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PurposeThe purpose of this paper is to summarize the advances in grey system theory research and various application achievements in science and engineering. At the same time, it commemorates the 40th anniversary of the birth of grey system theory and the 10th anniversary of Grey Systems–Theory and Application.Design/methodology/approachFirstly, the innovations of theoretical research in grey system theory were summarized and some of the widely recognized new results are briefly described. By searching and combing the research results of grey system theory in China national knowledge infrastructure (CNKI) database and Web of Science by Institute for Scientific Information (ISI), this paper shows the rapid development trend of grey system theory in the past 40 years, and the successful applications of grey system theory in the fields of social sciences, natural sciences and engineering technologies.FindingsMore than 227 thousands literature were found by input 10 phrases such as grey system, grey number and sequence operator etc. in CNKI database. After entering the new century, the number of grey system papers included in CNKI database is increasing rapidly. Since 2008, more than 10 thousands papers have been included per year and more than 15 thousands papers have been included per year since 2014. Grey system method and model are widely used in physics, chemistry, biology and other fields of natural science, as well as transportation, electric power, machinery and other fields of engineering technology, and a large number of valuable results have been achieved.Practical implicationsIt can be seen that the grey system theory plays an important role in promoting China’s scientific and technological progress, innovation and development and high-level talent training from tens of thousands of literatures marked with important national science and technology projects and a large number of grey system literatures published by China’s double first-class universities and double first-class discipline construction universities.Originality/valueBoth innovations of theoretical research and practical application play important role in the growth of new theory. The innovations of theoretical research provide methods and tools for practical application, which is conducive to improve application efficiency and broaden application fields. A large number of practical applications needs have become the source of theoretical innovation and the solid background for the birth of theoretical innovation achievements.

PurposeThe year 2022 marks the 40th anniversary of the establishment of the grey system theory (GST), which has been widely applied in the engineering field. This paper aims to systematically identify the achievements, hotspots, knowledge structure and emerging trends in this field.Design/methodology/approachA bibliometrics analysis was conducted on relevant publications retrieved from Web of Science (WoS) using CiteSpace and MapEquation. A statistical analysis of the collected 3,384 papers was completed. Three networks, including a co-occurrence network, cooperation network and co-citation network, were obtained to draw knowledge structure, hotspots and research frontiers.FindingsThe top four applied engineering fields are engineering electrical electronics, computer science artificial intelligence, engineering multi-disciplinary and automation control system. In total, 65 countries have engaged in this field, and China has occupied a leading position, with the largest number of articles published and the widest cooperation with other countries. The USA, United Kingdom (UK) and China Taiwan also contribute a lot. The Nanjing University of Aeronautics and Astronautics and Professor Liu Sifeng have a core position in the cooperation network. More hotspots appear in the last ten years. Regarding the emerging trends, the combination of theoretical models and practical engineering problems has attracted more attention. Besides, the application of GST in environment protection and the integration of the GST and intelligent algorithm became more popular.Originality/valueThe comprehensive bibliometrics analysis and visualization demonstration were conducted, presenting the interdisciplinary characteristics, major research topics and research frontiers in this field.

PurposeThe purpose of this paper is to summarize progress of grey forecasting modelling, explain mechanism of grey forecasting modelling and classify exist grey forecasting models.Design/methodology/approachGeneral modelling process and mechanism of grey forecasting modelling is summarized and classification of grey forecasting models is done according to their differential equation structure. Grey forecasting models with linear structure are divided into continuous single variable grey forecasting models, discrete single variable grey forecasting models, continuous multiple variable grey forecasting models and discrete multiple variable grey forecasting models. The mechanism and traceability of these models are discussed. In addition, grey forecasting models with nonlinear structure, grey forecasting models with grey number sequences and grey forecasting models with multi-input and multi-output variables are further discussed.FindingsIt is clearly to explain differences between grey forecasting models with other forecasting models. Accumulation generation operation is the main difference between grey forecasting models and other models, and it is helpful to mining system developing law with limited data. A great majority of grey forecasting models are linear structure while grey forecasting models with nonlinear structure should be further studied.Practical implicationsMechanism and classification of grey forecasting models are very helpful to combine with suitable real applications.Originality/valueThe main contributions of this paper are to classify models according to models' structure are linear or nonlinear, to analyse relationships and differences of models in same class and to deconstruct mechanism of grey forecasting models.

PurposeChina's population aging is gradually deepening and needs to be actively addressed. The purpose of this paper is to construct a novel model for analyzing the population aging.Design/methodology/approachTo analyze the aging status of a region, this study has considered three major indicators: total population, aged population and the proportion of the aged population. Additionally, the authors have developed a novel grey population prediction model that incorporates the fractional-order accumulation operator and Gompertz model (GM). By combining these techniques, the authors' model provides a comprehensive and accurate prediction of population aging trends in Jiangsu Province. This research methodology has the potential to contribute to the development of effective policy solutions to address the challenges posed by the population aging.FindingsThe fractional-order discrete grey GM is suitable for predicting the aging population and has good performance. The population aging of Jiangsu Province will continue to deepen in the next few years.Practical implicationsThe proposed model can be used to predict and analyze aging differences in Jiangsu Province. Based on the prediction and analysis results, identified some corresponding countermeasures are suggested to address the challenges of Jiangsu's future aging problem.Originality/valueThe fractional-order discrete grey GM is firstly proposed in this paper and this model is a novel grey population prediction model with good performance.

