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In recent years, domestic smog has become increasingly frequent and the adverse effects of smog have increasingly become the focus of public attention. It is a way to analyze such problems and provide solutions by mathematical methods.

This paper establishes a new gray model (GM) (1,N) prediction model based on the new kernel and degree of grayness sequences under the case that the interval gray number distribution information is known. First, the new kernel and degree of grayness sequences of the interval gray number sequence are calculated using the reconstruction definition of the kernel and degree of grayness. Then, the GM(1,N) model is formed based on the above new sequences to simulate and predict the kernel and degree of the grayness of the interval gray number sequence. Finally, the upper and lower bounds of the interval gray number are deduced based on the calculation formulas of the kernel and degree of grayness.

To verify further the practical significance of the model proposed in this paper, the authors apply the model to the simulation and prediction of smog. Compared with the traditional GM(1,N) model, the new GM(1,N) prediction model established in this paper has better prediction effect and accuracy.

This paper improves the traditional GM(1,N) prediction model and establishes a new GM(1,N) prediction model in the case of the known distribution information of the interval gray number of the smog pollutants concentrations data.

The purpose of this paper is to establish the algorithm rules of the interval grey numbers and propose a new ranking method of the interval grey numbers.

The definitions of “kernels” based on lower measure, upper measure or moderate measure are given according to the properties of the interval grey number problems. By means of the measurement error, the concept of the absolute degree of greyness and the relative degree of greyness corresponding to different “kernel” are given, and different simplified forms of the interval grey numbers are put forward.

The definitions of “kernel” and the degree of greyness in this paper not only take the upper limit, lower limit and the coverage of the interval grey numbers into account, but also avoid the inconsistency of the degree of greyness caused by the different universe of discourse.

Though the method proposed in this paper has some deficiencies, such as the definition of relative degree of greyness is meaningless when the kernel of the interval grey number is 0, it provides a new idea for calculating and sorting the interval grey numbers and is conducive to the further development of the grey system theory.

The method proposed in this paper can not only distinguish interval grey numbers in different situations, but also avoid the inconsistency of the degree of greyness caused by the different universe of discourse. In this basis, the interval grey number algorithm is established and a new ranking method of interval grey numbers is given.

This paper aims to propose an enhanced algorithm and used to decision-making that specifically focuses on the choice of a domain in the calculation of degree of greyness according to the principle of grey numbers operation. The domain means the emerging background of interval grey numbers, it is vital for the operational mechanism of such interval grey numbers. However, the criteria of selection of domain always remain same that is not only for the calculated grey numbers but also for the resultant grey numbers, which can be assumed as unrealistic up to a certain extent.

The existence of interval grey number operation based on kernel and the degree of greyness containing two calculation aspects, which are kernel and the degree of greyness. For the degree of greyness, it includes concepts of domain and calculation of the domain. The concepts of a domain are defined. The enhanced algorithm is also comprised of four deductive theorems and eight rules that are linked to the properties of the enhanced algorithm of the interval grey numbers based on the kernel and the degree of greyness.

Aiming to improve the algorithm of the degree of greyness for interval grey numbers, based on the variation of domain in the operation process, the degree of greyness of the operation result is defined in this paper, and the specific expressions for algebraic operations are given, which is relevant to the kernel, the degree of greyness and the domain. Then, these expressions are used to the algorithm of interval grey numbers based on the kernel and the degree of greyness, improving the accuracy of the operation results.

The enhanced algorithm in this paper can effectively reduce the loss of information in the operation process, so as to avoid the situation where the decision values are the same and scientific decisions cannot be made during the grey evaluation and decision-making process.

The purpose of this paper is to propose an extending correlation analysis method to deal with the correlation analysis between the sequences with incomplete information.

Based on the axiomatic definition of a grey number and its greyness degree in grey system theory, the whitenization mean, whitenization difference, whitenization covariance of sequences with interval grey numbers and their greyness degrees are defined in turn. In addition, the whitenization correlation coefficient and its greyness degree of sequences with interval grey numbers are also defined. By using the relationship between the greyness degree and kernel for a grey number, the transformation formula from the whitenization value and greyness degree of correlation coefficient to form of interval grey numbers are put forward further.

The whitenization value of correlation coefficient efficient of two arbitrary sequences with interval grey numbers have symmetry, with same greyness degree but without normalization in the interval [−1, 1]; the mean, difference, covariance and correlation coefficient defined in statistics are all the special cases of those in sequences with interval grey numbers.

Due to the complexity of operation of grey numbers, the reliability of correlation coefficient of interval numbers sequence is difficult to be tested by constructing statistics at present. The further research is needed.

The correlation analysis method of interval grey numbers can contribute to the further researches on the incomplete information system in the real world.

On the basis of grey system theory, a correlation analysis method for analyzing information incomplete sequences is proposed.

The purpose of this paper is to advance new rules about operations of grey numbers based on kernels and greyness and analyze the reliability of these new operations.

A grey number is usually represented as a closed interval or a discrete set of numbers. Based on the definition of traditional grey number, novel rules that grey numbers can be represented as their kernels and associated degrees of greyness are proposed. In new rules, the operation between kernels of grey numbers has significance in the application of grey numbers. The reliability of operations of grey numbers using their kernels and the degrees of greyness derived from kernel operations and their corresponding interval operations are studied.

The results show that the novel rules about operation of grey numbers satisfy to the concept of grey system properly. It is useful to calculate the grey degree of grey numbers and the process of calculating is easier than traditional operation rules.

The method exposed in the paper can be used to calculate each two grey numbers. The grey degree and the calculating results of two grey numbers can be given out easily. The method can also be used to calculate grey numbers more than two.

The paper succeeds in constructing novel operation rules of grey numbers. The reliability of novel operation rules is studied and it is new development of grey systems theory, undoubtedly.

