The Journal of Grey System

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1. Grey Exponential Cloud Decision Model for Monotonic Hesitant Fuzzy Linguistic Terms

Yuan Liu, Zhuozhuo Yang, Jinjin Zhu, Jingjing Hao, Xinwang Jiang

The Journal of Grey System 2021, 33 (1): 1-16.

摘要（269）

Monotonic natural language is widely used to express experts' subjective appraisal opinions, such as "not less than," "at least," "not more than," and "at most," which covey the information of expert hidden psychological preference. A novel quantitative computational method based on the grey exponential cloud model is proposed, transforming the expert's uncertainty appraisal information into decision data by systematically mining the experts' hesitant fuzzy preference on specific linguistic options. The comprehensive meaning of monotonic hesitant fuzzy linguistic terms is introduced, and the interval grey number is used to represent the expert's grey preference and indicator weight. Additionally, a programming model is constructed by the exponential cloud model and ABC classification analysis method, which can obtain the order of alternatives related to the optimal solution. Finally, a numerical study is conducted to show the advantages and effectiveness of the new method against other comparable methods is demonstrated.

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2. Linear Transformation Properties of Grey Model

Aiping Jiang, Xiaoling Li, Liang Zeng

The Journal of Grey System 2021, 33 (1): 17-29.

摘要（184）

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3. GRGAL: A Grey Relational Generative Adversarial Learning Method for Image Denoising#br#

Hongjun Li, Chaobo Li, Wei Hu, Junjie Chen, Shibing Zhang

The Journal of Grey System 2021, 33 (1): 30-42.

摘要（252）

Image denoising is a well-known problem in image processing. Deep networks can achieve state-of-the-art denoising results based on the quality or quantity of training samples. However, Deep networks face performance saturation when the interference of complex noise and few paired training samples. We introduce a grey relational generative adversarial learning method into the image denoising task. To solve the problem of deep network saturation caused by complex noises and the lack of paired training samples, an adversarial learning network is built to learn latent space distribution of noisy images and reconstruct the distribution of clear images. The grey relation analysis is introduced into the network to deal with the uncertainty of noisy images and improve the ability of adversarial learning. This network is optimized by a new loss function that combined the adversary and grey relation. The loss can reasonably measure the difference between the denoised images and clear images. Experiential results show the proposed method obtains the averaged PSNR of 29.67, which is 0.53 higher than the network without grey relation. Extensive experiments demonstrate the superiority of our approach in image denoising.

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4. Integrating two stages of Malmquist index and Grey forecasting to access industrial performance: A case of Vietnamese steel industry

Nhu-Ty Nguyen

The Journal of Grey System 2021, 33 (1): 43-58.

摘要（200）

Vietnam's steel industry is in its developing stage with significant progress since the openness of the economy. Many companies in this industry have adopted the development and advancement of technologies used in manufacturing. However, the productivity of the steel industry in Vietnam is still considered low compared to other countries, which leads to the decline in competitiveness in terms of Vietnam steel products to its competitors. Companies in the steel industry, as well as the government, need to know about productivity and performance in order to give out vital decisions for the development of the steel industry which is one of the prior and core industries in Vietnam. Thus, evaluation of the Vietnamese steel industry has become a significant issue. This study takes advantage of the integrated Data Envelopment Analysis model and Malmquist Productivity Index to evaluate the past-to-future performance of the Vietnam steel industry in two different timeframes; the first period was from 2014 to 2018, the second is from 2019 to 2023, which are the results from Grey system forecasting. Totally, the realistic financial reports of 16 companies are considered to be in this evaluation after the strict selection from the whole industry. Three factors are put into consideration, including efficiency change (catch-up), technical change (frontier-shift), and Malmquist productivity index are examined respectively, in each period mentioned. The results find that the performances of all companies have not shown many abrupt changes on their scores except some outstanding cases, which demonstrate the high applicable usability of the integrated methods.

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5. A Quality Overall Design Approach for Complex Products byintegrating Fuzzy QFD and Grey Relational Decision-making: A Quality Competitiveness Perspective

Huan Wang, Daao Wang, Zhigeng Fang, Xiaqing Liu

The Journal of Grey System 2021, 33 (1): 59-73.

摘要（182）

This paper aims to propose a novel quality overall design approach for complex products. The core parameter design and scheme optimization in the perspective of quality competitiveness are deeply addressed. Integrating the fuzzy QFD approach and the grey relational decision-making model, this paper explores the uncertain analysis and multiple attribute decision-making problems for the quality overall design of complex products. The overall quality design of a civil aircraft is presented as a numerical example, and conclusions and future work are discussed.

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6. Multi-stage Grey Intelligent Clustering Model

Dang Luo, Manman Zhang, Xiaolei Wang

The Journal of Grey System 2021, 33 (1): 74-97.

摘要（203）

In the process of grey clustering, the weights of indexes are always unknown and hard to obtain, and the decision paradox of the rule of "maximum value" often occurs. Aiming at the problems above, firstly, with the help of the Particle Swarm Optimization Algorithm, PSO-grey clustering coefficient vector is proposed to overcome the limitation of weights. Secondly, based on the theory of "entropy increasing theorem" and using the clustering weight vector group as an important tool, a multi-stage grey intelligent clustering model is established by introducing the entropy of clustering coefficient vector, which solves the decision paradox of rule of "maximum value" to a certain extent. To simplify the calculation process, the Matlab source code for this model is attached. Finally, by taking the drought risk assessment of irrigated agricultural areas in Henan Province as an example, the rationality and validity of the model are illustrated.