PurposeThe cumulative sum (Cusum) operator, also referred to as accumulating generation operator, is the fundamental of grey system models and proves to be successful in various real-world applications. This paper aims to uncover the advantages of the Cusum operator from a parameter estimation perspective, i.e. comparing integral matching with classical gradient matching.Design/methodology/approachGrey system models are represented as a state space form to investigate the effect of measurement errors on estimation performance; subsequently, gradient matching and integral matching are respectively formulated to estimate parameters from noisy observations and, then, their quantitative relationships are established by using matrix computation tricks.FindingsExtensive simulations, which are conducted on both linear and non-linear models under different sample size and noise level combinations, show that integral matching is superior to gradient matching, and, also the former is less sensitive to measurement error.Originality/valueThis paper explains why the Cusum operator is widely utilized in grey system models, thereby further solidifying the mathematical fundamentals of grey system models.

The file attached to this record is the author's final peer reviewed version. The Publisher's final version can be found by following the DOI link. Objectives The prediction and early warning of infectious diseases is an important work in the field of public health. This study constructed the grey self-memory system model to predict the incidence trend of infectious diseases affected by many uncertain factors. Study design The design of this study is a combination of the prediction method and empirical analysis. Methods By organically coupling the self-memory algorithm with the mean GM(1,1) model, the tuberculosis incidence statistics of China from 2004 to 2018 were selected for prediction analysis. Meanwhile, by comparing with the other traditional prediction methods, three representative accuracy check indexes (MSE, AME, MAPE) were conducting for error analysis. Results Owing to the multiple time-points initial fields, which replace the single time-points, the limitation of the traditional grey prediction model, which is sensitive to the initial value, is overcome in the self–memory equation. Consequently, compared with the mean GM model and other statistical methods, the grey self-memory model shows significant forecasting advantages, and its single-step rolling prediction accuracy is superior to other prediction methods. Therefore, the incidence of tuberculosis in China in the next year can be predicted as 55.30 (unit: 1/105). Conclusions The grey self-memory system model can closely capture the individual random fluctuation in the whole evolution trend of the uncertain system. It is appropriate for predicting the future incidence trend of infectious diseases and is worth popularizing to other similar public health prediction problems.

To explore the process of online social network information interaction, in this paper, we analyze the dynamics of a discrete Lotka–Volterra information diffusion model. Using the center manifold theorem, the conditions for transcritical bifurcation and flip bifurcation are obtained. With the help of approximation by a flow and Picard iteration, we explore the qualitative structures and stability of degenerate fixed point of the model with eigenvalues [Formula: see text]. What’s interesting is that our results reveal a new and complex qualitative structure for fixed point, which are different from the previous reports and called degenerate saddle point. Additionally, the qualitative structures provide a new idea for investigation the stability of degenerate fixed point. Meanwhile, near the maximum user density, the dynamic results of degenerate fixed point indicate that if the intervention rate is greater than the inverse of the maximum user density, then the higher user density decreases, the lower user density increases when intrinsic growth rates are small (between 0 and 2). However, when the intrinsic growth rate is greater than 2, the high user density will continue to increase until it approaches the maximum user density indefinitely, while the small user density will approach 0, which provide us with new insights into information diffusion. Finally, we show the results of the model by numerical simulations, and the characteristics of information diffusion near the degenerate fixed point are predicted by theoretical analysis.

while NGBM(1,1) model has been successfully used in various fields,there are still some limitations in NGBM(1,1) model: First, the existing NGBM(1,1) model family is all based on the assumption that some adjustable parameters are known, which reduced the prediction performance of the NGBM(1,1) model family. Then, the leap error from the differential to difference still exists in many NGBM(1,1) models.Moreover, most of the existing NGBM(1,1) models were constructed on the hypothesis that the raw data obey the homogeneous exponential trend. To address these problem, this paper aims to construct a novel self-adaptive nonlinear discrete grey model based on nonhomogeneous index sequence by combining nonhomogeneous discrete grey model with NGBM(1,1) model (abbreviated as SNNDGM(1,1)). Improvements in the proposed model have the following features: Firstly, the new proposed model can effectively eliminate the discretization error of NGBM(1,1) model.Secondly, The new proposed model effectively improves the prediction accuracy of the NGBM(1,1) model by removing the aforementioned modeling assumptions. Thirdly, a new multi-parameter simultaneous optimization scheme based on the firefly algorithm is designed to improve the parameter estimation accuracy. Last but not least, Sobol’s method is used to investigate the mechanism of the adjustable parameters in the SNNDGM(1,1) model, which provides an important basis for model parameter calibration.Two classic numerical cases are introduced to confirm the effectiveness of the new proposed model. Comparisons show that SNNDGM(1,1) model has the highest prediction accuracy. Based on industrial electricity consumption from 2010 to 2019, SNNDGM(1,1) model is constructed to predict the industrial electricity consumption of Anhui Province in China from 2020 to 2024. The results show that industry electricity consumption in Anhui Province maintains a continuous increase, reaching 165.25568 billion kwh in 2024, while the annual growth rate decreases to 3.45%, which provides important information for the decision-making of Anhui electric power department.

