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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 make multivariable gray model to be available for the application on interval gray number sequences directly, the matrix form of interval multivariable gray model (IMGM(1,m,k) model) is constructed to simulate and forecast original interval gray number sequences in this paper.

Firstly, the interval gray number is regarded as a three-dimensional column vector, and the parameters of multivariable gray model are expressed in matrix form. Based on the dynamic gray action and optimized background value, the interval multivariable gray model is constructed. Finally, two examples and comparisons are carried out to verify the effectiveness of IMGM(1,m,k) model.

The model is applied to simulate and predict expert value, foreign direct investment, automobile sales and steel output, respectively. The results show that the proposed model has better simulation and prediction performance than another two models.

Due to the uncertainty information and continuous changing of reality, the interval gray numbers are used to characterize full information of original data. And the IMGM(1,m,k) model not only considers the characteristics of parameters changing with time but also takes into account information on lower, middle and upper bounds of interval gray numbers simultaneously to make better suitable for practical application.

The main contribution of this paper is to propose a new interval multivariable gray model, which considers the interaction between the lower, middle and upper bounds of interval numbers and need not to transform interval gray number sequences into real sequences. According to combining different characteristics of each bound of interval gray numbers, the matrix form of interval multivariable gray model is established to simulate and forecast interval gray numbers. In addition, the model introduces dynamic gray action to reflect the changes of parameters over time. Instead of white equation of classic MGM(1,m), the difference equation is directly used to solve the simulated and predicted values.

The purpose of this paper is to research attribute reduction and decision making by gray dual‐information, taking into account the attribute reduction of attribute decision unknown for the interval gray numbers.

The authors obtain the attribute weights considering the consistency of experts’ judgment matrixes and the decision matrixes with gray information. They propose some experts’ attribute reduction ideas based on interval gray numbers of rough set. With the help of experts’ decision information, they consider attribute uncertainty ratio and attribute value ratio to reduce attribute. Finally, a numerical example shows its feasibility.

Some experts’ attribute reduction ideas are proposed based on interval gray numbers of rough set. With the help of experts’ decision information, attribute uncertainty ratio and attribute value ratio to reduce attribute can be considered.

Attribute reduction is keeping classified information systems under the same conditions and deleting redundant and irrelevant or unimportant attributes in order to solve the problem of decision making. This paper considers the attribute reduction based on gray dual‐information.

The purpose of this paper is to determine the project completion time and cost under non-deterministic conditions using interval gray numbers (IGNs).

The earned value management (EVM) method based on the IGN has been developed.

The EVM method based on the IGN has been verified by a numerical example that can be applied to construction projects.

The EVM method, based on the gray numbers, reduces the budget and time shortage risk. Also, using this method, the managers would not be restricted to provide very exact values in their progress reports in the non-deterministic conditions.

One notable and significant point in all projects during the execution process is to estimate the project completion time and cost. However, non-deterministic conditions for both planned and actual physical completion percentage of projects have not been considered for predicting the project completion time and cost in the literature. Therefore, the novelty of this paper is the prediction of project completion time and cost under non-deterministic conditions using IGN.

The purpose of this study is to construct a gray relational model based on information diffusion to avoid rank reversal when the available decision information is insufficient, or the decision objects vary.

Considering that the sample dependence of the ideal sequence selection in gray relational decision-making is based on case sampling, which causes the phenomenon of rank reversal, this study designs an ideal point diffusion method based on the development trend and distribution skewness of the sample information. In this method, a gray relational model for sample classification is constructed using a virtual-ideal sequence. Subsequently, an optimization model is established to obtain the criteria weights and classification radius values that minimize the deviation between the comprehensive relational degree of the classification object and the critical value.

The rank-reversal problem in gray relational models could drive decision-makers away from using this method. The results of this study demonstrate that the proposed gray relational model based on information diffusion and virtual-ideal sequencing can effectively avoid rank reversal. The method is applied to classify 31 brownfield redevelopment projects based on available interval gray information. The case analysis verifies the rationality and feasibility of the model.

This study proposes a robust method for ideal point choice when the decision information is limited or dynamic. This method can reduce the influence of ideal sequence changes in gray relational models on decision-making results considerably better than other approaches.

The statistical report published by the Iranian Social Security Organization in 2012 showed that, of all industries, the construction industry is associated with the highest number of work-related accidents. Furthermore, as this sector contains a large human workforce, identification of the factors contributing to the occurrence of such accidents is vital. The paper aims to discuss these issues.

Furthermore, as this sector contains a large human workforce, identification of the factors contributing to the occurrence of such accidents is vital. In the present study, such factors were initially identified, after which the most important of these, managerial factors, were selected. Subsequently, the identification of the causes of the managerial factors was carried out with the use of the fault tree analysis (FTA) method and application of OR and AND entries.

Since it is difficult to determine the probability of occurrence of events in this industry with certainty, and because gray numbers (numbers of which the exact value is unknown, and which represent uncertain information) have a strong relationship with human expressions, the probability of occurrence of the main undesired events was also evaluated using the gray numbers as input entries, in addition to ranking the probability of occurrence of intermediate events.

The findings revealed that FTA using gray numbers is a useful and effective tool in risk assessment.

This paper investigates a problem in a reverse logistics (RLs) network to decide whether to dispose of unsold goods in primary stores or re-commercialize them in outlet centers. By deducting the costs associated with each policy from its revenue, this study aims to maximize the profit from managing unsold goods.