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7. Grey System Model with Complex Order Accumulation

Zhengpeng Wu, Jianke Chen, Jianping Chai, Fangyuan Zhang

The Journal of Grey System 2021, 33 (1): 98-117.

摘要（7182）

Based on the classical GM(1,1) model of integer (fractional) accumulation, we propose the grey model of complex accumulation (which is denoted by CAGM z(1,1) for some complex number z ∈ C ). This formulation extends the choice of Grey model's parameter from the real axis to the complex plane. We claim that all complex accumulated generating operators admit a1 dimensional additive complex Lie group's structure, which is isomorphic to C. This construction brings Lie group's theory in Grey model theory for the first time, and lays a foundation for introducing Lie group's tools in the grey system. The method of nilpotent matrix E1 of index n and Taylor series could avoid any usage of existing popular Γ functions, which also provides an efficient way for computer programming. As a novel method, accumulated generating operators of complex order could adjust weights between old information and new information, between the real part and the imaginary part simultaneously, better simulation and prediction results could be expected. The advantages of CAGM z(1,1) model are discussed with several cases, better simulation and prediction results are presented.

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8. Evaluation and Analysis of Smart Community Elderly Care Service Quality Based on the Two-stage Decision Model with Contents Grey Synthetic Measures under Hesitant Fuzzy Situation

Lan Xu, Yu Zhang, Yeli Wei

The Journal of Grey System 2021, 33 (1): 118-137.

摘要（379）

To ensure smart elderly care service quality, this paper explores its multi-dimensional factors, and constructs an evaluation indicator system of smart community elderly care service quality based on intelligence, tangibles, responsiveness, security, reliability, and empathy. Aiming at the problem of the fuzziness of elderly's perception and the hesitation of experts’ evaluation of service quality in multi-attribute decision-making, the interval-valued intuitionistic fuzzy entropy (IVIFE) is introduced, and the evaluation method of the smart community elderly care service quality by combining the IVIFE and a two-stage decision model with grey synthetic measure is established. Four typical cities in Jiangsu Province were taken as examples to verify the feasibility of the proposed method, in order to provide a reference for promoting service standardization and reducing the difference in the level of smart community elderly care services between regions.

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9. Evolution Mechanism of Strategic Emerging Industrial Clusters Based on Hybridization of Grey Number and Optimized Scale-Free Network

Lirong Jian, Difei Wang, Daao Wang

The Journal of Grey System 2021, 33 (1): 138-155.

摘要（226）

Strategic emerging industrial cluster network as an important form of the current industrial development, with an intensive technological innovation, prominent economies of scale and knowledge spillovers, and other characteristics, is a symbol of strategic emerging industry formation and an effective model for its development. In this paper, the Logistic model is used to characterize the strategic emerging industry cluster’s life cycle. Then, the scale-free evolution model being optimized based on the growth rate of strategic emerging industrial clusters and gray number being introduced, an evolution model of the strategic emerging industrial network is constructed considering the growth of innovators flow, the evolution process, and the stability state of the strategic emerging industrial cluster network are analyzed. Finally, an example is given to simulate and comparatively analyze the characteristics of the inflection point with the change of relevant parameters in the evolution process of the cluster network. The research results can provide some theoretical reference and guidance for the cultivation of strategic emerging

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10. Grey Signal Predictor and Evolved Control for Practical Nonlinear Mechanical Systems

ZY Chen, Lucy Huang, Huakun Wu, Yahui Meng, Shunbo Xiang, Timothy Chen

The Journal of Grey System 2021, 33 (1): 156-170.

摘要（227）

To guarantee the asymptotic stability and improve the ride comfort of vehicles, this paper develops the fuzzy neural network (NN) evolved bat algorithm (EBA) adaptive backstepping controller with grey signal predictors. To keep track of these ideal signals, Lyapunov's theory is proposed to acquire the control final laws. The gray DGM (2.1) models are also used to design suspension movements so that commands can be executed before the movement to achieve timely control. The convergence and stability of the whole system is also proven by the Lyapunov-like theory. The control expands the practices of mechanical elastic wheels (MEW) and provides a good methodical basis for a new wheel adaptation.

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11. Multi-attribute Grey Relational Similarity Measure Evaluation Method for Weapon System Performance Based on Entropyweight

Wenguang Yang , Yunjie Wu, Shu Wang

The Journal of Grey System 2021, 33 (2): 1-13.

摘要（604）

In the current study, based on grey relational similarity measure and entropy-weight, a comprehensive grey relational similarity measure method for multi-attribute decision making (MADM) problem to evaluate the alternatives has been proposed. A new algorithm based on the same attribute value's comparison and different attribute value's synthesis for the MADM problems is also established. In this algorithm, the ideal attribute values are selected according to attribute properties. Later, the grey relational similarity measure method is used to calculate the grey value between the attribute value and the ideal attribute value in the same attribute. To compare the alternatives, the entropy-weight method is applied to determine the weights for different attributes objectively. Finally, we take the multi-attribute weapon system problem as an example to illustrate the effectiveness and validity of the proposed method. The example verification results show that the method constructed in this paper has higher credibility, and the rank reversal problem has also passed the test.