Purpose Accurate foreign tourist arrivals forecasting can help public and private sectors to formulate scientific tourism planning and improve the allocation efficiency of tourism resources. This paper aims to address the problem of low prediction accuracy of Chinese inbound tourism demand caused by the lack of valid historical data. Design/methodology/approach A novel hybrid Chinese inbound tourism demand forecasting model combining fractional non-homogenous discrete grey model and firefly algorithm is constructed. In the proposed model, all adjustable parameters of the fractional non-homogenous discrete grey model are optimized simultaneously by the firefly algorithm. Findings The data sets of annual foreign tourist arrivals to China are used to verify the validity of the proposed model. Experimental results show that the proposed method is effective and can be used as a useful predictor for the prediction of Chinese inbound tourism demand. Originality/value The method proposed in this paper is effective and can be used as a feasible approach for forecasting the development trend of Chinese inbound tourism.

PurposeThe airline industry has been significantly hit by the occurrence of the new coronavirus SARS-CoV-2, facing one of its worst crises in history. In this context, the present paper analyses one of the well-known boarding methods used in practice by the airlines before and during the coronavirus outbreak, namely back-to-front and suggests which variations of this method to use when three passenger boarding groups are considered and a jet bridge connects the airport terminal with the airplane.Design/methodology/approachBased on the importance accorded by the airlines to operational performance, health risks, and passengers' comfort, the variations in three passenger groups back-to-front boarding are divided into three clusters using the grey clustering approach offered by the grey systems theory.FindingsHaving the clusters based on the selected metrics and considering the social distance among the passengers, airlines can better understand how the variations in back-to-front perform in the new conditions imposed by the novel coronavirus and choose the boarding approach that better fits its policy and goals.Originality/valueThe paper combines the advantages offered by grey clustering and agent-based modelling for offering to determine which are the best configurations that offer a reduced boarding time, while accounting for reduced passengers' health risk, measured through three indicators: aisle risk, seat risk and type-3 seat interferences and for an increased comfort for the passengers manifested through a continuous walking flow while boarding.

CO2 emissions from the transportation sector occupy an increasingly important proportion in China’s carbon dioxide emissions. Measuring the accumulative impact of factors on carbon emissions over time is of great significance for formulating corresponding policies. This paper aims to propose a novel time-delay multivariate grey model to measure the CO2 emissions’ accumulating impact of China’s transportation sector. Firstly, the grey incidence model is used to identify time lags between the input and output variables, and also analyze the structure type of time-delay weights. Then, an accumulative time-delay multivariate grey prediction model is developed. In this model, a Gaussian formula is used to discretize the convolution integral of the time response function, and the particle swarm optimization algorithm is employed to determine the optimal weight coefficients. Finally, a case for CO2 emissions prediction is adopted to test the effectiveness and practicality of this model compared with the alternative models. The results show that the proposed model outperforms other six competing models in accordance with two measuring indices, and suggestions on emissions mitigation are proposed based on the prediction results.

Nonlinear grey Bernoulli model, abbreviated as NGBM model, has been validly used in real applications due to its high accuracy in nonlinear time series forecasting. However, there remain technical challenges to explain the mechanism of the accumulative sum operator in nonlinear grey modelling process and estimate structural parameters independent from the initial values. This paper aims to reconstruct the modelling process of the NGBM model so as to explain the modelling mechanism better by utilizing the integral matching approach, which consists of an integral formula and the numerical discretization-based least squares. First, the integral formula is employed to investigate the accumulative sum operator and further reconstruct the NGBM model to a generalized form, referred as to INGBM model. Then, a novel parameter estimation strategy, estimating structure parameters and initial values simultaneously, is developed by utilizing the numerical discretization-based least squares approach. Next, Monte Carlo simulation studies are designed to evaluate the finite sample performance of both models. Comparisons show that the INGBM model outperforms to the original one in terms of parameter estimation accuracy, forecasting accuracy and robustness to noise. Finally, we apply the INGBM model at a coal consumption in China study to further illustrate the usefulness of this model.

In the cloud manufacturing environment, integrated cross-supplier order and logistic scheduling can benefit both suppliers and third-party logistics, significantly reduce their production and trans...