A new mixed-integer linear programming model has been developed to address the problem, which considers the selling prices of products in primary and secondary stores and the costs of transportation, cross-docking and returning unwanted items. As a result of uncertain nature of the cost and time parameters, gray numbers are used to deal with it. In addition, an innovative uncertain solution approach for gray programming problems is presented that considers objective function satisfaction level as an indicator of optimism.

According to the results, higher costs, including transportation, cross-docking and return costs, make sending goods to outlet centers unprofitable and more goods are disposed of in primary stores. Prices in primary and secondary stores heavily influence the number of discarded goods. Higher prices in primary stores result in more disposed of goods, while higher prices in secondary stores result in fewer. As a result of the proposed method, the objective function satisfaction level can be viewed as a measure of optimism.

An integral contribution of this study is developing a new mixed-integer linear programming model for selecting the appropriate goods for re-commercialization and choosing the best outlet center based on the products' price and total profit. Another novelty of the proposed model is considering the matching percentage of boxes with secondary stores’ desired product lists and the probability of returning goods due to non-compliance with delivery dates. Moreover, a new uncertain solution approach is developed to solve mathematical programming problems with gray parameters.

In order to make full use of the generalized greyness of interval grey number, this paper analyzes the properties and its applications of generalized greyness.

Firstly, the static properties of generalized greyness in bounded background domain, infinite background domain and infinitesimal background domain are analyzed. Then, this paper gives the dynamic properties of generalized greyness in bounded background domain, infinite background domain and infinitesimal background domain and explains the dialectical principle contained in it. Finally, the generalized greyness is used to judge the effectiveness of interval grey number transformation.

The results show that the generalized greyness of interval grey number has relativity, normativity, unity, eternity and conservation. The static and dynamic properties of generalized greyness are the same in the infinite and infinitesimal background domain, while they are different in the bounded background domain. The generalized greyness can be used as an index to judge whether the grey number transformation is greyness or information preservation.

The research shows that the generalized greyness can be used as an index to judge the validity of the grey number transformation and also can be applied in grey evaluation, grey decision-making and grey prediction and so on.

The paper succeeds in realizing the mathematical principle of “white is black”, the “greyness clock-slow effect” of the value domain of interval grey number and the generalized greyness conservation principle, which provides a theoretical basis for the rational use of generalized greyness of interval grey number.

Aim to the limitations of grey relational analysis of interval grey number, based on the generalized greyness of interval grey number, this paper tries to construct a grey angle cosine relational degree model from the perspective of proximity and similarity.

Firstly, the algorithms of the generalized greyness of interval grey number and interval grey number vector are given, and its properties are analyzed. Then, based on the grey relational theory, the grey angle cosine relational model is proposed based on the generalized greyness of interval grey number, and the relationship between the classical cosine similarity model and the grey angle cosine relational model is analyzed. Finally, the validity of the model in this paper is illustrated by the calculation examples and an application example of related factor analysis of maize yield.

The results show that the grey angle cosine relational degree model has strict theoretical basis, convenient calculation and is easy to program, which can not only fully utilize the information of interval grey numbers but also overcome the shortcomings of greyness relational degree model. The grey angle cosine relational degree is an extended form of cosine similarity degree of real numbers. The calculation examples and the related factor analysis of maize yield show that the model proposed in this paper is feasible and valid.

The research results not only further enrich the grey system theory and method but also provide a basis for the grey relational analysis of the sequences in which the interval grey numbers coexist with the real numbers.

The paper succeeds in realizing the algorithms of the generalized greyness of interval grey number and interval grey number vector, and the grey angle cosine relational degree, which provide a new method for grey relational analysis.

The purpose of this paper is to summarize the progress in grey system research during 2000-2015, so as to present some important new concepts, models, methods and a new framework of grey system theory.

The new thinking, new models and new methods of grey system theory and their applications are presented in this paper. It includes algorithm rules of grey numbers based on the “kernel” and the degree of greyness of grey numbers, the concept of general grey numbers, the synthesis axiom of degree of greyness of grey numbers and their operations; the general form of buffer operators of grey sequence operators; the four basic models of grey model GM(1,1), such as even GM, original difference GM, even difference GM, discrete GM and the suitable sequence type of each basic model, and suitable range of most used grey forecasting models; the similarity degree of grey incidences, the closeness degree of grey incidences and the three-dimensional absolute degree of grey incidence of grey incidence analysis models; the grey cluster model based on center-point and end-point mixed triangular whitenization functions; the multi-attribute intelligent grey target decision model, the two stages decision model with grey synthetic measure of grey decision models; grey game models, grey input-output models of grey combined models; and the problems of robust stability for grey stochastic time-delay systems of neutral type, distributed-delay type and neutral distributed-delay type of grey control, etc. And the new framework of grey system theory is given as well.

The problems which remain for further studying are discussed at the end of each section. The reader could know the general picture of research and developing trend of grey system theory from this paper.

A lot of successful practical applications of the new models to solve various problems have been found in many different areas of natural science, social science and engineering, including spaceflight, civil aviation, information, metallurgy, machinery, petroleum, chemical industry, electrical power, electronics, light industries, energy resources, transportation, medicine, health, agriculture, forestry, geography, hydrology, seismology, meteorology, environment protection, architecture, behavioral science, management science, law, education, military science, etc. These practical applications have brought forward definite and noticeable social and economic benefits. It demonstrates a wide range of applicability of grey system theory, especially in the situation where the available information is incomplete and the collected data are inaccurate.

The reader is given a general picture of grey systems theory as a new model system and a new framework for studying problems where partial information is known; especially for uncertain systems with few data points and poor information. The problems remaining for further studying are identified at the end of each section.