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12. A novel grey model for multi-regional macro-data forecasting by considering spatial correlation and actual-state rolling

Ying Zhu, Yaping Li, Tangbin Xia, Lifeng Xi

The Journal of Grey System 2021, 33 (2): 14-28.

摘要（287）

Accurate prediction of regional development trend is important to regional planning and coordinated development in China. It provides a basis for decision-makings on the resource balance in multi-regional integration. However, due to the limitation of macro data and the influence of multi-regional correlation, the prediction accuracy of the existing forecasting methods in multi-regional macro-data forecasting is reduced. To overcome these problems, an improved grey model is proposed in this study. Firstly, a new spatial weight matrix is constructed based on the grey correlation analysis to define the spatial effect of multiple regions. Then, an actual-state rolling spatial-effect weighted grey model (ARSWGM) is developed considering the spatial interactions and the actual-state rolling mechanism. Finally, the proposed model is validated by the forecasting of manufacturing quality level of representative provinces in the process of regional coordinated development in China. The result shows that the proposed model demonstrates the best predicting performance compared with the classical grey forecasting models, indicating the advantages of this proposed model in multi-regional macro-data forecasting. Furthermore, this model can also be applied for a broader range of multi-regional limited macro-data forecasting.

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13. On The Model of Industrial Structure Coordination Degree and Optimization Planning of Industrial Structure in Jiangsu Province and China

Sifeng Liu, Jing Deng

The Journal of Grey System 2021, 33 (2): 29-38.

摘要（230）

Industrial structure is the result of the allocation of economic resources of a country or a region, and to a large extent determines the efficiency of the use of economic resources. The process of national or regional economic growth is the dynamic evolution process of industrial structure from uncoordinated to coordinated, from lower-level coordination to higher-level coordination, that is, the process of continuous optimization of industrial structure. The concepts of industrial structure deviation and industrial structure coordination degree are proposed in this paper. The weighted average values of the industrial structures of the developed countries such as the USA, UK, France, Germany and Japan in the year when their per capita GDP reaches US $10000, US $20000 and US $30000 was calculated respectively. Then, the standard industrial structure was calculated with the weighted average values of the industrial structures of the five developed countries as reference frame. And the definition of industrial structure deviation and industrial structure coordination degree is put forward accordingly. At last, the coordination degree of industrial structure in China, Jiangsu Province, Southern Jiangsu, Central Jiangsu and Northern Jiangsu is calculated according to the data of actual industrial structure in 2019. And the industrial structure optimization and adjustment plan of China and Jiangsu Province is worked out according to the set target structure

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14. Positive and Inverse Degree of Grey Incidence Estimation Model of Soil Organic Matter Based on Hyper-spectral Data

Hao Zhong, Li Li, Xican Li , Haoran Zhai , Xuesong Cao

The Journal of Grey System 2021, 33 (2): 39-57.

摘要（228）

To improve the estimation accuracy of soil organic matter based on hyper-spectral data when using degree of grey incidence, this paper first proposes the concept of the positive and inverse degree of grey incidence considering the limitation of the degree of grey incidence for estimating problems. Then, two new models of positive and inverse degrees of grey incidence are established. Thereafter, the properties of positive and inverse degree of grey incidence are analyzed. Moreover, the estimation model of soil organic matter using positive and inverse degree of grey incidence is established based on hyper-spectral data, and detailed calculation steps are given. Finally, the validity of the model is verified by taking 76 soil samples collected from Zhangqiu District, Jinan City, Shandong Province. The application results show that using the positive and inverse degree of grey incidence, the hyper-spectral estimation model of soil organic matter has high precision. The mean relative error (MRE) of 16 samples to be estimated is 5.312% and the determination coefficient (R2) is 0.930. The research shows that the positive and inverse degree of grey incidence proposed in this paper effectively expands the application of the degree of grey incidence. It is feasible and effective for hyper-spectral estimation of soil organic matter content, and meanwhile, it provides a new way to solve uncertain prediction problems.

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15. Research on Grey Forecasting of The Peak and Its Application

Jinluan Yang , Yang Li, Chaoqing Yuan

The Journal of Grey System 2021, 33 (2): 58-73.

摘要（191）

GM(1,1) is mainly used for monotone sequence prediction. But when used for sequences with peaks, such as the total population, total energy consumption, etc, GM(1,1) is not a good consideration. This paper proposes a peak prediction algorithm based on the model GM(1,1) (GPPA) to forecast the development trend of the system in the long run. According to the developing law of the system, the growth inflection point is analyzed and determined, and from which the growth rate of the system will be smaller and its trend can be decomposed into two parts based on the logistic curve of population growth: one is the original growth trend and the other one is the restraining trend of the retarding factors. Metabolic GM(1,1) is used to model both the two trends respectively and get the predictions of them. The difference between the predictions is used to forecast the development of the system. In addition, the energy consumption and population are taken as examples to prove the applicability of the method.

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16. The Economic Growth Effect of the Blue Economic Zone Based on a GRAM-DID Model

Rui Han, Kedong Yin , Xuemei Li

The Journal of Grey System 2021, 33 (2): 74-94.

摘要（270）

The marine economy is a new economic growth target under the new paradigms of China's economy. Since 2011, China has established successive marine economy demonstration zones. However, there is no conclusion yet drawn as to the effect of the marine economy demonstration zones. In this paper, the earliest Shandong Peninsula Blue Economic Zone is set as our research object, the GRAM-DID model is used to evaluate the net effect of the establishment of the Blue Economic Zone, and the marine economy of Shandong Province input-output model is compiled to estimate the economic growth effects of other industries induced thereby, which makes the assessment of the net effect more accurate and comprehensive. The results show that the net economic growth arising from establishment of the Blue Economic Zone is 304204.7 million yuan, which induces significant economic growth in the circulation of goods and services and the manufacturing sector.

