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Parameter Estimation of Integro-differential Equation-based Grey Predator-prey Model From Noisy Data  Zhaoya Zhang, Naiming Xie, Lu Yang, Xiaolei Wang The Journal of Grey System    2024, 36 (2): 79-89.   Abstract （62）            Knowledge map    The grey predator-prey model is widely applied in many fields for its effectiveness. While most of the research on the grey predator-prey model focuses on the different model forms, less attention has been paid to identifying parameters and initial value from noisy data. In this work, considering the measurement noise in practice, we propose a two-stage approach to estimate the structural parameters and initial value of the grey predator-prey model with integro-differential equations(IDEs). First, by introducing an integral operator, we propose another form of the grey predator-prey model, namely the IDE-based grey predator-prey model. Next, we develop a parameter estimation approach based on the idea of a two-stage in the presence of measurement noise, where in the first stage we smooth time series using cubic B-Spline and in the second stage estimate structural parameters and initial conditions by the separable nonlinear least squares. Then, the finite sample performance of the IDE-based model is investigated with Monte Carlo numerical experiments. Results demonstrate that the IDE-based model has better performance in terms of accuracy than the Cusum-based model. Finally, we use a case to further verify the accuracy and stability of the proposed method.   Related Articles | Metrics

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A Novel Logistic Multivariate Grey Prediction Model for Energy Consumption: A case study of China Coal  Sihao Chen, Yongshan Liu, Huiming Duan The Journal of Grey System    2023, 35 (4): 132-153.   Abstract （170）            Knowledge map    Coal consumption plays a pivotal role in the national economic growth, and the coal-based energy supply system as the main guarantee of China's energy is impossible to change in the short term. In order to ensure the security of China's energy supply, accurate coal consumption forecast can provide important theoretical basis for the development of scientific and effective energy planning and decision-making. Starting from the classical Logistic model, this paper introduces a variety of related factors such as energy consumption, economic growth rate and carbon dioxide emission growth rate to expand the modeling objects of the Logistic model and improve the performance of the model by relying on its ability to capture the historical trend of the data model and accurately predict the future value. At the same time, a new logistic multivariate grey prediction model of energy consumption is established by introducing the principle of grey variance information organically combined with the logistic model. The modeling steps of the model are obtained by using mathematical methods such as least squares estimation of parameters and number multiplication transformation. Finally, the new model is applied to the prediction of Chinese coal consumption, and the validity of the model is verified from different perspectives of three cases, showing that the fitted and predicted data of the new model have good consistency with the actual results. The new model has a high accuracy for China's coal short-term forecast, and uses simulation and prediction effects of 1.68220% and 1.29866%, respectively, to effectively forecast China's coal consumption in 2022-2026, and points out the development trend of Chinese coal consumption, and provides a basis for China to make scientific and effective energy planning and decision-making. Related Articles | Metrics

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Stock Movement Prediction With Sentiment Analysis Based on Grey Exponential Smoothing Method: A Case Study on Colombo Stock Exchange, Sri Lanka D.M. K. N. Seneviratna , M.V.D.H.P Malawana , R. M. K. T. Rathnayaka The Journal of Grey System    2023, 35 (4): 1-18.   Abstract （85）            Knowledge map    Sentiment  Analysis  is  an  innovative  development  technique  that  uses  natural language processing techniques to derive people's emotions under positive, negative,and neutral based on public opinions of information. The main objective of this study is  to  introduce  a  novel  stock  market  prediction  method  based  on  the  Grey Exponential Smoothing method for analyzing social media data within a big-data distributed environment. The empirical investigation of this study is mainly carried out based on the stock market price indices parallel to the extracted Tweets collected during the three selected politically important moments that happened in Sri Lanka during the past ten years; the first case study is based on the political background after  the  ending  of  the  thirty  years  of  civil  war  in  years  2009.  In  the  year  2015,Maithripala Sirisena ended the dynastic rule of Mahinda Rajapaksa. So, the second case  study  has  based  the Tweets  on  the  political  reforms  done  after  the  2015 presidential  election;  the  third  study  is  based  on  the  Sri  Lankan  political  and economic  background  after  the  Rajapaksas  rose  again  in  2020.  For  validations purpose, K Nearest Neighbour, Decision Tree Model, Support Vector Machine, Grey Exponential Smoothing model, and Multinomial Naïve Bayes machine learning were considered.  According  to  the  empirical  findings,  the  new  proposed  Hybrid  Grey Exponential Smoothing model is highly accurate with the lowest RMSE error values in one-head forecasting. Furthermore, the key finding of this research suggested that the  hybrid  Grey  Exponential  Smoothing  model  performs  well  in  sentiment classification-based financial predictions than traditional methods, especially with non-stationary behavioral backgrounds.  Related Articles | Metrics

