme j, or the generalized weighted greyness distance for short.

In order to solve the proximity σj of the scheme j to the best scheme, j=1,2,⋯,n, the following objective function is established.

(30)min{F(σj)=∑j=1n(Dja2+Djb2)=∑j=1n[σj2dja2+(1−σj)2djb2]}

Equation (30) means that the square sum of the generalized weighted greyness distance from all decision schemes to the best scheme and the worst scheme is the minimum. It can be seen that the objective function has clear mathematical and physical meaning.

Theorem 7.

If the proximity σj of the scheme j satisfies equation (30) of the objective function, then the optimal solution of the proximity σj is as follows.

(31)σj=11+{∑i=1mλiα∑l=13[g∘(⊗wli)|g∘(⊗rlij)−g∘(⊗ali)|]α∑i=1mλiα∑l=13[g∘(⊗wli)|g∘(⊗rlij)−g∘(⊗bli)|]α}2α

In equation (31), α is the distance parameter, usually taken as α=1 or α=2.

Proof Take the first order derivative of equation (31) and make the derivative equal to zero, and we can get the following equation.

∂F∂σj=2σjdja2−2(1−σj)djb2=0

(32)σj=djb2dja2+djb2

Substitute equations (27) and (28) into equation (32), the following equation is obtained.

(33)σj=11+{∑i=1mλiα∑l=13[g∘(⊗wli)|g∘(⊗rlij)−g∘(⊗ali)|]α∑i=1mλiα∑l=13[g∘(⊗wli)|g∘(⊗rlij)−g∘(⊗bli)|]α}2α

Therefore, theorem 7 holds.

Equation (32) or (33) is called the multi-attribute grey decision-making model based on generalized greyness, or grey decision-making model for short.

Theorem 8.

The grey decision-making model (32) has the following properties:

1. If dja=0,djb≠0, then σj=1, σjc=0.

2. If dja≠0,djb=0, then σj=0, σjc=1.

3. If dja=djb≠0, then σj=0.5 and σjc=0.5.

Proof From equation (32), theorem 8 is obviously true.

Theorem 8 shows that if the generalized greyness distance between the decision scheme and the best scheme is zero, then the proximity between the decision scheme and the best scheme is 1. If the greyness distance between the decision scheme and the best scheme is equal to the greyness distance between the decision scheme and the worst scheme, then the proximities of the decision scheme to the best scheme and the worst scheme are all 0.5. At this time, the decision scheme is in the most uncertain (grey) state.

Using equation (32) or (33), the proximity σj of the scheme j to the best scheme can be obtained, j=1,2,⋯,n. Then the proximity vector of n decision schemes is as follows.

(34)σ=(σ1,σ2,⋯,σn)

From equation (34), according to the principle that the greater the proximity, the better the scheme, n decision schemes can be ranked and the optimal decision scheme can be selected.

3.3 Simplified forms of the multi-attribute grey decision-making model

The grey decision model proposed in this paper provides technological support for the decision-making problem in which both attributive weights and attributive values are interval grey numbers, and does not need to solve the exact values of weights. In practice, the attributive weights and attributive values might not all be interval grey numbers. Therefore, depending on whether the attributive weight or the attributive value is a real number or an interval grey number, the simplified forms of the grey decision-making model are given below.

Theorem 9.

If the attributive weight and the attributive value are both interval grey numbers and each attribute has the same generalized greyness vector of weight, then the grey decision-making model (33) can be simplified as follows.

(35)σj=11+{∑i=1m∑l=13[g∘(⊗wli)|g∘(⊗rlij)−g∘(⊗ali)|]α∑i=1m∑l=13[g∘(⊗wli)|g∘(⊗rlij)−g∘(⊗bli)|]α}2α

Proof If the generalized greyness vectors of weights of each attribute are the same, suppose g∘(⊗w1i)=g1, g∘(⊗w3i)=g3, i=1,2,⋯,m, then

(36)λiα=g°(⨂w1i))αg°(⨂w1i))α+g°(⨂w3i))α=g1αg1α+g3α=λα

Substituting equation (36) into equation (33), then equation (35) is obtained. Therefore, theorem 9 holds.

Theorem 10.

If the attributive values are interval grey numbers and the attributive weights are real numbers, then the grey decision-making model (33) can be changed as follows.

(37)σj=11+{∑i=1m∑l=13[wi|g∘(⊗rlij)−g∘(⊗ali)|]α∑i=1m∑l=13[wi|g∘(⊗rlij)−g∘(⊗bli)|]α}2α

Proof When the attributive values are interval grey numbers and the attributive weights are real numbers, suppose the weighted greyness distance from the j th scheme to the best scheme ⊗A and the worst scheme ⊗B are as follows, respectively.

(38)dja={∑i=1mλ∑l=13[wi|g∘(⊗rlij)−g∘(⊗ali)|]α}1/α

(39)djb={∑i=1mλ∑l=13[wi|g∘(⊗rlij)−g∘(⊗bli)|]α}1/α

in equation (38) and (39), λ represents the conversion coefficient between the greyness distance and real number distance, generally taken λ=0.5. wi is the weight of the i th attribute, 0≤wi≤1, and ∑i=1mwi=1.

Substituting equations (38) and (39) into equation (32), and the following equation is obtained.

(40)σj=11+{∑i=1m∑l=13[wi|g∘(⊗rlij)−g∘(⊗ali)|]α∑i=1m∑l=13[wi|g∘(⊗rlij)−g∘(⊗bli)|]α}2α

Therefore, theorem 10 holds.