PurposeThis study examines the potential of contracts as one of the supply chain coordination mechanisms under competitive conditions. It also investigates a two-echelon supply chain model with two manufacturers and two retailers to develop a competitive structure in grey stochastic demand.Design/methodology/approachSupply chain demand is considered as a stochastic phenomenon depending on the selling price of the product. Also, products can be replaced by market manufacturers. Each retailer faces the pricing of products from two manufacturers, leading to competition between downstream retailers. In the present study, the duopoly supply chain model was presented based on the wholesale price contract, revenue-sharing contract and quantity discount contract separately.FindingsGrey optimization and analysis of their coordination were presented. The results showed the high performance of revenue-sharing contracts in the supply chain. Thus, manufacturers will give the next priority to quantity discount contracts.Originality/valueOrdering is the main factor contributing to competitive decision-making. Meanwhile, decision-making along with ordering and pricing will be required due to the nature of the demand.

Purposepurpose of this paper is providing a solution for flexible flow shop scheduling problem with uncertain processing time in aeronautical composite lay-up workshop.Design/methodology/approachA flexible flow scheduling model and algorithm with interval grey processing time is established. First, according to actual needs of composite laminate shop scheduling process, interval grey number is used to represent uncertain processing time, and interval grey processing time measurement method, grey number calculation and comparison rules, grey Gantt chart, and other methods are further applied. Then a flexible flow shop scheduling model with interval grey processing time (G-FFSP) is established, and an artificial bee colony algorithm based on an adaptive neighbourhood search strategy is designed to solve the model. Finally, six examples are generated for simulation scheduling, and the efficiency and performance of the model and algorithm are evaluated by comparing the results.FindingsResults show that flexible flow shop scheduling model and algorithm with interval grey processing time can provide an optimal solution for composite lay-up shop scheduling problems and other similar flow shop scheduling problems.Social implicationsUncertain processing time is common in flexible workshop manufacturing, and manual scheduling greatly restricts the production efficiency of workshop. In this paper, combined with grey system theory, an intelligent algorithm is used to solve flexible flow shop scheduling problem to promote intelligent and efficient production of enterprises.Originality/valueThis paper applies and perfects interval grey processing time measurement method, grey number calculation and comparison rules, grey Gantt chart and other methods. A flexible flow shop scheduling model with interval grey processing time is established, and an artificial bee colony algorithm with an adaptive domain search strategy is designed. It provides a comprehensive solution for flexible flow shop scheduling with uncertain processing time.

This study seeks to explicate the philosophical foundations and theoretical outlines of grey systems theory by focusing on human perception, cognition, and understanding processes and by considering their functions in the process of producing knowledge. Primarily, the study investigates the processes of perception, cognition, and understanding, as well as their dynamicity. Then, it is explained how knowledge is produced through the interpretation/understanding of information and data and through the dynamicity governing this process. The findings reveal that human perception, cognition, and understanding perpetually remain grey and imperfect, while lost sensatory data mark the very first aspect of greyness in such human capabilities. Furthermore, any piece of data, as a discrete and single entity, represents a determinate fact, symbol or signal from the world, while any set of data, no matter how large, remains incomplete and imperfect, and human beings always interpret inadequate sets of data in the world. Therefore, information and knowledge always remain grey and uncertain because they deeply rely on subjective understandings/interpretations and work with imperfect inputs and sets of data. The impact of knowledge on action and interaction and their effects on the world are some factors adding to the dynamicity of the world. Constant change in world data, outdated knowledge, defects in datasets, and human biases in understanding all confirm the infinitely grey nature of human knowledge of the world.

PurposeThe 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.Design/methodology/approachIn 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.FindingsAs 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.Originality/valueThe 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.

PurposeThe purpose of this paper is to propose an uncertainty representation and information measurement method for characterizing grey numbers, estimating their internal laws and solving how to generate them based on available information data in the real world.Design/methodology/approachThis paper attempts to present a new mathematical methodology in the field of grey numbers. The generalized grey number is defined at first with the concept of information elements and information samples. Then, the probability function of a grey number is proposed to describe the internal law of the grey number. By finding the feasible information elements from information samples, the probability calculation method for the true value of a grey number is presented. Finally, some numerical examples and comparisons are carried out to assess the efficiency and performance.FindingsThe results show that the uncertainty representation and information measurement method is effective in characterizing and quantifying grey numbers based on available information data.Practical implicationsUncertain information is widespread in practical applications. In this manuscript, the grey number is represented and its information is measured through some existing data in discrete or interval forms, which provides a grey information concept that utilizes information elements to represent uncertainty in the real world.Originality/valueThe proposal presents a novel data-driven method to generate a grey number representation from available data rather than the classical whitening weight function constructed from experience, and the dynamic evolution process of a grey number is measured by the increase of information samples.

