Glossary | Grey System

Posted Sep 22, 01:21 AM

Glossary of Important Terminologies and Models

Glossary of Grey Models

Dynamic Grey Relational Analysis

Dynamic Grey Relational Analysis (DGRA) method is a generalized form of Deng’s GRA model. Unlike Deng’s GRA model, the DGRA model does not assumes the value of the Distinguishing Coefficient to be static (e.g., 0.5). In fact, in the DGRA the Distinguishing Coefficient is dynamic and it evolves as the problem evolves. The Dynamic Distinguishing Coefficient of the DGRA is a true representation of the dynamicness of a grey system, a system with partially known information that evolves as new information becomes available.

It can be used for both Multiple Criteria Decision Analysis (MCDA) and grey correlational analysis between multiple factors. It can be used for nonparametric analysis of the relationship between dependent variable and multiple independent variables. The software to apply the DGRA model will be released soon. Please keep visiting the website.

The model has been positively rated in the contemporary literature (see, e.g., Delcea and Cotfas, 2023). Recently, it has seen application in apparel industry, information and communication technologies (ICT), and e-commerce.

Grey Ordinal Priority Approach

Grey Ordinal Priority Approach (OPA-G) method is a Grey Number-based Ordinal Priority Approach to Multiple Criteria Decision Analysis. The method is suitable for decision-making under uncertainty. The method can estimate the weights of criteria, alternatives and experts simultaneously.

The pioneering work on the OPA-G has been recognized by the De Gruyter Handbook of Responsible Project Management. Recently, the OPA-G model has seen applications in Electric Vehicle, and agriculture industries.

Posterior-Variance Test (for forecasting)

Forecasting of a variable should not end at forecast error reporting. It is also important that whether the forecasting model used was qualified to forecast the given variable reliably or not. For this purpose, the posterior-variance test (also called, posterior-error test) provides a useful technique for the forecasters, especially the ones using grey forecasting models. A very detailed introduction to the PVT has recently been published here.

Recently, it has seen application in digital economy, and methane emissions forecasting.

Optimized Discrete Grey Forecasting Model, DGM (1, 1, θ)

The classical discrete grey forecasting model DGM (1,1) does not produces reliable results under some cases, e.g., when the historic data contains significant amount of non-linearity. To solve this issue DGM (1, 1, θ) was proposed. Generally, DGM (1, 1, θ), which is a generalized case of DGM (1,1), is more accurate than EGM (1, 1, α, θ).

Recently, it has seen application in energy sector and ISO/IEC 27001 Certifications analysis.

Optimized Even Grey Forecasting Model, EGM (1, 1, α, θ)

The classical grey forecasting model EGM (1,1) does not produces reliable results under some cases, e.g., when the historic data contains significant amount of non-linearity. To solve this issue EGM (1, 1, α, θ) was proposed.

Recently, it has seen applications in environmental science, finance, automobile industry, international trade (exports volume prediction; trade deficit prediction), tourism sectors of Malawi and Indonesia, and ISO/IEC 27001 Certifications analysis.

Multivariate Hybrid Grey Forecasting Model Verhulst-GM(1,N)

V-GM(1,N) is a multivariable hybrid of Grey Verhulst and the classical GM (1,N) models. The models enjoys the strengths of both models and is a suitable approach to forecasting using multiple variables.

Grey Linear Programming

Linear programing under uncertainty is no easy task. Grey Linear Programming (GLP) method is a convenient and effective method to solve linear programming problems where uncertainty can be represented through interval grey numbers.

Framework to Predict the State of an Equipment

In manufacturing industry, predicting the state of a system (i.e., a system is performing normally or abnormally) is of critical importance. Two-stage multi-level equipment grey state prediction model is an effective decision support system to deal with such problems and can effectively serve as an early warning tool. The system can also help the managers in reducing the equipment’s major failure risk and maintenance costs.

Grey Assessment (for linguistic input and output)

When the members of a group are ranked using linguistic expressions (e.g., a teacher may grade her students as @excellet’, “good”, “average,” etc.) the overall (mean) performance of the group can’t be assessed by applying traditional method of calculating mean or average of the individual scores of its members. What is the average of “excellent” and “very bad”? Certainly, it’s not easy to answer. Even if we convert these linguistic/qualitative expressions into numeric scores and then take average what is the qualitative meaning of that mean value? For example, if “excellent” is 7, and “very bad” is 2, and mean is 4.5, then what does this 4.5 signifies? Good or very good or something else? Voskoglou’ Grey Assessment method is a very convenient and effective approach to handle such problems.

Glossary of Keywords

Distinguishing Coefficient

The Distinguishing Coefficient (often denoted as ξ or ρ) is a key parameter in Grey Relational Analysis (GRA) , specifically in the formula that calculates the grey relational coefficient. It is also called resolution coefficient. Today, its standard form is known as the Dynamic Distinguishing Coefficient.

The works by Dr. S. A. Javed, who proposed the first formal framework to conceptualize the distinguishing coefficient, are essential for understanding the properties of the Distinguishing Coefficient.