Smog seriously affects the ecological environment and poses a threat to public health. Therefore, smog control has become a key task in China, which requires reliable prediction.

This paper establishes a novel time-lag GM(1,N) model based on interval grey number sequences. Firstly, calculating kernel and degree of greyness of the interval grey number sequence respectively. Then, establishing the time-lag GM(1,N) model of kernel and degree of greyness sequences respectively to obtain their values after determining the time-lag parameters of two models. Finally, the upper and lower bounds of interval grey number sequences are obtained by restoring the values of kernel and degree of greyness.

In order to verify the validity and practicability of the model, the monthly concentrations of PM2.5, SO2 and NO2 in Beijing during August 2017 to September 2018 are selected to establish the time-lag GM(1,3) model for kernel and degree of greyness sequences respectively. Compared with three existing models, the proposed model in this paper has better simulation accuracy. Therefore, the novel model is applied to forecast monthly PM2.5 concentration for October to December 2018 in Beijing and provides a reference basis for the government to formulate smog control policies.

The proposed model can simulate and forecast system characteristic data with the time-lag effect more accurately, which shows that the time-lag GM(1,N) model proposed in this paper is practical and effective.

Based on interval grey number sequences, the traditional GM(1,N) model neglects the time-lag effect of driving terms, hence this paper introduces the time-lag parameters into driving terms of the traditional GM(1,N) model and proposes a novel time-lag GM(1,N) model.

The purpose of this paper is to advance new rules about operations of grey numbers.

The paper first puts forward the definitions of basic element of grey number and general grey number. The operation axiom, operation rules of general grey numbers and a new algebraic system for general grey numbers are built based on the “kernel” and the degree of greyness of grey numbers.

Up to now, the operation of general grey numbers has been transformed to operation of real numbers; thus, the difficult problem for set up operation of general grey numbers has been solved to a certain degree.

The method exposed in the paper can be used to integrate information from a different source. The operation of general grey numbers could be extended to the case of grey algebraic equation, grey differential equation and grey matrix which includes general grey numbers. The operation system of general grey numbers also opened a new passageway for research on grey input‐output and grey programming, etc.

The new conception of a basic element of grey number and general grey number was given for the first time in this paper. The novel operation rules of general grey numbers were also constructed.

The degree of greyness may be regarded as a measure of cognitive uncertainty. Therefore, it is a part of the epistemological core of the grey systems theory. The theoretical importance of the degree of greyness concept is also due to its application in a range of uncertainty modelling methods: predictive, relational and decision-making methods. Greyness, being a result of cognitive uncertainty, was recently subjected to axiomatization in the form of grey space with the use of the classical sets theory. The purpose of this article is to develop a new approach to the degree of greyness, being consistent with the grey space concept.

In order to realise the article’s goals, the research is divided into three stages described in particular sections. The first section of the article presents a theoretical framework of the degree of greyness and the grey space. The second part includes the assumptions of the new degree of greyness concept, along with the mathematical models for the first, the second and the third degree of greyness. The third section contains numerical examples for each degree of greyness.

As a result of the research, a concept of a degree of greyness was created and it was linked with a concept of grey space. This new approach to the issue of the degree of greyness has allowed the analysing of this category in three dimensions dependent on an accepted reference base. As a result, a concept of concrete and abstractive grey numbers was introduced and relationships between these categories of numbers and the degree of greyness were determined.

The proposed approach to the issue of the degree of greyness is a theoretical unification of the previous considerations in this area. The proposed three dimensions of greyness degree will be derived from the grey space, so they will also be a function of quantity. Thus, the degree of greyness was linked with a classical set theory. An original input in this article is also a differentiation of concrete and abstractive grey numbers, which give a basis for deliberations connected with interpretation of grey numbers in the context of real applications.

The purpose of this paper is to propose a multi-scale extended grey target decision method for dealing with multi-attribute decision-making problems with interval grey numbers whose value distribution information is asymmetrical.

First, the whitenization weight function (WWF) was adopted to show the value distribution information of interval grey numbers. The definitions of kernel, degree of greyness, relative kernel and whitenization standard deviation of interval grey numbers were given based on the WWF. Then, the relative kernel grey target and whitenization standard deviation grey target were constructed to take full advantage of the owned decision information. Finally, the relative bull’s-eye coefficient was proposed to rank the preference order of all alternatives.

The relative bull’s-eye coefficient reflects the influence of the decision information on decision results with respect to the mean level and value distribution of attribute values. Thus, the decision-maker could set the return and risk adjustment coefficient according to their preferences and select the optimal alternative with a high expected return and low risk.

The paper considers the valve distribution information of interval grey numbers, and a novel definition for kernels, degrees of greyness, relative kernels and whitenization standard deviations, which are given based on the WWF. The paper not only considers the influence of mean levels of decision information over decision results, but also takes the value distribution information into account.

The purpose of this paper is to study distance measuring and sorting method of general grey number.

First, the concept of generalised grey number based on grey system theory is given in this paper. Second, from the perspective of kernel and degree of greyness of general grey number, the method of measuring the distance of general grey number and its properties are given. At the same time, the concepts of the kernel expectation and the kernel variance of the general grey number are proposed.

Up to now, the method of measuring the distance and sorting of general grey number is established. Thus, the difficult problem for set up sorting of general grey number has been solved to a certain degree.

The method exposed in this paper can be used to integrate information form a different source. Distance measuring and sorting method of general grey number could be extended to the case of grey algebraic equation, grey differential equation and grey matrix which includes general grey numbers, etc.

The concepts of the kernel expectation and the kernel variance of the general grey number are proposed for the first time in this paper; the novel sorting rules of general grey numbers were also constructed.