Today’s challenges to sustainability are explored through a complex combination of interdisciplinary topics that explore various interactions between economic, social, and environmental systems that further contribute to existing uncertainties. Solving complex/dynamic sustainability constraints does not demand exclusively technical and practical methods, as it is equally important to have a profound conceptual understanding of the origins of such challenges. The purpose of this study was to investigate the challenges to the sustainable development process from the perspective of the philosophy of grey systems theory (GST). GST considers inherent defects and shortcomings in human understanding/knowledge and identifies the roots of uncertainty. The study concentrates on the sustainable development process, highlighting the ways in which GST explains the causes and sources of uncertainty in this process. It is emphasized that sustainability cannot be achieved without intentional human intervention, and that international collaboration is vital in solving sustainability problems. Uncertainty and challenges to sustainable development stem from human grey understanding and knowledge. This problem makes it difficult for humans to understand and model dynamicity, to strike a balance between different spheres of science, and to have an objective view of reality due to the dependence of knowledge on thinking paradigms and values. These shortcomings ultimately bring about value conflicts, different understandings of risks, and impediments to international collaboration and agreement. Finally, the study explains that uncertainty arises from incomplete understanding and grey knowledge, and that uncertainty undermines the prediction of outcomes. Furthermore, delays inherent in interactions and the impacts of diverse systems on the world increase uncertainty and complicate decision- and policymaking in improvement projects. In their efforts to implement their decisions and policies, humans also encounter various limitations in terms of their capacities, resources, and facilities. The application of GST-based approaches to the operational area is also discussed.

Every scientific or intellectual movement is founded upon basic assumptions and hypotheses that shape its specifically formulated philosophy. This study seeks to explore and explicate the basic philosophical underpinnings of grey systems theory (GST), as well as the paradigm governing its postulates. The study, more specifically, scrutinizes the underlying principles of GST from the perspective of postmodern philosophy. To accomplish this, the epistemology, ontology, human nature, and methodology of GST are substantially investigated in the light of postmodern philosophy. The study draws on Burrell and Morgan’s framework to reveal the paradigm underlying the philosophy of GST. Results demonstrate that GST is an anti-realistic, anti-positivistic, and non-deterministic theory which is inherently pluralistic and ideographic. Based on the principles of GST, change is an indispensable dimension of human speculation about the world and systems, and knowledge is ceaselessly reproduced as new information is collected. As a result, knowledge, narratives, theories and scientific laws are dynamically changed. GST, then, is remarkably compatible with the foundations of postmodern thought and it could be regarded as a postmodern theory governed by a humanistic paradigm.

The solution of Bernoulli differential equation can be described as a generalized Logistic curve function. Weibull cumulative distribution function is also an analytic solution of one variable coefficient differential equation. But, however because of the exact analytical solution problem of the equation, up to now, the developing coefficient of all the grey forecasting models is still defined as a constant. So, the aim of this paper is just to construct a novel grey differential equation model by combining NGBM(1,1) and Weibull cumulative distribution function. The proposed model(WBGM(1,1)) has the advantages of NGBM(1,1) and Weibull cumulative distribution. Where there are double shape parameters, the developing coefficient of the grey forecasting model is extended to be a variant. Property analysis of WBGM(1,1) shows that the fitting accuracy is higher and the applicable confines are wider. Finally, this paper gives an optimization method for the parameters of WBGM(1,1). A classic example and a practical case are studied for confirming the effectiveness of WBGM(1,1). The case study is the prediction for the number of invention patents of integrated circuit(IC) filed in China from 2007 to 2017. Results of the example and case study are compared to other forecasting models, including GM(1,1), NGBM(1,1), Holt exponential smoothing and ARIMA. Results show that WBGM(1,1) is a more general and more efficient model in grey prediction theory.

Human resources have played an important role in the production systems of manufacturing enterprises. At the same time, human resource allocation, as a scheduling problem, has attracted more and more attention from the industry and academia, with the increasing complexity of technology and the rising cost of workforce. However, the existing scheduling methods fail to fully consider the relationship between tasks and employees. In this paper we consider a single workgroup scheduling problem with the assumption of human resources variability. Jobs must be assigned to employees, where the number of processing personnel and the processing time are variable for one job. We model the problem as a combinatorial optimization problem and the objective considered is to minimize the maximum finish time which is called makespan. This problem is transformed into a two-dimension rectangle strip packing problem and we propose a hybrid optimization algorithm which combines scheduling algorithm with packing algorithm. Finally, different size cases are adopted to test the model. Results show the proposed strategy and algorithm are suitable for solving single workgroup scheduling very well.