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Research on Grey Clustering Model Based on NDEA for Equipment System-of-Systems Configuration Selection Decision Jingru Zhang, Zhigeng Fang, Shuyu Xiao, Luyue Zhang The Journal of Grey System    2023, 35 (4): 91-107.   Abstract （54）            Knowledge map    Resources (e.g., development budget, equipment performance) is not infinite for the plan and development of equipment system-of-systems (ESoS). Decision makers (DMs) must determine the priority of the ESoS configuration scheme under many constraints. Aiming for this problem, a structure and operation logic modeling of ESoS is analyzed. The network DEA approach describes each ESoS as a n-phase network decision unit with inputs and outputs. Secondly, the performance and cost of single equipment and ESoS combat effect are all considered. Based on this, we calculate the input-output efficiency of ESoS and consider two situations regarding the development budget. Then, with phased efficiency as evaluation indexes, the grey clustering evaluation based on the possibility function is applied to measure the ESoS configuration from the perspective of DMs. Finally, a case study verifies the feasibility and efficacy of the proposed methodology via selection decision results. The proposed method can aid DMs throughout the decision process for ESoS.  Related Articles | Metrics

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Adaptive Fluctuation Grey Model withAK Fractional Derivative for Short-term Traffic Flow Prediction Quntao Fu, Shuhua Mao The Journal of Grey System    2023, 35 (4): 108-131.   Abstract （93）            Knowledge map    Short-term traffic flow prediction is an essential component of intelligent transportation systems. Shallow and deep pattern learning methods have been widely applied to short-term traffic flow prediction. However, shallow learning methods struggle with highly volatile data and models are usually constant-coefficient. On the other hand, deep learning methods require significant computational resources and time. In this paper, we propose a new adaptive fluctuation grey model for short-term traffic flow prediction. We combine the fractional differential equation and fractional accumulation generation operator, and expand the GM(1,1) model using trigonometric functions. Furthermore, we improve the Harris hawks algorithm by optimizing the distribution of the initial population with Cauchy mutation operator and introducing boundary constraint handling techniques to enhance the model parameter search capability. Finally, we apply the model to short-term traffic flow parameter prediction and compare it with the benchmark model. Results indicate that the new model shows better accuracy performance and better extraction of fluctuation information.  Related Articles | Metrics

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A Conformable Fractional Non-homogeneous Grey Forecasting Model with Adjustable Parameters CFNGMA(1,1,k,c) and its Application  Wenqing Wu , Xin Ma , Bo Zeng , Peng Zhang The Journal of Grey System    2024, 36 (2): 1-12.   Abstract （121）            Knowledge map    The inconsistency between the whitening differential equation and the grey basic form of the non-homogeneous continuous grey model CFNGM(1,1,k,c) will result in internal errors. Thus this paper proposes a CFNGMA(1,1,k,c) model with adjustable parameters, which improves the accuracy of the CFNGM(1,1,k,c). This paper first elucidates reasons for the internal errors generated by the continuous grey model CFNGM(1,1,k,c), and explains the classic method, the discrete grey forecasting model, of eliminating internal errors. On the basis of an in-depth analysis of the modeling mechanism of CFNGM(1,1,k,c) model, a new parameter adjustable grey forecasting model is proposed by introducing parameter adjustment factors to modify model’s parameters. Finally, the new model is applied to explore the gross regional product of Chengdu and Deyang in the Chengdu metropolitan area. The calculation results indicate that the newly proposed model can obtain more accurate results.  Related Articles | Metrics