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17. Grain Yield Prediction Based on the Metabolic Grey - Markov Integration Model

Chao Fan, Fangfang Chen, Hao Lin, Litao Yang, Ashley Ndlovu

The Journal of Grey System 2021, 33 (2): 95-108.

摘要（232）

forecasting model based on the weighted Markov method is proposed. Since the grain yield is affected by many uncertain factors, the grain yield is predicted by the error amended metabolism grey model. Later, considering that the grain yield is also affected by the outputs over the years, the state transition probability matrixes are calculated, and the yield influence weights of the past years are decided. Lastly, combining the predicted yield and the influence weights, the final yield is corrected by recent years' yields, and the metabolic grey model is constructed. By using above procedures, the yields of 2016 to 2020 are predicted based on the data of 2005-2015, the results show that the forecasting error is less than 2% for all predicted years, and the mean error for 5 years achieves to 1.04%, which can be used to predict the grain yield accurately in the medium and short term.

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18. A Graph model for Conflict Resolution based on a Grey Multi-criteria Preference Ranking Approach

Jian Li , Wanming Chen , Huanhuan Zhao , Renshi Zhang

The Journal of Grey System 2021, 33 (2): 109-127.

摘要（331）

PDF （368KB）（400）

Preference ranking is a vital issue in the process of graph model for conflict resolution (GMCR). Concerning the ranking problem of uncertainty preference, we propose a grey multi-criteria preference ranking approach based on grey interval numbers to represent the uncertainty preference of decision-makers, and calculate the comprehensive grey incidence degrees between DMs. And then, based on maximum entropy, we construct a multi-objective optimization model to minimize uncertainty. Finally, we exploit a real-world conflict incident of "Taihu Lake water pollution" to illustrate the feasibility and effectiveness of the proposed model

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19. Analysis of Policy Effects on New-energy Vehicles

Zhen Chen, Kanghui Zhang∗, Shuwei Jia, Dongyong Zhang

The Journal of Grey System 2021, 33 (2): 128-149.

摘要（304）

This study used system dynamics to draw a stock-and-flow diagram for new-energy vehicles. A new approach called a "reverse even grey model (1,1)" was developed, and SPSS was used to process the data. Furthermore, the synthetic degree of incidence was used to examine the feasibility of the model. Four hypothetical scenarios for new-energy vehicles development were analyzed to find the ideal solution for China. These analyses showed the following: (1) Cadmium has certain environmental risks. At present, battery recycling is the main problem related to new-energy vehicles in China. (2) Total energy consumption is influenced by the dominant vehicle type in the market (new-energy vehicles and traditional fuel vehicles). (3) The government's purchase policy can promote sales of new-energy vehicles, but its effect is limited. Finally, (4) a "green paradox" effect exists in new-energy vehicles in China. Based on the findings, suggestions are made for the reasonable development of new-energy vehicles in China. Hence, this research has certain health and environmental benefits.

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20. The Optimal Solution for Grey Fuzzy Flexible Linear Programming Problems Based on The Feasibility and Efficiency Concepts

M. Asghari , Bing Yuan Cao , S.H. Nasseri

The Journal of Grey System 2021, 33 (2): 150-165.

摘要（227）

The purpose of this paper is to extend the newly established α- feasibility and α- efficiently concept for grey flexible fuzzy linear programming, so as to present some important new concepts, models, methods, and a new framework of grey system theory in mathematical programming. In this paper, we concentrate on Grey Fuzzy Flexible Linear Programming (GFFLP) problems as a reasonable extension of GLP models that adapt more to real situations. For this aim, after defining the classical GFFLP model, we first introduce a new concept of α ̅-feasibility and α ̅- efficiency to these problems, and then we propose a two-phase approach to solve the mentioned problems. Furthermore, we give some fundamental theorems and constructive results to support and verify the proposed solving process. This approach will be open a new window to the modeling of the problems in the real world under flexibility conditions. A lot of successful practical applications of the new models to solve various problems have been found in many different areas and disciplines such as agriculture, decision sciences, diet problem, ecology, economy, geology, earthquake, industry, material science sports, medicine, management, transportation, and etc. Because of the ability to deal with poor, incomplete, or uncertain problems with grey systems, most real-world processes in decision problems are in the grey stage due to lack of information and uncertainty. However, the flexibility assumption in decision making is more comfortable for the Decision Maker (DM), hence in this paper, we concentrate on Grey Fuzzy Flexible Linear Programming (GFFLP) problems as a reasonable extension of GLP models in which is more adept with the real situations, 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 when the available information is incomplete and the collected data is inaccurate. In this study, a general picture of grey mathematical programming under flexibility conditions is given as a new model and a new framework for various real problems where partial information is known; especially for uncertain decision systems with few data points and poor information.

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21. A Novel Grey Incidence Decision-making Method Embodying Development Tendency and Its Application

Heng Ma, Peng Yu, Yingjie Yang, Liangyan Tao, David Mba

The Journal of Grey System 2021, 33 (3): 1-15.