The purpose of this paper is to evaluate, analyse and select the best suppliers for Satrec Vitalait Milk Company, operating in Senegal, based on criteria obtained from economic, environmental and social dimensions of sustainable supply chain management, through the application of Deng’s grey relational analysis (GRA) model, absolute GRA model (ADGRA) and a novel second synthetic GRA (SSGRA) model, combined with one decision making under the uncertainty-based model, namely, the Hurwicz criteria.,The research adopts a new synthetic GRA model and highlights its reliability on small sample gathered from four senior experts of the company who administered a total number of 28 specialists operating in four departments of the company, through the employment of a self-administered questionnaire designed based on criteria identified from the literature that were refined via a Q-sort model.,The outcomes of the research methodology designated that all the selected five suppliers present a degree of attaining sustainability due to the fact that supplying unprocessed milk does not require the use of polluting methods for stocking and transportation. The undertaken study specifies that all the socio-environmental criteria play a crucial role in shaping the sustainability level of Satrec Vitalait’s suppliers and demonstrates the accuracy of the results obtained through the second synthetic degree of grey relation analysis for ranking the suppliers. Supplier 2 was found to be the best supplier for the company and, as result, a model for other suppliers to mimic.,Future researchers can replicate the GRA-based supply chain model proposed in the current study in different environments especially in the context of green supply chain. Also, in future the SSGRA model, while using the bidirectional ADGRA instead of the conventional ADGRA, should also be tested, especially when the data sequences associated with different supply chain parameters have inconsistent directions. Also, comparative analysis of SSGRA-based results with that of modern statistical methods like structural equation modelling can also be used for future explorations. Furthermore, the current study is built upon the data associated with the Satrec Vitalait Milk Company (Senegal); therefore, the findings should be generalised with caution.,The study can be seen as a first-stepping stone for gauging and selecting the best sustainable supplier for Satrec Vitalait using grey system theory. For purpose of attaining the research goal, the SSGRA was exploited as an innovative experimental approach to estimate relationships between criteria with regard to the sustainability level of the company’s suppliers. Under this scope, relationships between criteria themselves and their goal were depicted by Deng’s degree of GRA and AGRA, respectively. The research is innovative by means of the framework of its methodology and data analysis.

PurposeThe purpose of this paper is to propose a framework for data analytics where everything is grey in nature and the associated uncertainty is considered as an essential part in data collection, profiling, imputation, analysis and decision making.Design/methodology/approachA comparative study is conducted between the available uncertainty models and the feasibility of grey systems is highlighted. Furthermore, a general framework for the integration of grey systems and grey sets into data analytics is proposed.FindingsGrey systems and grey sets are useful not only for small data, but also big data as well. It is complementary to other models and can play a significant role in data analytics.Research limitations/implicationsThe proposed framework brings a radical change in data analytics. It may bring a fundamental change in our way to deal with uncertainties.Practical implicationsThe proposed model has the potential to avoid the mistake from a misleading data imputation.Social implicationsThe proposed model takes the philosophy of grey systems in recognising the limitation of our knowledge which has significant implications in our way to deal with our social life and relations.Originality/valueThis is the first time that the whole data analytics is considered from the point of view of grey systems.

Purpose The purpose of this paper is to establish a new evaluation system to assess the degree of integration between industry and the internet. And use the gray correlation matrix method to evaluate the “internet + industry” integration degree of China’s provinces. Design/methodology/approach This paper establishes a new evaluation system to assess the degree of integration between industry and the internet using the matrix gray relational analysis method. Findings The main indexes and its rankings of the provinces’ integration degree and the rankings of the provinces’ integration degree are obtained. Practical implications The ranking of the degree of integration of various provinces in the country has certain guiding significance in promoting the development of “internet +” and “industry 4.0.” Originality/value Establishing a new model for the quantitative assessment of the degree of fusion, this method has a positive impact on the quantitative assessment of “internet + industrial” integration.

Nonlinear grey system models, serving to time series forecasting, are extensively used in diverse areas of science and engineering. However, most research concerns improving classical models and developing novel models, relatively limited attention has been paid to the relationship among diverse models and the modelling mechanism. The current paper proposes a unified framework and reconstructs the unified model from an integro-differential equation perspective. First, we propose a methodological framework that subsumes various nonlinear grey system models as special cases, providing a cumulative sum series-orientated modelling paradigm. Then, by introducing an integral operator, the unified model is reduced to an equivalent integro-differential equation; on this basis, the structural parameters and initial value are estimated simultaneously via the integral matching approach. The modelling procedure comparison further indicates that the integral matching-based integro-differential equation provides a direct modelling paradigm. Next, large-scale Monte Carlo simulations are conducted to compare the finite sample performance, and the results show that the reduced model has higher accuracy and robustness to noise. Applications of forecasting the municipal sewage discharge and water consumption in the Yangtze River Delta of China further illustrate the effectiveness of the reconstructed nonlinear grey models.

Grey theory-based time series models are widely used in various fields and disciplines. While most of the research is focused on the development and improvement of novel discrete-time models, very limited attention has been paid to the relationships among diverse models. The current paper proposes a methodological and practical framework to unify the single-variable, multi-variable, and multi-output discrete-time grey models, making it easier for practitioners to select an appropriate model for a given time-series forecasting problem. The recursive extrapolation strategy is present to generate multi-step ahead forecasts and four model families are deduced within the universal framework. Large-scale simulation studies are conducted to evaluate the finite-sample fitting and multi-step ahead forecasting performance. The proposed approach is illustrated using application examples from the indirect measurement of the tensile strength of materials.