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Grey Clustering Methods With Universal Possibility Functions Long Wang, Zhigeng Fang, Qin Zhang, Sifeng Liu The Journal of Grey System    2023, 35 (4): 19-33.   Abstract （64）            Knowledge map    The traditional possibility functions are always assumed to be linear functions. The preferences of decision-makers are not considered. It might not be proper to analyze all kinds of indicators with the traditional possibility functions. Therefore, we consider the preferences and first develop the universal possibility functions. The decision-makers can obtain the appropriate universal possibility functions by adjusting the clustering preference. Then, the related properties are revealed by the proof. Next, grey clustering methods with universal possibility functions are proposed. Finally, the effectiveness of the suggested methods is verified via the case illustration and comparative analysis. Related Articles | Metrics

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A Novel Modeling Method of Extended Grey EGM(1,1,∑e^(ck)) Model and Its Application in Predictions Maolin Cheng, Bin Liu The Journal of Grey System    2023, 35 (4): 55-75.   Abstract （65）            Knowledge map    In the grey models, the GM(1,1) model is an important type of prediction model. The traditional grey GM(1,1) model has good prediction results in the case the original data show exponential variations at a slow rate. However, in practical problems, although showing exponential variations or approximately exponential variations, original data vary very fast sometimes. In these cases, the traditional grey GM(1,1) model tends to have poor prediction accuracy, mainly because the data fails to meet the laws presented by the traditional model. Therefore, the paper makes improvements in the following two aspects: first, the paper transforms the traditional accumulated generating sequence of original data; second, the paper extends the traditional grey model's structure, i.e., building a grey EGM(1,1,∑e^(ck)) model. The paper offers the parameter optimization method of the grey EGM(1,1,∑e^(ck)) model. Using the novel modeling method proposed, the paper builds the grey EGM(1,1,∑e^(ck)) models for China's total electricity consumption and China's GDP per capita, respectively, in the final section. Results show that the models built with the proposed modeling method have high simulation precision and prediction precision. Related Articles | Metrics

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A Multi-attribute Decision-making Method Based on Grey Correlation Lirong Sun, Chi Zheng, Chenkai Jiang, Yinghua Tian, Yujing Ye The Journal of Grey System    2023, 35 (4): 76-90.   Abstract （93）            Knowledge map    Aiming at the grey feature problem of ' small sample and poor information ', this paper extends the traditional analytic hierarchy process, entropy method and ' vertical and horizontal ' scatter degree method to the field of grey number, and proposes a multi-attribute decision-making method based on grey correlation. Firstly, the applicable form of index weight is enriched, and the determination method of index weight in grey number form is given systematically. Secondly, aiming at the problem that the traditional evaluation method can not be directly applied to the comprehensive evaluation with grey characteristics, a comprehensive evaluation model in the form of grey number is proposed. Finally, through the interval grey number integration method and the ' kernel and grey number ' integration method, the evaluation values under each index are formed into a comprehensive evaluation value, and the evaluation results are sorted. Compared with the traditional evaluation method, the proposed method more reflects the rationality and dynamics of the evaluation results. Related Articles | Metrics

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Reinforcement Model for Unmanned Combat System of Systems Based on Multi-Layer Grey Target  Xueting Hao, Zhigeng Fang, Jingru Zhang, Fei Deng, Ankang Jiang, Shuyu Xiao The Journal of Grey System    2024, 36 (2): 54-66.   Abstract （71）            Knowledge map    In the future battlefield, unmanned combat mode will be crucial. Its attack strategy formulation is a significant and complex task. To address the issue of decisionmaking in unmanned combat system of systems(UCSoS) striking strategy, this paper begins by analyzing the characteristics of UCSoS. By utilizing the GERT concept, an unmanned combat A-GERT network is developed to provide parameter support for evaluating its effectiveness. Secondly, for effectiveness evaluation, a multi-layer gray target model built on the A-GERT network is proposed. This model is utilized to formulate the optimal striking scheme for the UCSoS. And Agent technology is employed to solve the intelligent learning decision-making problem for UCSoS based on multi-layer grey target model. Finally, a case study illustrates the efficiency and effectiveness of the reinforcement model for UCSoS based on multi-layer grey target. Related Articles | Metrics