It should be noted that when the attributive values are interval grey number and the weights are real numbers, it cannot be directly simplified according to equation (33). Because in the generalized greyness vector g∘(⊗wi)=(g∘(⊗w1i),g∘(⊗w2i),g∘(⊗w3i)) of the real number weight, there exist g∘(⊗w2i)≡0, if using equation (33) to simplify, this will lose the difference information of the attributive value of interval grey number, that is, |g∘(⊗r2ij)−g∘(⊗a2i)| and |g∘(⊗r2ij)−g∘(⊗b2i)|.

Theorem 11.

If the attributive values are real numbers within the interval [0, 1], and the attributive weights are grey weight ⊗wi∈[wi1,wi2], 0≤wi1< wi2≤1, then the grey decision-making model (33) can be changed as follows.

(41)σj=11+{∑i=1m[wi1α+wi2α+(wi2−wi1)α]|rij−ai|α∑i=1m[wi1α+wi2α+(wi2−wi1)α]|rij−bi|α}2α

Proof Suppose the i th attributive value of the j th scheme is rij, because rij is the real number within the interval [0, 1], then 0≤rij≤1, i=1,2,⋯,m, j=1,2,⋯,n. Suppose the i th attributive values of the best scheme A and worst scheme B are ai,bi respectively, and 0≤bi<ai≤1. If the grey weight is ⊗wi∈[wi1,wi2], 0≤wi1< wi2≤1, then

(42)g∘(⊗w1i)=wi1,g∘(⊗w2i)=wi2−wi1,1−g∘(⊗w3i)=1−(1−wi2)=wi2

Since any real number k (k>0) multiplies an interval grey number ⊗ is interval grey number, then

⊗rij′=⊗wi⋅rij=⊗[wi1rij,wi2rij],0≤wi1rij<wi2rij≤1

⊗ai′=⊗wi⋅ai=⊗[wi1ai,wi2ai],0≤wi1ai<wi2ai≤1

⊗bi′=⊗wi⋅bi=⊗[wi1bi,wi2bi],0≤wi1bi<wi2bi≤1

From definition 1, the generalized greyness vectors of interval grey numbers ⊗rij′,⊗ai′,⊗bi′ in the background domain Ω=[0,1] (μ(Ω)=1) are as follows.

(43)g∘(⊗rij′)=(g∘(⊗r1ij′),g∘(⊗r2ij′),g∘(⊗r3ij′))=(wi1rij,(wi2−wi1)rij,1−wi2rij)

(44)g∘(⊗ai′)=(g∘(⊗a1i′),g∘(⊗a2i′),g∘(⊗a3i′))=(wi1ai,(wi2−wi1)ai,1−wi2ai)

(45)g∘(⊗bi′)=(g∘(⊗b1i′),g∘(⊗b2i′),g∘(⊗b3i′))=(wi1bi,(wi2−wi1)bi,1−wi2bi)

Imitating equation (4), suppose after weighting the indicators, the weighted greyness distances from the j th scheme to the best scheme ⊗A and worst scheme ⊗B are as follows.

(46)dja={∑i=1mλ∑l=13[|g∘(⊗rlij′)−g∘(⊗ali′)|]α}1/α

(47)djb={∑i=1mλ∑l=13[|g∘(⊗rlij′)−g∘(⊗bli′)|]α}1/α

In equation (46) and (47), λ represents the conversion coefficient, usually taken λ=0.5. According to equations (43), (44) and (45), organizing equations (46) and (47), then

(48)dja={∑i=1mλ[wi1α+wi2α+(wi2−wi1)α] |rij−ai|α}1/α

(49)djb={∑i=1mλ[wi1α+wi2α+(wi2−wi1)α]|rij−bi|α}1/α

Substitute equations (48) and (49) into equation (32), then

(50)σj=11+{∑i=1m[wi1α+wi2α+(wi2−wi1)α]|rij−ai|α∑i=1m[wi1α+wi2α+(wi2−wi1)α]|rij−bi|α}2α

Therefore, theorem 11 holds.

It should be noted that when the attributive values are real numbers and the attributive weights are interval grey numbers, it cannot be directly simplified according to equation (33). Otherwise, the part information of grey weight will be lost. From equations (48) and (49), it can be seen that the model (50) fully utilizes all the greyness information of the grey weight.

Theorem 12.

If the attributive value and attributive weight are all real numbers within the interval [0,1], then equation (33) of the grey decision-making model can be changed as follows.

(51)σj=11+{∑i=1m[wi|rij−ai|]α∑i=1m[wi|rij−bi|]α}2α

rij represents the i th attributive value of the j th scheme, 0≤rij≤1, j=1,2,⋯,n. wi represents the weight, 0≤wi≤1, and ∑i=1mwi=1. ai and bi represent the i th attributive value of the best scheme A and the worst scheme B respectively, and 0≤ai≤1, 0≤bi≤1, i=1,2,⋯,m. Based on the relativity of decision-making, generally taken as ai=∨j=1nrij, bi=∧j=1nrij, or ai=1,bi=0, i=1,2,⋯,m.

Proof Suppose the grey weight is ⊗wi∈[wi1,wi2], 0≤wi1< wi2≤1, if the grey weights become real number weights, then w1i=w2i=wi, i=1,2,⋯,m. According to equation (50), when w1i=w2i=wi, then

(52)wi1α+wi2α+(wi2−wi1)α=2wiα

Substituting equation (52) into equation (50), then equation (51) is obtained. Therefore, theorem 12 holds. In addition, theorem 12 can also be proven by inference 1.