Since most of the research about grey forecasting models is focused on developing novel models and improving accuracy, relatively limited attention has been paid to the modelling mechanism and relationships among diverse kinds of models. This paper aims to unify and reconstruct continuous-time grey models, highlighting the differences and similarities among different models. First, the unified form of grey models is proposed and simplified into a reduced-order ordinary differential equation. Then, the integral matching that consists of integral transformation and least squares, is proposed to estimate the structural parameter and initial value simultaneously. The cumulative sum operator, an essential element in grey modelling, proves to be the discrete approximation of the integral transformation formula. Next, grey models are reconstructed by the integral matching-based ordinary differential equations. Finally, the existing grey models are compared with the reconstructed models through extensive simulation studies, and a real-world example shows how to apply and further verify the reconstructed model.

The grey prediction model, as a time-series analysis tool, has been used in various fields only with partly known distribution information. The grey polynomial model is a novel method to solve the problem that the original sequence is in accord with a more general trend rather than the special homogeneous or non-homogeneous trend, but how to select the polynomial order still needs further study. In this paper the tuned background coefficient is introduced into the grey polynomial model and then the algorithmic framework for polynomial order selection, background coefficient search and parameter estimation is proposed. The quantitative relations between the affine transformation of accumulating sequence and the parameter estimates are deduced. The modeling performance proves to be independent of the affine transformation. The numerical example and application are carried out to assess the modeling efficiency in comparison with other conventional models.

Job processing time is not always given accurately as a time quota in a production system, particularly in the complex products manufacturing process. It is meaningful to study novel model and algorithm based on uncertainty processing time so as to solve uncertainty job shop scheduling problems. This paper defines inaccurate time quota as interval grey processing time and subsequently proposes a novel uncertainty job shop scheduling model with the objective of minimizing the interval grey makespan. By defining the arithmetic operations and Gantt chart of interval grey processing time, the elitism genetic algorithm coupling elitism strategy in external memory is designed. Finally, the proposed algorithm is tested with different size cases of composite components job-shop scheduling. Results show the proposed algorithm is suitable for solving the uncertainty job shop scheduling problem with interval grey processing time.

Purpose Considering current development of Grey Systems Theory (GST), we can come up with the following thesis: practical applications are a dominant subject of research. Thus, what seems to be symptomatic for relatively young knowledge disciplines, the authors observe the presence of imbalance between the development of GST application tools and theory’s epistemological and methodological background. As for GST, epistemological and methodological problems are becoming visible especially in the issues of determining a clear criterion of demarcation of this kind of a theory from others. In other words, this problem can be reduced to the issue of a precise determination of what the category of a grey system and grey information is. This problem is of great importance for further development and popularisation of GST in the world of science. Realising its significance, the purpose of this paper is to create a general overview of Grey Systems epistemology and afterwards create axiomatic and formal frames for a category of greyness. Design/methodology/approach In order to achieve set goals, two research approaches were accepted. In the area of inference about epistemology of GST an approach characteristic of an analytical philosophy was used, whereas in the case of axiomatic and formal frames for a category of greyness the authors referred to terms of a set theory and the principles of a pragmatic logic. Findings The result of research is to formulate a concept of a grey system and a concept of grey information in the context of a process of cognition. Moreover, a function of greyness and other fundamental categories of GST will be defined in an axiomatic way. Originality/value The paper presents a new consistent frame for the issues of methodological and epistemological backgrounds of GST. An original concept is to refer in considerations to a newly proposed grey space. This space was used for a formal justification of such elementary categories as grey numbers, a weight function of whitenization or grey sequences. The value of achievements shown in the paper is underlined by the fact that proposed theoretical constructions require further development and they can potentially open up new research trends in the GST.

Purpose This paper aims to analyze general development trend of China’s population and to forecast China’s total population under the change of China’s family planning policy so as to measure shock disturbance effects on China’s population development. Design/methodology/approach China has been the most populous country for hundreds of years. And this state will be sustained in the forthcoming decade. Obviously, China is confronted with greater pressure on controlling total scale of population than any other country. Meanwhile, controlling population will be beneficial for not only China but also the whole world. This paper first analyzes general development trend of China’s population total amount, sex ratio and aging ratio. The mechanism for measurement of the impact effect of a policy shock disturbance is proposed. Linear regression model, exponential curve model and grey Verhulst model are adopted to test accuracy of simulation of China’s total population. Then considering the policy shock disturbance on population, discrete grey model, DGM (1, 1), and grey Verhulst model were adopted to measure how China’s one-child policy affected its total population between 1978 and 2015. And similarly, the grey Verhulst model and scenario analysis of economic developing level were further used to forecast the effect of adjustment from China’s one-child policy to two-child policy. Findings Results show that China has made an outstanding contribution toward controlling population; it was estimated that China prevented nearly 470 million births since the late 1970s to 2015. However, according to the forecast, with the adjustment of the one-child policy, the birth rate will be a little higher, China’s total population was estimated to reach 1,485.59 million in 2025. Although the scale of population will keep increasing, but it is tolerable for China and sex ratio and trend of aging will be relieved obviously. Practical implications The approach constructed in the paper can be used to measure the effect of population change under the policy shock disturbance. It can be used for other policy effect measurement problems under shock events’ disturbance. Originality/value The paper succeeded in studying the mechanism for the measurement of the post-impact effect of a policy and the effect of changes in China’s population following the revision of the one-child policy. The mechanism is useful for solving system forecasting problems and can contribute toward improving the grey decision-making models.