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Improved Fractional Order Single Optimization Parameter Grey Model Jiangtao Wei, Yonghong Wu The Journal of Grey System    2023, 35 (4): 154-171.   Abstract （245）            Knowledge map    Some grey prediction models suffer from outliers and overfitting, and their prediction performance can be improved. Based on the fractional grey model (FGM) and the fractional time delayed grey model (FTDGM), an improved fractional single optimization parameter grey model (IFSGM) is proposed in this paper. A timetranslation power term is used to reduce model overfitting. The data is pre-processed to reduce the influence of outliers based on data transformation in cumulants and summations. The convolutional computational formulation is used to perform the prediction and improve the prediction effect. The genetic optimization algorithm is used to optimize the order of the fractional cumulants, find the optimal value of the order, and improve the model fit and prediction effect. In 6 data sets, the IFSGM and the 7 grey models are compared and tested. The experimental results show that IFSGM achieves excellent prediction performance.  Related Articles | Metrics

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On Grey Weighted Central Moving Average Model and Its Application Sifeng Liu, Zurun Xu, Liangyan Tao, Yingjie Yang The Journal of Grey System    2023, 35 (4): 172-182.   Abstract （60）            Knowledge map    The idea and method of weight vector group of kernel clustering have been combined with the central moving average model and grey system prediction model in response to the problems existing in the traditional moving average formula. A grey-weighted central moving average model was proposed in this paper. In the modeling process of the grey-weighted center moving average model, the number of moving average terms should be determined first, and the weight vector should be set according to the rules of weight setting for each component of the weight vector group of kernel clustering. Next, the weighted center moving average formula can be used to calculate the moving average simulation value, and the simulation errors were analyzed. Then, the grey system prediction model is applied to obtain a set of predicted values for the studied time series data. Finally, based on actual data and the predicted values of the grey system model, the required predicted values are calculated using the weighted center moving average formula. The new model can effectively solve the problem of serious lag in the simulation and prediction results of the simple moving average formula and the weighted moving average formula and also overcome the shortcomings of the center moving average model and the weighted center moving average model, which cannot be used to predict future changes due to the need for data on both sides of the "center" yt during calculation. From the simulation and prediction results of China's invention patent authorization volume, it can be seen that the new model has obvious advantages.  Related Articles | Metrics

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Interaction-based nonlinear INDGM(1,N) model and its application Ye Li, Dongyu Liu, Junjuan Liu, Meidan Xiao The Journal of Grey System    2024, 36 (1): 56-62.   Abstract （81）            Knowledge map    Multivariate grey models are commonly used to evaluate the independent effects of related factors, but they may fail to account for any nonlinear interactions that could exist between them. To address this limitation, this study introduces the INDGM(1,N) model, which considers the nonlinear interactions between related factors. Simultaneously, to depict the nonlinear impact of both the system behavior sequence and corresponding factor sequences more accurately, varying power parameters have been incorporated into the model proposed in this paper. Moreover, by adjusting the parameter values of the INDGM(1,N) model, it can be converted into several other models, such as the DGM(1,N) model, GM(1,N) model, DGM(1,1) model, or GM(1,1) model. This study uses a genetic algorithm to obtain the time response of the INDGM(1,N) model by solving its nonlinear characteristic parameters. We then use the model to simulate and forecast China's CO2 emissions and compare its performance with that of other models. The results show that the INDGM(1,N) model provides more accurate simulation and prediction accuracy than other models, highlighting its effectiveness.  Related Articles | Metrics