From equations (33), (35), (37), (41) and (51), it can be seen that the grey decision-making model (33) is the general form of the extended model of ordinary decision-making model (51), and the grey decision-making model (35), (37), (41) are the simplified form of the model (33). Equations (33), (35), (37) and (41) are collectively called the grey decision model based on generalized greyness. They can be used not only for decisions when attributive weights and attributive values are all interval grey numbers, but also for decisions when interval grey numbers and real numbers coexist. Therefore, the grey decision-making model proposed in this article is an extended form of ordinary decision-making model, and it has broader application value.

Similarly, using equations (35), (37), (41) or (51), the proximity σj of the scheme j to the best scheme can be calculated, j=1,2,⋯,n. According to the principle that the greater the proximity, the better the scheme, n decision schemes can be ranked and the optimal decision scheme can be selected.

3.4 The steps of multi-attribute grey decision making model

The calculation steps of the multi-attribute grey decision making model proposed in this paper are as follows.

1. According to the collected information, construct the decision matrix ⊗X of interval grey number.

2. Determine the background domain Ω of each attribute and the distance parameter α.

3. According to equation (1), transform the decision matrix ⊗X into the generalized greyness matrix G(⊗R).

4. Determine grey weight vector ⊗W.

5. According to equation (1), transform the grey weight vector ⊗W into the generalized greyness matrix G(⊗W).

6. Select the appropriate formula to calculate the proximity σj, j=1,2,⋯,n.

7. Rank the decision-making schemes and select the optimal schemes according to the proximity.

8. If the results match the reality, the calculation ends. Otherwise adjust the weight and repeat steps from (5) to (7).

The flowchart of the above calculation steps is shown in Figure 1.

4. Application example

In order to illustrate the validity of the model proposed in this paper, the evaluation of regional technological innovation ability of 6 provinces in China and tow calculation examples are given as follows, and the model in this paper is compared with the commonly used methods.

4.1 Evaluation of regional technological innovation ability

Science and technology are the primary productive forces, and the technological innovation ability is a powerful driving force for social and economic development. Scientific evaluation of the technological innovation ability of a country or region is an important basis for decision-making. In this example, six provinces “Shanxi, Hebei, Henan, Shandong, Anhui, Jiangsu” of China are selected as the evaluation object to form the decision set: U={u1,u2,⋯,u6}, j=1,2,⋯,6. And five indexes are determined to form the evaluation index set: V={v1,v2,⋯,v5}, i=1,2,⋯,5. v1 represents the technology market turnover (unit: 100 million yuan RMB), v2 represents the number of R&D personnel (unit: 10,000 people), v3 represents the R&D expenditure (unit: 100 million yuan RMB), v4 represents the number of students in university (unit: 10,000 people), v5 represents the number of public library (unit: one). The data comes from the “China science and technology statistical yearbook”.

For the dynamic evaluation, the average value of 2017 and 2018 data are used as the lower limit value of the value domain of interval grey number, and the average value of 2018 and 2019 data are used as the upper limit value of the value domain of interval grey number, forming the evaluation index of the attribute with interval grey number, as shown in Table 1. the five evaluation indexes are all benefit-type indicators.

4.1.1 Evaluation results based on grey decision model

From Table 1, the evaluation indexes have both interval grey numbers and real numbers, so it is suitable to use the grey decision-making model to evaluate the technological innovation ability of six provinces from July 2018 to June 2019. The calculation steps are as follows.

• Step 1 Build the decision matrix ⊗X of the schemes with interval grey number, as shown in Table 1 (omitted).

• Step 2 Convert the decision matrix ⊗X of interval grey number into the generalized greyness matrix G(⊗R). According to the relativity of generalized greyness and the data in Table 1, the background domain Ωi=[pi,qi] of each index is reasonably determined as follows, i=1,2,⋯,5.

Ω1=[0,1300],Ω2=[0,50],Ω3=[0,2200],Ω4=[0,230],Ω5=[0,180]

• Step 3 Calculate the greyness of the lower bound domain, value domain and upper bound domain of each index by equation (1), as shown in Table 2.

• Step 4 Determine the grey weight and its generalized greyness. Suppose the background domain of the interval grey number of grey weight is Ωw=[0,1], for the convenience of calculation, the grey weights of each evaluation index are the same, that is ⊗wi∈[0.15,0.25], i=1,2,⋯,5. Obviously, there exist a whitening weight value ⊗ˆwi and it satisfies ∑i=15⊗ˆwi=1.

Calculate the generalized greyness vector g∘(⊗wi) of grey weight by equation (1), i=1,2,⋯,5, that is

g∘(⊗wi)=(g∘(⊗w1i),g∘(⊗w2i),g∘(⊗w3i))=(0.15,0.10,0.75)

• Step 5 Determine the best scheme and the worst scheme and their generalized greyness. As the five evaluation indexes are all benefit-type indicators, according to definition 7, determine the generalized greyness vector g∘(⊗ai) and g∘(⊗bi) of the evaluation index of the best scheme ⊗A and worst scheme ⊗B, i=1,2,⋯,5, namely

g∘(⊗ai)=(g∘(⊗a1i) g∘(⊗a2i) g∘(⊗a3i))=(1,0,0)

g∘(⊗bi)=(g∘(⊗b1i) g∘(⊗b2i) g∘(⊗b3i))=(0,0,1)

• Step 6 Determine the generalized greyness matrix of evaluation index for each evaluation object. According to the data in Table 2, the generalized greyness matrix of evaluation index for the evaluation object u1 (Shanxi province) is as follows.