Purpose The purpose of this paper is to construct a grey clustering model based on the nonlinear whitenization weight function and to assess the safety of civil aircraft by using a quantitative method. Design/methodology/approach According to the running stage of civil aircraft safety assessment issues, first the civil aircraft safety evaluation index system is constructed by using a qualitative method. Taking the information duplication between indicators, the grey relational analysis method is used to filter the key indicators, then the grey clustering evaluation model of nonlinear whitening right function is built to evaluate the safety of civil aircraft and the algorithm steps of the evaluation model are given. Finally, the model is validated by collecting the parameters of nine different civil aircrafts at home and abroad. Findings The results show that the safety level of different types of aircraft is different due to the different index parameters, and to some extent, explain the rationality and scientificity of the method proposed in this paper to solve the problem. Practical implications This paper gives a complete set of security assessment methods, which can be used to evaluate the security of civil aircraft in the operational phase quantitatively, scientifically and reasonably. Furthermore, it can be extended to other complex system security or stability assessment issues. Originality/value It not only provides the supplement and perfection of the safety assessment method in the theoretical system to a certain extent, but also provides a theoretical guidance to solve the problem of civil aircraft system safety assessment of civil aircraft manufacturing enterprise all over the world. At the same time, the nonlinear grey clustering evaluation model constructed in this paper is an improvement of the traditional model, which is, to some extent, the improvement of the grey clustering evaluation theory.

PurposeThe purpose of this paper is to combine the interval grey number algorithm with a project scheduling model, so as to solve the interval grey number based project scheduling problem.Design/methodology/approachAccording to the grey system theory, if we could not clearly collect the right value of an activity while obtaining the upper and lower boundaries of the activity duration time, we can find activity duration time as interval grey number. This paper tries to combine interval grey number with project scheduling problem. The general definition of interval grey number was given. And then the linear operations of independent interval grey number and non-independent interval grey number were further defined. Interval grey number based project scheduling model and its algorithm were further proposed. Finally, a numerical case was adopted to test the effectiveness of the proposed method.FindingsThe results show that grey project scheduling and grey critical path method could be established under interval grey number measured activities.Practical implicationsScheduling problem was widely used in project planning and control. This manuscript attempted to combine interval grey number with project scheduling so as to explain the flexible deadline in a special project. It is truly in accordance with the requirements for real applications.Originality/valueThis paper proposed linear operations of independent and non-independent interval grey numbers, and proposed a interval grey number based project scheduling model. Valuable contribution is to combine interval grey numbers with project scheduling and it is meaningful to make a useful strategy in a flexible condition of project scheduling.

The vehicle routing problem with simultaneous delivery & pickup and real-time traffic information (VRPSDPTI) is a dynamic problem of combinatorial network optimisation in logistics and supply chain management. It is also a typical NP-hard problem that has plagued enterprises with reverse logistics operations for many years. The main objective of this research is to determine an optimal vehicle routing plan for the VRPSDPTI problem. To achieve this goal, a mixed integer programming (MIP) model is constructed, with the objective of minimising the total travel cycle for the VRPSDPTI problem. A hybrid algorithm of the ant colony system and virtual transformation method (ACS-VTM) is then designed to explore the vehicle routing plan for the VRPSDPTI problem; it adopts a flexible split of the time period for re-optimisation and an improved partial service policy for customer demand. Based on it, a practical case of the VRPSDPTI problem is used to verify the proposed model and algorithm. One major finding is proposed through the study of the practical case that adopting the hybrid ACS-VTM algorithm can effectively reduce the total travel cycle for the VRPSDPTI problem. The main contributions of this study are that it can provide an efficient decision tool to solve the VRPSDPTI, and can not only reduce the total travel cycle, but also improve the efficiency of vehicle utilisation.

Batch scheduling involves a machine that can process several jobs simultaneously. The existing literature mainly focuses on the batch-size-dependent setup time. However, setup time depends not only on the batch size but also on technological characteristics. This paper investigates a single batch scheduling problem with dual quantitative and technological setup times extracted from autoclave molding production. A mixed-integer linear programming model (MILP) is established, in which the objective function is to minimize the makespan. The quantitative setup time is linear with the batch size, whereas the technological setup time is nonlinear with the occupied area of all jobs (jobs are represented by rectangles) in a batch. Three lower bounds are presented for evaluating the proposed algorithms. Then, a two-stage approximate algorithm is designed to solve the problem. The first stage concerns the normal batch processing time. In the second stage, an iterative local search method focuses on all terms in the objective function and improves the current solution’s quality. Finally, different scales of instances are designed to test the effectiveness and efficiency of the algorithms. Statistical analysis shows that the proposed algorithm in this paper can handle the problem well compared to other algorithms.