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Time-Delay TLDBGM(1,N) model with dynamic background value and its application  Dang Luo, Xinqing Qiao The Journal of Grey System    2024, 36 (1): 63-78.   Abstract （67）            Knowledge map    Since the traditional multivariable grey prediction model has insufficient consideration of the coupling effect of background value and time-delay accumulative term, which leads to the low prediction accuracy of the model. Based on this, we propose a new multivariable time-delay grey prediction model with dynamic background value. The model adds dynamic background value coefficient, time-delay parameters, linear correction term, and grey action quantity term to the traditional GM(1,N) model. First, the delay periods of driver factors are determined by using the grey time-delay correlation analysis method. Second, the parameter estimation method of the model is discussed and the direct solution of the TLDBGM(1,N) time response function is given by defining the derived form of the TLDBGM(1,N) model. Finally, the model background value coefficient and time-delay parameters are identified and optimized based on the differential evolutionary algorithm. The model is applied to the problem of grain yield prediction in Henan Province. Result shows that the simulation and prediction accuracy of TLDBGM(1,N) are better than other multivariable grey prediction models. The model is theoretically more generalized. And it is shown that GM(1,1), GM(1,N), and TLGM(1,N) models are all special forms of the model for different parameter values.   Related Articles | Metrics

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On Grey Real Number and Its Operation Rules Xican Li, Li Li The Journal of Grey System    2024, 36 (1): 32-44.   Abstract （69）            Knowledge map    In order to reveal the mathematical mechanism of generating grey numbers and the law of grey number evolution, the extension principle of grey set is proposed in this paper firstly, which provides a theoretical basis for grey number operation. Secondly, based on the possibility function of grey set, the concept and properties of grey convex set are given. According to grey convex set, the definitions of grey real number, universal grey real number and interval grey number are put forward. Thirdly, the basic operation rules of grey real number are given based on the extension principle, and the operations of data block expression are further given to meet the reversibility of grey real number operation. Finally, the sufficient and necessary condition for the reversibility of universal grey real number operation is given, and it is proved that the grey real number operations using data block expression are reversible. Examples show that the grey real number operations using data block expression proposed in this paper are feasible and effective. The research results enrich the grey system theory and provide a theoretical basis for grey algebra research.   Related Articles | Metrics

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A study on the grey relational weighted evaluation model for the selection of leading industries in the airport economic zone  Hongqing Liao, Zhigeng Fang, Qin Zhang, Xiaqing Liu, Chuanhui Wang, Ding Chen, Xiaochao Qian The Journal of Grey System    2024, 36 (1): 45-55.   Abstract （54）            Knowledge map    The evaluation of leading industries involves the use of evaluation index data that exhibit multi-source uncertainty. This implies that the evaluation index values encompass various types of numbers, such as grey numbers, white numbers, fuzzy numbers, interval value fuzzy numbers, and more. The evaluation index system possesses a multi-level structure with interconnections among the indicators. In this study, we propose a generalized grey relational weighted evaluation model based on multi-source uncertainty for assessing leading industries. To begin with, the generalized grey number is used to represent all uncertain data, and the weights of multi-level indicators are calculated by combining the generalized grey number with the information entropy model. Subsequently, the correlation between the indexes is examined utilizing the theorem of the generalized grey absolute relational degree model. This analysis leads to the construction of a generalized grey absolute relational degree matrix, from which eigenvalues and eigenvectors are derived. Based on the generalized grey weight of the multi-level indexes, the eigenvalue and eigenvector of the generalized grey absolute relational matrix, and the grey relational weighted evaluation model for leading industry evaluation are established. Finally, the effectiveness and feasibility of the proposed model are validated through a case study involving the evaluation of the leading industry in the airport economic zone.   Related Articles | Metrics