g∘(⊗R1)=[⊗R11⊗R21⊗R31⊗R41⊗R51 ]=[g∘(⊗r111) g∘(⊗r121) g∘(⊗r131) g∘(⊗r141) g∘(⊗r151)g∘(⊗r211) g∘(⊗r221) g∘(⊗r231) g∘(⊗r241) g∘(⊗r251)g∘(⊗r311) g∘(⊗r321) g∘(⊗r331) g∘(⊗r341) g∘(⊗r351)]=[0.094 0.055 0.055 0.332 0.7110.006 0.004 0.006 0.009 0.0000.900 0.941 0.939 0.659 0.289]

⊗Ri1 represents the generalized greyness vector of the i th evaluation index of the evaluation object u1 (Shanxi province), the specific data is shown in Table 2. For example, ⊗R11 is as follows.

⊗R11=(g∘(⊗r111) g∘(⊗r211) g∘(⊗r311))T=(0.094,0.006,0.900)T

Similarly, the generalized greyness matrix g∘(⊗Rj) of the evaluation index of other evaluation objects can be determined, j=1,2,⋯,6. To save space, they are not listed.

• Step 7 Calculate the greyness distance from each evaluation object to the best scheme and the worst scheme. If the distance parameter is taken α=1, the greyness distances from each evaluation object to the best scheme and worst scheme are calculated by equations (27) and (28), then

d1a=0.5602,d2a=0.4334,d3a=0.3765,d4a=0.2157,d5a=0.4575,d6a=0.1194

d1b=0.1906,d2b=0.3226,d3b=0.3799,d4b=0.5472,d5b=0.2979,d6b=0.6448

• Step 8 Calculate the proximity of each evaluation object to the best scheme. The proximity vector calculated by equations (32) or (33) is as follows.

(53)σ=(σ1,σ2,σ3,σ4,σ5,σ6)=(0.1038,0.3565,0.5044,0.8655,0.2977,0.9699)

• Step 9 Evaluation and decision. Due to σ6>σ4>σ3>σ2>σ5>σ1, the ranking of the six evaluation objects is as follows.

u6≻u4≻u3≻u2≻u5≻u1

Therefore, the evaluation object u6 is the best.

If the distance parameter is taken α=2, repeat the above calculation steps, and the proximity vector is as follows.

(54)σ=(σ1,σ2,σ3,σ4,σ5,σ6)=(0.1710,0.4017,0.5096,0.8522,0.3265,0.9559)

Due to σ6>σ4>σ3>σ2>σ5>σ1, then the ranking of the six evaluation objects is as follows.

u6≻u4≻u3≻u2≻u5≻u1

Obviously, the evaluation object u6 is the best, and it is consistent with the results when α=1. Therefore, the technological innovation ability of six provinces in China is ranked from strong to weak: Jiangsu, Shandong, Henan, Hebei, Anhui, and Shanxi.

From Table 1, Jiangsu province ranks first because of its significant advantages in the technology market turnover, R&D personnel and R&D expenditure. Shandong province ranks second because of its great advantages in the technology market turnover, R&D personnel, R&D expenditure and the number of students in university. Shanxi province ranks sixth because of its obvious disadvantages in the R&D personnel, R&D expenditure and the number of students in university. Henan province has significant advantages in the number of students in university, and it has certain advantages in the R&D personnel and R&D expenditure compared with Hebei province, so it ranks third before Hebei province. Anhui province has advantages over Hebei province in terms of technology market turnover, R&D personnel and R&D expenditure, while it is significantly weaker than Hebei province in terms of the number of students in university and the number of public library. Therefore, Anhui province ranks behind Hebei province, but there is little difference between them. Therefore, the evaluation results using the model proposed in this paper are in line with reality.

4.1.2 Sensitivity analysis of grey decision model

From equation (33), in the case of a given attribute value, the proximity depends on the distance parameter, the grey weight and the background domain. The effects of the distance parameters, grey weights and background domain on proximity are analyzed as follows.

• (1) Sensitivity analysis of changing distance parameter

When the grey weights of each evaluation index are the same, that is ⊗wi∈[0.15,0.25], i=1,2,⋯,5, according to the data in Table 2 and equation (33), the computed proximity with changing distance parameter is shown in Table 3 and Figure 2.

As can be seen from Table 3 and Figure 2, the proximity of the six objects varied with the change of the distance parameter. When the distance parameter is within the range of interval [0.2, 3.0], the proximity of the evaluation object u1, u2, u3 and u5 has a minimum point respectively, while there is a maximum point for the evaluation object u4 and u6, respectively. When the distance parameter α=0.6, the standard deviation of the proximities of six evaluation objects reaches the maximum value, that is s=0.3434. Although the distance parameter influences the value of proximity, the ranking of the 6 objects is not changed. This shows that the evaluation results have good robustness to the influence of distance parameters.

• (2) Sensitivity analysis of when the grey weight of each evaluation index is different

Assuming that the grey weight of each evaluation index is different, if the greyness matrix of the grey weight is as below.

g∘(⊗w)=[0.30   0.10   0.600.20   0.15   0.650.15   0.15   0.700.12   0.10   0.780.10   0.10   0.80]

Then the proximities calculated by equation (33) are as follows.

When α=1, σ1=0.050, σ2=0.203, σ3=0.282, σ4=0.806, σ5=0.210, σ6=0.971.

When α=2, σ1=0.055, σ2=0.176, σ3=0.214, σ4=0.783, σ5=0.182, σ6=0.972.