摘要（418）

In grey incidence decision-making models, the development tendency of each indicator value for the evaluated object is rarely considered, and the degree of discrimination between evaluation values is not high enough sometimes. In view of this, a novel grey incidence decision-making method embodying development tendency is proposed, which can guide the evaluated objects to a better direction in the future and can also distinguish the evaluation results to the greatest extent. Firstly, the development factor is defined, which can exert an effect on the development tendency of each indicator value over time. Secondly, guided by an exponential function, the weighted degree of grey incidence based on exponential function is constructed by combining the maximizing deviation and grey entropy in assigning weights to the indicators. Thirdly, the weights of the time series are delivered by the combined weighting method based on level difference maximization. Hence, the dynamic evaluation values are produced for ranking the evaluated objects. Finally, a practical example of the transformation and upgrading of the manufacturing industry in the Yangtze River Delta (YRD) demonstrates the effectiveness and application of the proposed model.

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22. Modelling Principles of Grey Matrix Incidence Analysis for Panel Data

Decai Sun, Dang Luo, Huihui Zhang

The Journal of Grey System 2021, 33 (3): 16-30.

摘要（315）

Grey incidence analysis (GIA), a branch of grey system theory, is commonly used in a broad range of scientific disciplines, from natural to social sciences. Since most current research on GIA models for panel data focuses on improving them, the mathematical principles and physical interpretations receive relatively limited attention. The principles of grey matrix incidence analysis (GMIA), which allows for both cross-sectional and time-series characteristics of panel data, are proposed in this paper. The panel data is first represented as a matrix, and then the matrix incidence operators are presented, along with theoretical properties and physical interpretations. The modeling principles, including the normativity, closeness, and column permutation independence, are articulated mathematically in a concise manner. The unified representation of GMIA models is then suggested, and the comprehensive procedures for expanding the GIA models for time series into the GMIA models for panel data are illustrated using the generalized GIA model as an example. Finally, the findings of the two examples indicate that the proposed solution has interpretability and robustness advantages over the compared approaches.

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23. Discrete Grey DGMFP(1,1,r) Model with Fractional Polynomial and Its Application

Jun Zhang , Chong Liu , Tongfei Lao , Zhanbo Chen

The Journal of Grey System 2021, 33 (3): 31-42.

摘要（242）

Discretization is an effective tactic to improve the accuracy of grey prediction model. In order to further improve the accuracy of the discrete grey prediction model, based on the discrete grey DGMP(1,1,N) model with polynomial, the degree of polynomial is expanded from integer to fraction, and the discrete grey DGMFP(1,1,r) model with fractional polynomial is proposed in the present study. To determine the best DGMFP(1,1,r) model, the mean absolute percentage error (MAPE) is established as an objective function of the optimization model, and a quantum genetic algorithm is used to calculate the optimal degree of fractional polynomials in DGMFP(1,1,r) model. Finally, the empirical results from two application cases indicate that, compared with other discrete grey models, DGMFP(1,1,r) model has a higher simulation and prediction accuracy and can overcome the restrictions of DGMP(1,1,N) model class ratio test, and has stronger generalization ability and wider adaptability

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24.

Applying Grey absolute degree of incidence and TOPSIS to evaluate Financial Performance: Case of Companies of Automotive Industry and Auto-Parts Manufacturing Group in Tehran Stock Exchange

Ehsan Javanmardi , Sadaf Javanmardi , Naiming Xie , Chaoqing Yuan

The Journal of Grey System 2021, 33 (3): 43-66.

摘要（298）

This study seeks to create an optimal investment portfolio by applying grey principal components analysis (GPCA) to financial performance evaluation. The GPCA model concomitantly relies on the advantages of grey systems theory (requiring no definite range of data distribution and using limited data) and those of principal components analysis (reducing variable dimensionality, assigning fitted weights to variables, providing multivariate evaluation). This study uses 25 financial indicators to evaluate the financial performance and determine optimal investment portfolios in 28 companies in Tehran Stock Exchange within five years from 2015 to 2019. The grey relations matrix is created through grey relational analysis and replaces the covariance matrix in the principal components analysis method. To verify the model, TOPSIS is used, and a correlation coefficient test is conducted between the results of the two models across the five years. The significant correlation between the techniques confirms the validity of the model. Furthermore, to decide the most important financial ratios affecting the companies’ evaluation, the correlation between each of the ratios and the results of the model solution is computed. The findings show that total assets, return on total assets, net working capital, current ratio, the price at the end of the period, and return on common stockholders are the most important financial ratios in the ranking of the companies.

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25. The New Axiomatization of the Grey Shapley Value

Osman Palancı

The Journal of Grey System 2021, 33 (3): 67-77.

摘要（242）

The grey Shapley value is a solution concept in cooperative games where the coalitional values are interval grey numbers. Recently, much attention has been paid on this value in Operations Research models and methods. The purpose of this study is to characterize this value on cooperative grey games. The grey Shapley value is characterized by following some axioms. Our axioms are g-efficiency, g-triviality , g-coalitional strategic equivalence, and g-fair ranking. These axioms give us a new perspective on the characterization of this value. Finally, some examples and applications of cooperative grey games are also given. It is hoped that this study will inform readers for axiomatic characterization of this value.

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26. Preservation Behavior Research on Perishable Products Supply Chain Based on Grey Game

Qing Zhang , Xinyu Ma , Zhichao Zhang

The Journal of Grey System 2021, 33 (3): 78-99.