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Memorabilia of the Establishment and Development of Grey System Theory  Sifeng Liu, Liangyan Tao, Wei Tang The Journal of Grey System    2024, 36 (1): 1-3.   Abstract （106）            Knowledge map    This article summarizes and records important historical events in the establishment and 40 year development process of continuing innovation and dissemination of grey system theory, providing reference for scholars who pay attention to the evolution laws of grey system theory, a new branch of uncertainty system research, as well as colleagues engaged in grey system theory research. If there are any important omissions, we sincerely welcome readers to supplement and improve.  Related Articles | Metrics

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Modeling and Predicting the Socio-Economic Performance of Countries Using Grey Relational Analysis and K-NN Algorithm Hande Hakan, Ecem Coşar Canlıer, Çiğdem Özarı, Esin Nesrin Can The Journal of Grey System    2024, 36 (1): 4-15.   Abstract （136）            Knowledge map    The main purpose of this study is to forecast the countries’ socio-economic performance with the fewest possible parameters. To do this, we propose a model consisting of methods from Multi-Criteria Decision Making and Machine Learning. Since the existence of different classifications of countries and several socioeconomic parameters, it becomes difficult to make a prediction of their belonging group and compare countries based on these parameters. Using the Grey Relational Analysis and the Critic method, we classify the countries into four different subgroups based on several socio-economic dimensions. K-Nearest Neighbor (K-NN) algorithm with basic macro-economic parameters is implemented to predict the countries' socioeconomic groups. The results rank the countries according to their socio-economic performance and predict the countries’ development levels for the future. The main findings indicated that the proposed approach can be used for similar research questions. The highest prediction percentages are accurate for small values of k. This study provides a convenient and effective method for grouping countries at different levels of development using basic economic parameters and provides a simple and practical method to predict the belonging group.   Related Articles | Metrics

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An Exponential-Polynomial Matrix Model Based on the Accumulation Generation of Ternary Interval Number Series and Its Application in Forecasting China's GDP by Region Lihua Ning, Fangli He, Xiangyan Zeng, Yunjie Mei, Haoze Cang The Journal of Grey System    2023, 35 (4): 34-54.   Abstract （71）            Knowledge map    Ternary interval number includes the total GDP amount in a certain period and its change range. Comprehensive information is more conducive to management decision-making. Affected by regional characteristics and national macro-control, the development trend of GDP in various regions of China in the past 15 years has been different. Some central regions grew rapidly in the early stage and fell back in the later stage, showing a saturated growth trend. Some coastal economically developed areas showed exponential growth. While some regions show an unstable upward and downward fluctuation trend. In order to predict the development trend of different GDPs, a matrix model based on exponential and polynomial regression, which can directly model the ternary interval number, is proposed in this paper. In order to eliminate the random fluctuation of data, the original ternary interval number sequence is accumulated based on the data preprocessing method in the grey model, which makes the general non-negative sequence show quasi-exponential growth so that it can be applied to the exponential-polynomial matrix model. The particle swarm optimization algorithm and the least square method are combined to estimate the parameters of the new model. The new model, quadratic polynomial, GM (1, 1), and exponential function are used to predict the GDP of 31 regions in China from 2005 to 2020. The results show that the effect of the new model is better than other models in predicting GDP for 20 regions.  Related Articles | Metrics

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Discovering the mechanism of grey forecasting models from the perspective of dynamic system modelling   Xiaolei Wang, Naiming Xie The Journal of Grey System    2024, 36 (2): 90-99.   Abstract （65）            Knowledge map    Grey forecasting models have found extensive applications across various domains, but the connection between their theory and practice has not yet been fully revealed. This paper seeks to discuss the modelling mechanism of grey forecasting models from the perspective of dynamic system modelling and illustrate how to establish grey forecasting models to address real-world challenges. Firstly, we outline the grey forecasting models under the traditional and direct frameworks. Then, the problem description and model assumptions of grey forecasting models are discussed by incorporating model characteristics and Prof. Deng's original concepts. The complete process of model establishment as well as the purpose and tasks of each step is elaborated in detail. Ultimately, taking the inventory of perishable products as a case study, this article discusses the utilization of grey forecasting models in inventory management and elucidates the application process using citrus as a specific example.  Related Articles | Metrics