Obviously, when α=1 or α=2, due to σ6>σ4>σ3>σ5>σ2>σ1, then the ranking of the six evaluation schemes is u6≻u4≻u3≻u5≻u2≻u1, and the evaluation scheme u6 is the best. Due to the small difference between Anhui province u5 and Hebei province u2, when the grey weight is changed, the ranking of the two is reversed, which is in line with reality and reasonable. It can be seen that when the grey weight is changed, the grey weight can effectively adjust the ranking of the evaluation schemes, but the best evaluation scheme remains unchanged. This shows that the grey decision model proposed in this paper is sensitive to the change of weight.

• (3) Sensitivity analysis of when the background domain of each evaluation index is changed

In the case that the generalized greyness vector of the grey weight is g∘(⊗wi)=(0.15,0.1,0.75), i=1,2,⋯,4; if the background domain Ω of each evaluation index is changed, the proximity of each scheme is shown in Table 4.

From Table 4, when the grey weight of each index is the same, change the background domain of the index, take the greyness distance parameter α=1 or α=2, then the ranking of the six evaluation schemes is u6≻u4≻u3≻u2≻u5≻u1, and the evaluation scheme u6 is the best. This shows that the range-valued change of the background domain of each index will affect the value of proximity, but generally will not affect the ranking of the evaluation schemes, which reflects the relativity of generalized greyness. In addition, the closer the upper limit value of the background domain of each index is to the maximum value of the evaluation index, the greater the averaged value of the proximity and the greater the variance, which is conducive to decision-making. Therefore, in practical application, the upper limit value of the background domain of each index should take the maximum value of each index or a value slightly larger than the maximum value.

4.1.3 Evaluation results based on the commonly used decision methods

In order to further illustrate the effectiveness of the method in this paper, the generalized greyness in Table 2 are converted to the standardized interval grey number according to theorem 2, then, Deng’s Grey Relational Degree model (DGRD) (Luo and Liu, 2005), Weighted “Kernel” model (WK) (Liu et al., 2017), TOPSIS model (Li et al., 2023), and Grey Comprehensive Evaluation model (GCE) (Li and Li, 2022b, 2023c) are used to the evaluation. As the first three models are more difficult for calculation with grey weights, so they use equal weight to calculate the evaluation values, that is, taken wi=0.2, i=1,2,⋯,5. The last model is calculated by equal grey weight, that is, taken ⊗wi=⊗[0.15,0.25], i=1,2,⋯,5. The results are shown in Table 5, and the results of equation (54) are shown in Table 5.

From Table 5, according to the principle that the larger the comprehensive value, the better the decision scheme, the ranking results of the DGRD, WK, TOPSIS and GCE models are all the u6≻u4≻u3≻u2≻u5≻u1, and the evaluation object u6 is the best. Obviously, the evaluation result is the same as the evaluation result of the model using grey weights in this paper, which shows that the method proposed in this paper is valid. However, when the attributive weight and the attributive value are all interval grey number, the first three models are difficult to calculate, which are not based on the scientific expression and not make use of all the information of interval grey number. Although the GCE model can makes full use of the information of grey indexes and grey weights, the calculation of comparing interval grey number is also troublesome. In addition, it can be seen from Table 5 that the synthetic decision values calculated by the method in this paper have larger dispersion and are easy to identify.

4.2 Comparison and analysis of calculation cases

Because the essence of the generalized greyness of interval grey number is the measurement of interval, in principle, the grey decision-making model in this paper is also suitable for the decision-making in which the attribute values are interval number, but its corresponding connotation is different from the grey decision-making model at this time (We’ll discuss it separately). In order to show the extended application of grey decision model in this paper, two decision cases are given as follows.

• Case 1 We consider the selection of air-conditioning system in a municipal library. There are 5 alternative schemes uj (j=1,2,⋯,5), and 8 evaluation indexes vi (i=1,2,⋯,8). The partial information of weights is given, and the attributive values are all interval (grey) numbers, and the specific data are seen in the reference (Xu, 2004). Try to determine the optimal scheme.

Solution Suppose the background domains for the eight indicator values are as follows respectively.

Ω1=[0,5],Ω2=[0,7],Ω3=[0,10],Ω4=[0,80]

Ω5=[0,10],Ω6=[0,100],Ω7=[0,10],Ω8=[0,10]

After converting the grey weights and attributive values into the generalized greyness (measurement), according to equation (33), the following results are obtained.

When the distance parameter is taken α=1, the proximity vector is as follows.

σ=(σ1,σ2,σ3,σ4,σ5)=(0.8176,0.5896,0.7842,0.9289,0.8511)

When the distance parameter is taken α=2, the proximity vector is as follows.

σ=(σ1,σ2,σ3,σ4,σ5)=(0.8424,0.6510,0.8249,0.9361,0.8729)

Obviously, when α=1 or α=2, there is σ4>σ5>σ1>σ3>σ2, so the ranking of the five schemes is u4≻u5≻u1≻u3≻u2, and the scheme u4 is the best.

The evaluation result is the same as that of the reference (Xu, 2004). But the method of the reference is troublesome to calculate the exact value of weight by single objective programming method and rank the interval numbers by possibility degree matrix, while the method in this paper does not need to calculate the exact value of the weight.

To further illustrate the effectiveness of the method in this paper, an example of grey dynamic evaluation of transformer state is given as follows.