摘要（223）

This paper discusses the preservation behavior of participants in the perishable products supply chain by modeling two preservation methods, ecological preservation, and traditional preservation, and listing eight combinations of preservation strategies under non-cooperation and cooperation. We grey-quantize indescribable preservation efforts to establish grey interval functions of profits under different situations from three perspectives to obtain optimal decision-making, and then make conclusions through numerical analysis: (1) in the view of grower profits-oriented, cooperative preservation is always the best choice, which is completely contrary to the marketing side perspective; (2) making more efforts to preserve products doesn’t always bring high profits;(3) in most cases, ‘cooperation’ and ‘ecological preservation’ are relatively beneficial. This paper provides some enlightenment on choosing the most suitable preservation methods and efforts to better coordinate the relationships among the perishable products supply chain.

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27. Multi-variable DGMTP(1,N,α ) Prediction Model with Time Polynomial

Ye Li, Yuanping Ding , Bin Liu

The Journal of Grey System 2021, 33 (3): 100-115.

摘要（245）

The multi-variable grey prediction model represented by the GM(1,N) model is an important causal relationship forecasting model. However, the GM(1,N) and its improved models believe that the development trend of the dependent variable sequence is only related to its own lag term and independent variables while ignoring the development trend of the dependent variable sequence with time. For this, a multi-variable DGMTP(1,N,α ) prediction model with time polynomial is proposed, and the value of parameter α is solved by debugging method. It is theoretically proved that the DGMTP(1,N, α ) model can achieve mutual transformation with the multi-variable GM(0,N) model, GM(1,N) model, DGM(1,N) model and the uni-variable GM(1,1) model, DGM(1,1) model, NDGM(1,1) model by adjusting the parameter values. To illustrate the performance of the DGMTP(1,N,α ) model, the new model is used to simulate and predict the air quality index in Zhengzhou city. The simulation and prediction results of the DGMTP(1,N,α ) model are compared with those of other grey and non-grey prediction models. Results show that the DGMTP(1,N,α ) model has evidently superior performance to other prediction models; this is because the DGMTP(1,N,α ) model avoids the large sample requirement of the non-grey prediction model in the modeling, avoids the jumping error in parameter estimation and application, and considers the time development trend of dependent variable sequence, which fully proves that the structure of the DGMTP(1,N,α ) model is reasonable and practicable.

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28. Automatic Lung Parenchyma Segmentation of CT Images Based on Matrix Grey Incidence

Caixia Liu , Wanli Xie

The Journal of Grey System 2021, 33 (3): 116-129.

摘要（195）

Accurate lung parenchyma segmentation plays an important role in lung disease diagnosis, which contributes to improving the survival rate and prognostic conditions. However, image noises, complex thorax tissue structures, large individual differences, and so on make lung segmentation a complex task. In this paper, an automatic lung parenchyma segmentation algorithm based on superpixels and matrix degree of grey incidences is presented to address the problem. Lung CT image is first preprocessed with a group of morphological operations and then divided into a set of superpixels. Then, matrix grey incidence is utilized to classify the superpixels of the thorax into lung tissues and pleural tissues after the reference superpixels were extracted. Finally, the segmentation results are refined with a contour correction approach based on corner detection and convex hull to facilitate accurate lung contours. The segmentation results of our algorithm are compared with ground truths, and experimental results show that the proposed algorithm achieves high accuracy, and the average Jaccard's similarity index is more than 92%.

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29. Method for Robust Multiple Criteria Decision Making Based on Grey Relational and Information Extension

Baohua Yang , Haidan Zhao, Jinshuai Zhao

The Journal of Grey System 2021, 33 (3): 130-149.

摘要（216）

Several multiple criteria decision making(MCDM) techniques have been used to assist decision-makers (DMs) in selecting better alternatives for various problems. However, it has been observed that with the addition of new alternatives or the deletion of existing ones, the rank of available alternatives will present a problem referred to as rank reversal. An improved grey relational analysis method is proposed in which the information expansion method and virtual ideal scheme are used to prevent changes of extreme value in standardization when there are alterations in the alternative set. These are the main reasons for rank reversal. When comparing literature on results from case studies and simulated cases, it is clear that the new method can maintain a robust rank of alternatives. This indicates that the proposed method is capable of preventing the rank reversal phenomenon, which arises out of changes in available alternatives.

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30. A Novel Time-Varying Multivariable Nonlinear Grey Model and Its Application

Sandang Guo, Yaqian Jing, Qian Li

The Journal of Grey System 2021, 33 (3): 150-163.

摘要（319）

This study develops a novel time-varying multivariable nonlinear grey model, namely TVNGM(1,N), which can capture the nonlinear and potential features of dynamic development trends. The novel multivariable nonlinear grey model has introduced a linear time-varying driving coefficient to replace the proposed model's constant parameter and added adjustment coefficient. The new model can be completely compatible with a single variable and multivariable grey models by adjusting different parameter values. For furtherly improving forecasting accuracy, the particle swarm optimization (PSO) algorithm is used to efficiently optimize the model’s parameters. Then, estimated parameters and the connotative prediction formula of the TVNGM(1,N) model are deduced by using the difference equation. To this end, two case studies are selected to prove the practicality of the method and compare it with other models. The results demonstrate that the proposed model has superior performance.

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31. A Grey Target Decision-Based Risk Evaluation and Prioritization Method for FMEA with Interval Grey Number

Dandan Wang, Lirong Jian, Sifeng Liu, Shuaishuai Fu

The Journal of Grey System 2021, 33 (4): 1-15.

摘要（164）

As an effective risk evaluation method, the Failure mode and effect analysis (FMEA) has been widely adopted to assist in controlling risk and enhancing the reliability of the system in a variety of workplaces. Nevertheless, the current risk prioritization approach for FMEA is insufficient to fully handle the deviation among risk evaluation information from experts with different professional backgrounds in the existing literature. With the intention of remedying this gap, this study put forward a novel risk evaluation and prioritization method for FMEA based on grey theory. Firstly, in order to cope with the deviation in risk evaluation information, which is caused by extreme information from group decision-making, the interval grey number is used as language variables for risk evaluation by experts with different professional backgrounds. Secondly, consider the correlation among risk factors, the important weight of these risk factors is determined by grey relational analysis and entropy measurement method. Thirdly, the grey target decision approach is adopted to decide the priority of these failure modes. Finally, an instance of an aircraft landing system is used to verify the viability and application of this proposed method.

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32. Assessment of China’s Aerospace Industry Sustainable Development Capability

Chaoqing Yuan, Xiuqin Wang

The Journal of Grey System 2021, 33 (4): 16-31.

摘要（138）

The aerospace industry is very important due to its characteristic of dual-use. The aerospace industry is facing greater difficulties in terms of sustainability of both resource input and market expansion than the early years. This paper defines the aerospace industry's sustainable development capability, and its connotations are analyzed from four aspects. On this basis, the aerospace industry sustainable development capability assessment index system is constructed, consisting of 15 indexes whose weights are obtained by using the AHP method. With this index system, the sustainable development capability of China's aerospace industry from 2012 to 2018 is assessed by using the method of grey fixed-weight clustering where grey classes including High, Medium, and Low, and it is found that the sustainable development capability of China's aerospace industry belongs to High or Medium grey classes, but the problems of industrial ecology and industrial system flexibility are rather prominent. Accordingly, some suggestions are put forward.

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33. Evaluation and Analysis of Key Influencing Factors of Scientific Research Efficiency of "Double First-Class" Universities in China

Hongwei Li, Haiyang Jiang

The Journal of Grey System 2021, 33 (4): 32-45.

摘要（151）

Scientific research efficiency is an important index of the ability of an institution to comprehensively utilize scientific research resources, especially in "Double First-Class" Universities in China. It is significant for guiding the optimal allocation of universities’ scientific research resources as they are constructed. In this study, an evaluation index system for "Double First-Class" Universities’ scientific research efficiency is constructed based on the characteristics of their scientific research activities. Their efficiency is evaluated scientifically through an output-oriented two-stage super efficiency Data Envelopment Analysis model, and key influencing factors of efficiency are identified using the improved Grey Correlation Analysis method. 27 "Double First-Class" Universities’ scientific research efficiencies are calculated based on data from 2016 to 2018. The results show that: (1) The two-stage DEA model considers the hysteresis of scientific activities and can calculate the decomposed efficiency. (2) The overall scientific research efficiency of "Double First-Class" Universities in China is low, and there is a wide range of efficiencies among the universities. (3) Number of highly cited papers and the total number of published papers are the key influencing factors of scientific research efficiency. Based on these results, some suggestions are put forward to improve the scientific research efficiency of "Double First-Class" Universities in China.

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34. Predictive Model for the Number of Elevator Failures Based on the Residual Error Correction Model and a GM (1,1)-Markov Chain

Xin Feng, JunCheng Jiang, WenFeng Wang, YueGui Feng

The Journal of Grey System 2021, 33 (4): 46-60.

摘要（127）

The rapid increase in the number of elevators in China demonstrates the indispensable role in daily life. We propose a prediction model based on the residual error correction model and the Markov chain for forecasting the number of elevator failures. First, we developed a traditional grey model (GM) (1,1) by using data from January to October; we then used the residual errors to correct the GM(1,1), thereby creating an error-correction GM (C-GM). The experimental results indicated that after two corrections, this model achieved a variance ratio C was 0.26, and a small error probability P was 1.0. Finally, we created a Markov-C-GM by combining the C-GM with the Markov chain to predict the number of elevator failures for the subsequent 2 months. We also compared this model with the autoregressive integrated moving average (ARIMA) model regarding their ability to predict the number of elevator failures in December; the results demonstrated that the Markov-C-GM outperformed the ARIMA model in terms of its ability to process small-sample data.

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35. The Influence of Random Disturbance on DGM(1,1) Model

Lingling Pei, Junjie Yan, Yuanyuan Guo

The Journal of Grey System 2021, 33 (4): 61-74.

摘要（197）

At present, the DGM(1,1) model has many applications, which makes the model have a lot of improvement in accuracy and robustness. However, for these current applications, the impact of random disturbances on the model is not considered. A white noise is introduced in the traditional DGM(1,1) model by referring to tthe idea of random difference. The iterative method is used to solve the DGM(1,1) model which introduces the white noise sequence, and the corresponding analysis formula is obtained. According to this analysis, different values of β_1 will cause random disturbances to have different effects on the model. The cases with different β_1 values are used for comparative analysis. In the more general case of this model, what will the properties of the model differ after adding the random disturbance? The results show that when |β_1|<1, the variance of the model converges gradually under the influence of random disturbance, and the model is relatively stable, while when |β_1|>1, the variance increases exponentially, and the model is in an unstable state.