• Case 2 Suppose the evaluation object set is composed of 4 transformers, that is S={s1,s2,s3,s4}, j=1,2,3,4; the 10 indicators make up the indicator set V={v1,v2,⋯,v10}, i=1,2,⋯,10. The monitoring index values (interval grey number) of 4 transformers in 4 periods are collected, and the specific data are seen in Table 3 of the references (omitted) (Zhou et al., 2020). Try to evaluate the operation state of 4 transformers by using the grey decision method.

Solution According to the normalized data in Table 3 f the references, let the background domains of the 10 indexes be Ωi=[pi,qi]=[0,0.5], i=1,2,⋯,10. According to equation (1), the evaluation index values are converted into the generalized greyness vectors of interval grey number. Since the weights of the indexes are real numbers, the evaluation calculation in this paper adopts equation (37) and takes the distance parameter α=2. The specific calculation process is omitted for saving space.

Firstly, the operating states of 4 transformers in 4 periods are sorted according to the calculated proximity, respectively. Then the operating states of 4 transformers are sorted according to the comprehensive proximity, and the results are shown in Table 6. For comparative analysis, the evaluation results of the reference (Zhou et al., 2020) are listed in Table 7.

It can be seen from Tables 6 and 7 that the order of the total evaluation results of the method in this paper is consistent with the results of the reference (Zhou et al., 2020), but there are some differences in the evaluation results using the two methods in different time periods. For example, in period 1, the order of 4 transformer states evaluated by the method in this paper is s4≻s1≻s3≻s2, while the order of 4 transformer states evaluated by grey target decision method is s1≻s4≻s3≻s2. According to the specific data shown in Table 3 of the references, among the 10 evaluation indexes, 7 indexes of transformer s4 are better than that of transformer s1, so the transformer s4 in the first row is realistic. The reasons for the differences can be analyzed from the following two aspects.

On the one hand, because there is no constraint on the degree of expected bull’s-eye and degree of edge bull’s-eye, the calculated relative degree of bull’s-eye is distorted. The degree of expected bull’s-eye and degree of edge bull’s-eye of the transformer s1 and s4 in the reference are as follows, respectively.

γ+(s1)=0.6658,γ−(s1)=0.4839,γ+(s4)=0.6962,γ−(s4)=0.5622

Then the relative degree of bull’s-eye of the transformer s1 and s4 is as follows, respectively.

f(s1)=γ+(s1)γ−(s1)+γ+(s1)=0.5791,f(s4)=γ+(s4)γ−(s4)+γ+(s4)=0.5572

Although the degree of expected bull’s-eye of transformer s4 is greater than that of the transformer s1, the degree of edge bull’s-eye of transformer s4 is also greater than that of transformer s1, so the relative degree of bull’s-eye of transformer s4 is less than that of transformer s1. Therefore, this results in distortion of the evaluation results in the reference.

On the other hand, the reference uses the distance formula of interval number to calculate the distance of interval grey numbers, and only uses the information of the characteristic points of interval (grey) number. Suppose the two interval numbers a∼=[a1,a2] and b∼=[b1,b2], the commonly used distance formula for interval numbers is as follows.

(55)d(a∼,b∼)=12[(a1−b1)2+(a2−b2)2]

If a1=a2=a, b1=b2=b, then d(a∼,b∼)=|a−b|. Obviously, equation (55) is an extension of the ordinary distance formula, but it is not the only extension. If

(56)d′(a∼,b∼)=13[(a1−b1)2+(a2−b2)2+(c1−c2)2]

c1,c2 represent the median number of interval number a∼,b∼, that is c1=(a1+a2)/2 and c2=(b1+b2)/2.

If a1=a2=a, b1=b2=b, then c1=a, c2=b, d′(a∼,b∼)=|a−b|. Obviously, equation (56) is also an extension of the ordinary distance formula, but in generally d′(a∼,b∼)≠d(a∼,b∼).

Suppose a∼=[a1,a2]=[4,8], b∼=[b1,b2]=[24,30], then c1=(a1+a2)/2=6, c2=(b1+b2)/2=27. According to equations (55) and (56), then

d(a∼,b∼)=12[(4−24)2+(8−30)2]=488

d′(a∼,b∼)=13[(4−24)2+(8−30)2+(6−27)2]=47213

Obviously, d′(a∼,b∼)≠d(a∼,b∼). Therefore, equation (55) is only an approximate distance of interval numbers or the distance of characteristic points based on interval numbers, and not a distance model of interval numbers in strict mathematical sense.

In contrast, the grey decision model proposed in this paper is based on the scientific expression of interval grey number, and the greyness distance can make full use of all the connotative information of interval grey number and identify the differences of evaluation objects more accurately. This is one of the advantages of the model in this paper.

In addition, in order to overcome the shortage of grey target decision model, the reference (Zhou et al., 2020) further uses the state differential information of transformers in different period to make decision analysis, but does not give the concrete formula to calculate the differential information. Therefore, this paper does not make the comparative analysis of decision results based on differential information.

4.3 Discussion

Compared with the existing decision-making methods, the multi-attribute grey decision-making model in this paper has some innovations and advantages. The analysis is as follows.

1. When the attributive values are interval grey numbers, most of the existing literature treat them as interval numbers and process them in some way. One way is weighted aggregation according to the operation rules of interval numbers, that is, the calculated synthetic value is also an interval number (Xu, 2004; Qian et al., 2021). The second method is that the interval number is transformed into the distance measure to the referenced scheme (Sun, 2020; Zhou et al., 2020; Li et al., 2023). However, there are some problems as follows. Firstly, the interval grey number is confused with the interval number. Secondly, the distance measure of interval numbers is only an approximate value, which cannot make full use of all the information of interval grey number. In this paper, we use the generalized greyness to express the connotative information of interval grey number. It not only condenses all the information of interval grey number, but also provides a theoretical basis for processing interval grey number and real number.

2. When the partial information of attributive weights is known, most of the existing literature solve the exact value of the weights by single-objective programming method (Xu, 2004; Zhou et al., 2020; Luo et al., 2021), but the calculation is complicated and the all information of the weights is not used. In contrast, the decision-making model proposed in this paper does not need to calculate the exact value of attributive weight (or grey weight) and attributive value. It is not only simple and easy to calculate, but also can effectively use the certainty and uncertainty information of the weight and attribute, and realize the scientific decision-making under uncertain environment.

3. The standardization of attributive values and the selection of positive and negative ideal schemes are two important tasks of the existing decision method, and when the aggregated values are interval grey numbers, it also need to compare the size of interval grey numbers (Yan et al., 2023; Gu et al., 2023). In this paper, the generalized greyness of interval grey number can automatically standardize the attributive values and generate the positive and negative ideal schemes, and the proximity of the decision model can be directly used to rank schemes and select the optimal schemes.

4. The decision-making model proposed in this paper is suitable for the decision-making problems where the attributive weight and attributive value are real numbers, interval grey numbers and the coexistence of real number and interval grey number. Therefore, it is an extension of the ordinary decision model, which not only has extensive adaptability and application value, but also can save decision cost effectively. For example, the model in this paper does not need to seek the exact values of attributive weights and attributive values, which will greatly reduce the cost of data acquisition and reduce the efforts of experts.

5. Conclusions

Aiming at the problem that attributive value and attributive weight are both interval grey numbers, this paper constructs a grey decision model based on the generalized greyness of interval grey number. The model not only has strict theoretical basis, clear mathematical and physical meaning, convenient calculation and easy programming, but also can make full use of the certainty and uncertainty information of interval grey number, and realize the scientific decision-making under the uncertain environment of “insufficient and incomplete information”. The decision-making model proposed in this paper does not need to seek the optimal solution of the attributive weight and the attributive value, and can save the decision-making labor and capital investment. The application example shows that the decision-making model proposed in this paper is feasible and effective. The research results not only enrich the grey system theory, but also provide a new way to deal with the decision-making problem that attributive weights and attributive values are both interval grey numbers and the coexistence of interval grey numbers and real numbers.

Figures

Figure 1

The flowchart of the calculation steps

Figure 2

The change chart of proximity with distance parameter

Table 1

Evaluation indexes of technological innovation ability of 6 provinces in China

Evaluation object	Evaluation indexes
	v1	v2	v3	v4	v5
Shanxi u1	⊗ [122.4, 130.1]	⊗ [2.74, 2.95]	⊗ [121.7, 134.7]	⊗ [76.4, 78.4]	128
Hebei u2	⊗ [182.4, 328.6]	⊗ [7.25, 7.40]	⊗ [366.5, 410.3]	⊗ [130.6, 140.8]	173
Henan u3	⊗ [113.1, 190.6]	⊗ [12.58, 13.42]	⊗ [500.6, 568.8]	⊗ [207.3, 223.0]	⊗ [159, 162]
Shandong u4	⊗ [665.8, 965.0]	⊗ [21.74, 23.78]	⊗ [1314.7, 1491.1]	⊗ [202.8, 211.2]	154
Anhui u5	⊗ [285.4, 385.5]	⊗ [10.52, 11.56]	⊗ [466.7, 536.9]	⊗ [114.3, 119.0]	⊗ [125, 127]
Jiangsu u6	⊗ [884.9, 1231.5]	⊗ [45.55, 48.19]	⊗ [1929.2, 2115.3]	⊗ [178.7, 184.0]	⊗ [116, 117]

Source(s): Authors work

Table 2

Generalized greyness of the evaluation index of technological innovation ability of 6 provinces in China

Evaluation object	Generalized greyness vector of evaluation index
	v1	v2	v3	v4	v5
Shanxi u1	(0.094, 0.006, 0.900)	(0.055, 0.004, 0.941)	(0.055, 0.006, 0.939)	(0.332, 0.009, 0.659)	(0.711, 0.000, 0.289)
Hebei u2	(0.140, 0.113, 0.747)	(0.145, 0.003, 0.852)	(0.167, 0.020, 0.813)	(0.568, 0.044, 0.388)	(0.961, 0.000, 0.039)
Henan u3	(0.087, 0.060, 0.853)	(0.252, 0.017, 0.731)	(0.228, 0.031, 0.741)	(0.901, 0.068, 0.031)	(0.883, 0.017, 0.100)
Shandong u4	(0.512, 0.230, 0.258)	(0.435, 0.041, 0.524)	(0.598, 0.080, 0.322)	(0.881, 0.037, 0.082)	(0.856, 0.000, 0.144)
Anhui u5	(0.220, 0.077, 0.703)	(0.210, 0.021, 0.769)	(0.212, 0.032, 0.756)	(0.497, 0.020, 0.483)	(0.694, 0.012, 0.294)
Jiangsu u6	(0.681, 0.267, 0.052)	(0.918, 0.046, 0.036)	(0.877, 0.085, 0.038)	(0.777, 0.023, 0.200)	(0.644, 0.006, 0.350)