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36. Interval Number Time Series Forecasting Based on GM (1, 1) and Nonlinear Regression

Xiangyan Zeng, Yunjie Mei, Shuli Yan

The Journal of Grey System 2021, 33 (4): 75-88.

摘要（163）

GM (1, 1) model is suitable for the series with exponential growth and small fluctuation, but the prediction accuracy of the series with parabolic or saturated development trends is not high. Parabolic growth sequences are widely found in practical problems. For example, the annual GDP of some provinces in China grew rapidly in the early stage but slowed down in the later stage, presenting a parabolic or saturated development trend. In order to improve the prediction effect of grey model on exponential and parabolic sequences, the sum of a quadratic polynomial and GM (1, 1) model is proposed as a new model (NRGM (1, 1)). Furthermore, the matrix model (MINRGM (1, 1)) of NRGM (1, 1) is proposed, which is directly applicable to interval number sequences. The prediction formula of the model is obtained based on Cramer's law. The MINRGM (1, 1) model is used to forecast China's railway passenger volume, civil aviation passenger volume, total passenger volume, and the GDP of Hebei province, and the GDP of Hebei province presents a parabolic development trend. Compared with the competition models, the MINRGM (1, 1) model achieves better prediction results.

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37. A Grey Information Evolution Theory From the Perspective of Topological Structures

Jianghui Wen, Chaozhong Wu, Xinping Xiao, Yu Shi

The Journal of Grey System 2021, 33 (4): 89-101.

摘要（111）

For analyzing the evolution process of grey information, this paper aims to establish a mathematical structure theory of grey information which is limited and insufficient. Firstly, grey space theory is built to describe the connotation and extension of grey information, proving that grey information during evolution satisfies the grey kernel attribute invariance and grey information measure non-increasing principles. Secondly, the dynamic evolution of grey information is calculated based on grey information measurement. Then, complete steps of the grey clustering method based on grey information evolution theory are given. Our method can be extended to a general uncertain system.

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38. Game Network and Equilibrium Analysis of Mixed-Ownership Asset Supervision in a Grey Information Context

Jing Wang, Yufeng Song, Xiaoyu Yang, Na Zhang, Zhigeng Fang

The Journal of Grey System 2021, 33 (4): 102-128.

摘要（125）

The supervision of state-owned assets is not only an important issue in the reform of state-owned assets but also the key to the preservation and appreciation of state-owned assets. This research combines the generality of modern enterprise governance with the Chinese characteristics of adhering to the Party’s leadership over state-owned enterprises and proposes a new regulatory framework for state-owned assets. By combining the principal-agent chain relationship of mixed-ownership enterprises, a grey principal-agent model under the incomplete information on agents’ actions is constructed, and based on the constructed model, and an optimization algorithm of the grey principal-agent problem is designed by employing the cooperative game equilibrium solution. The results show that: under the condition of constant supervision intensity, the optimal effort level of public and non-public managers of state-owned asset appreciation is inversely proportional to their effort cost coefficient, that is, the lower the effort cost coefficients of the public and non-public managers, the higher the optimal effort levels, thereby the higher the level of asset appreciation. In addition, the asset appreciation incentive coefficient of public and non-public managers always decreases with increasing supervision intensity. With the increase in supervision intensity, the income of the enterprise always increases, but the net profit decreases after exceeding the critical point of optimal supervision intensity, which means that the optimal supervision intensity to maximize the interests cannot guarantee the maximization of asset appreciation. By introducing the collaborative regulatory framework, the greyness of the three participants’ income decreases, and the minimum incomes of the three participants under supervision can be guaranteed to be higher than that without supervision over a wide range.

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39. The Superiority Analysis of Two Kinds of Discrete Grey Model with Fractional-Order Accumulation

Jiefang Liu, Pumei Gao, Shanshan Li, Bingjun Li

The Journal of Grey System 2021, 33 (4): 129-137.

摘要（110）

The basis of the grey forecasting model is the accumulation of original data. The rule of system change can be highlighted through the accumulation of data. The fractional-order accumulation discrete grey forecasting model presents a fractional-order accumulation method for raw data. The stability of the model can be effectively increased by fractional-order accumulation. The fractional-order reverse accumulation discrete grey forecasting model can make full use of the new information of the system, which is more consistent with the principle of new information priority. In this paper, the characteristics and advantages of the two models are compared theoretically and verified by an example. We found that when the old data of the original data were disturbed, the stability of the discrete grey forecasting model with reverse fractional-order accumulation was good.

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40. Multi-attribute Annular Grey Target Decision-making Based on Density Weighted Operators

Sandang Guo, Bingjun Li, Fenyi Dong

The Journal of Grey System 2021, 33 (4): 138-149.

摘要（100）

An annular grey target decision-making method based on density weighted operator is proposed in view of the present situation that the existing grey target decision-making methods don’t divide the grey target into different groups and couldn't scientifically solve the decision problems that the distribution of the evaluation information is inconsistent. According to different dentistry, the evaluation information is divided into different categories by orderly incremental method, and a nonlinear programming model is established for calculating the weight vector. On this basis, a grey decision-making method with one bull's-eye and multiple annular gaps is defined, and the corresponding weight vector is given. It is made the evaluation results more reasonable and practical by using twice aggregation of the evaluation information inside the rings and between the rings. Finally, two cases are given to illustrate the feasibility and validity of the proposed method.

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