Source(s): Authors work

Table 3

The effect analysis of distance parameter on proximity

Distance parameter α	The value of proximity						Standard deviation	Ranking objects
	u1	u5	u2	u3	u4	u6
0.2	0.0749	0.3017	0.3814	0.5417	0.8331	0.9475	0.3305	u6≻u4≻u3≻u2≻u5≻u1
0.4	0.0718	0.2901	0.3603	0.5309	0.8508	0.9598	0.3411	u6≻u4≻u3≻u2≻u5≻u1
0.6	0.0788	0.2887	0.3517	0.5193	0.8602	0.9651	0.3434	u6≻u4≻u3≻u2≻u5≻u1
0.8	0.0903	0.2922	0.3516	0.5101	0.8645	0.9669	0.3415	u6≻u4≻u3≻u2≻u5≻u1
1.0	0.1038	0.2977	0.3565	0.5044	0.8655	0.9669	0.3372	u6≻u4≻u3≻u2≻u5≻u1
1.2	0.1180	0.3037	0.3642	0.5018	0.8645	0.9657	0.3318	u6≻u4≻u3≻u2≻u5≻u1
1.4	0.1322	0.3098	0.3732	0.5018	0.8623	0.9638	0.3257	u6≻u4≻u3≻u2≻u5≻u1
1.6	0.1459	0.3156	0.3828	0.5034	0.8593	0.9614	0.3195	u6≻u4≻u3≻u2≻u5≻u1
1.8	0.1589	0.3212	0.3924	0.5062	0.8559	0.9587	0.3133	u6≻u4≻u3≻u2≻u5≻u1
2.0	0.1710	0.3265	0.4017	0.5096	0.8522	0.9559	0.3073	u6≻u4≻u3≻u2≻u5≻u1
2.2	0.1823	0.3316	0.4106	0.5132	0.8483	0.9531	0.3015	u6≻u4≻u3≻u2≻u5≻u1
2.4	0.1926	0.3365	0.4189	0.5170	0.8444	0.9503	0.2960	u6≻u4≻u3≻u2≻u5≻u1
2.6	0.2021	0.3411	0.4267	0.5206	0.8405	0.9476	0.2909	u6≻u4≻u3≻u2≻u5≻u1
2.8	0.2109	0.3455	0.4339	0.5241	0.8367	0.9450	0.2861	u6≻u4≻u3≻u2≻u5≻u1
3.0	0.2819	0.3498	0.4406	0.5274	0.8330	0.9425	0.2815	u6≻u4≻u3≻u2≻u5≻u1

Source(s): Authors work

Table 4

Proximity of different index background domains

Background domain Ω of the index					Distance parameter α	The value of proximity						Eigenvalues of proximity
Ω1	Ω2	Ω3	Ω4	Ω5		σ1	σ2	σ3	σ4	σ5	σ6	Average value	Standard deviation
[0,1300]	[0,50]	[0,2200]	[0,230]	[0,180]	α = 1	0.104	0.356	0.504	0.865	0.298	0.967	0.516	0.337
					α = 2	0.171	0.402	0.510	0.852	0.327	0.956	0.536	0.307
[0,1500]	[0,75]	[0,2500]	[0,250]	[0,200]	α = 1	0.079	0.269	0.385	0.729	0.215	0.858	0.423	0.307
					α = 2	0.138	0.329	0.425	0.719	0.249	0.860	0.453	0.281
[0,2000]	[0,100]	[0,3000]	[0,350]	[0,300]	α = 1	0.034	0.118	0.176	0.418	0.098	0.568	0.235	0.210
					α = 2	0.061	0.161	0.227	0.434	0.115	0.579	0.263	0.202

Source(s): Authors work

Table 5

Comprehensive evaluation results of each evaluation object

Models	The comprehensive decision value of the evaluation object						Standard deviation
	u1	u2	u3	u4	u5	u6
DGRD	0.3932	0.4855	0.5969	0.6773	0.4442	0.8138	0.1587
WK	0.2520	0.4141	0.4894	0.6951	0.3828	0.8219	0.2116
TOPSIS	0.0281	0.2442	0.4030	0.6486	0.2183	0.8397	0.3002
GCE	0.0214	0.0894	0.1931	0.4119	0.0823	0.5229	0.2024
Model in this paper	0.1710	0.4017	0.5096	0.8522	0.3265	0.9559	0.3073

Source(s): Authors work

Table 6

The dynamic evaluation results of transformer state using grey decision model when α=2

Assessment approach	Proximity				Ranking transformer state
	s1	s2	s3	s4
Period 1	0.5353	0.4610	0.4713	0.5794	s4≻s1≻s3≻s2
Period 2	0.4701	0.5666	0.5137	0.5043	s2≻s3≻s4≻s1
Period 3	0.4415	0.5184	0.6091	0.4482	s3≻s2≻s4≻s1
Period 4	0.4449	0.5133	0.5136	0.5585	s4≻s3≻s2≻s1
Comprehensive evaluation	0.4730	0.5148	0.5269	0.5226	s3≻s4≻s2≻s1

Source(s): Authors work

Table 7

The dynamic evaluation results of transformer state using grey target decision method

Assessment approach	The relative degree of bull’s-eye				Ranking transformer state
	s1	s2	s3	s4
Period 1	0.5791	0.4036	0.4964	0.5572*	s1≻s4≻s3≻s2
Period 2	0.4502	0.5194	0.5204	0.4927	s3≻s2≻s4≻s1
Period 3	0.4116	0.4883	0.6300	0.3929	s3≻s2≻s1≻s4
Period 4	0.4578	0.5302	0.5201	0.5411	s4≻s2≻s3≻s1
Comprehensive evaluation	0.4747	0.4854	0.5417	0.4960*	s3≻s4≻s2≻s1

Note(s): *The data in the reference are incorrect

Source(s): From the reference