References:
Javed, S. A. (2019). A novel research on Grey Incidence Analysis models and its application in Project Management (Doctoral dissertation). Nanjing, China: Nanjing University of Aeronautics and Astronautics. https://lib.nuaa.edu.cn/

Javed, S. A., Gunasekaran, A., & Mahmoudi, A. (2022). DGRA: Multi-sourcing and Supplier Classification through Dynamic Grey Relational Analysis Method. Computers & Industrial Engineering, 173, 108674. https://doi.org/10.1016/j.cie.2022.108674

Dynamic Distinguishing Coefficient

The Dynamic Distinguishing Coefficient (often denoted as ξ(k) or ρ(k)) is an extension of the classic distinguishing coefficient used in Grey Relational Analysis (GRA). It is central to Dynamic Grey Relational Analysis (DGRA), a model that generalizes Deng’s traditional GRA method.

The core idea is that in a complex, evolving/dynamic system, a single static value for the distinguishing coefficient is insufficient. The Dynamic Distinguishing Coefficient optimizes the coefficient individually for each data point, providing a more adaptive, responsive and granular analysis.

The works by Dr. S. A. Javed, who proposed the first formal framework to conceptualize the distinguishing coefficient, are essential for understanding the properties of the Dynamic Distinguishing Coefficient.

Grey Forecasting

Grey forecasting (or grey prediction) is a time-series forecasting methodology designed to build models from small, incomplete, or uncertain data. There are several types of grey forecasting models, e.g., univariate (Javed et al., 2025), multivariate (Ofosu-Adarkwa et al., 2020), etc.

References:
Javed, S. A., Mahmoudi, A., Tao., L., & Dong, W. (2025). Electric vehicle stock forecasting and planning in the USA. Environment, Development and Sustainability. https://doi.org/10.1007/s10668-025-06659-6

Ofosu-Adarkwa, J., Xie, N., & Javed, S. A. (2020). Forecasting CO2 emissions of China’s cement industry using a hybrid Verhulst-GM(1,N) model and emissions’ technical conversion. Renewable and Sustainable Energy Reviews, 130, 109945. https://doi.org/10.1016/j.rser.2020.109945

Grey Information

Grey information is simply information that is partially known and partially unknown. It’s the raw material of a grey system.

Grey information isn’t random or fuzzy—it has a known boundary but an unknown exact value. This bounded uncertainty is its defining characteristic. It often appears in forms like:

Interval numbers: The company’s revenue is between 5M and 9M. (You know the range, not the precise figure).

Linguistic terms with a range: “High performance” might be defined as a score between 80 and 100.

Partially known structures: You know the influencing factors of a system but not the exact mathematical relationships between them.

Grey Number

A grey number is a number whose exact value is unknown, but the interval or range within which the value lies is known. It is typically denoted by the symbol “⊗” and represents incomplete or uncertain information in contrast to a white number (fully known) or a black number (completely unknown).

In more precise terms: If a and a are known real numbers with a ≤ a ≤ a , then a grey number ⊗ a ∈ [a , a ] indicates that the true value of ⊗ a lies somewhere within this closed interval, but its exact value is not specified.

Example: ⊗t ∈ [29, 31] — the temperature is somewhere between 29°C and 31°C, but the exact value is unknown.

References:
Candra, C. S. (2022). Evaluation of Barriers to Electric Vehicle Adoption in Indonesia through Grey Ordinal Priority Approach. International Journal of Grey Systems, 2(1), 38-56. https://doi.org/10.52812/ijgs.46

Mahmoudi, A., Liu, S., Javed, S.A., & Abbasi, M. (2018). A Novel Method of Solving Linear Programming with Grey parameters. Journal of Intelligent & Fuzzy Systems, 36(1), 161–172. https://doi.org/10.1007/s40815-020-00827-8

Voskoglou, M. G. (2023). Grey Assessment. International Journal of Grey Systems, 3(2), 5-7. https://doi.org/10.52812/ijgs.77

Grey Relational Analysis

Grey relational analysis (GRA) is a technique that measures the degree of relationship between different sequences of data, especially when the information is incomplete, the sample size is small, or the underlying system’s structure is unclear. In essence, it quantifies how closely one data curve follows another—turning grey, uncertain relationships into a crisp “grey relational grade” that you can compare and rank. Today, the standard form of GRA is known as the Dynamic GRA.

Javed (2019) reported confusion in the literature and was the first to note that Grey Relational Analysis, Grey Incidence Analysis, and Grey Correlational Analysis are names for the same methodology. The works by Dr. S. A. Javed are essential for understanding the theory and methodology of GRA.

Grey System

Grey system is a system whose information is partially available.

For example, the economy is a grey system, which is why there are multiple perspectives on how to conceptualize and manage it. Other interesting examples of grey systems include new products (e.g., ChatGPT) and complex systems (e.g., the human brain and behavior). If complete information is labelled ‘white’ and a complete lack of information is labelled ‘black’, then ‘grey’ implies partially complete and partially incomplete information.

The Grey Systems Theory classifies all systems into three classes:

White system – a system with completely known information, e.g., a simple pendulum.

Black system – a system with completely unknown information, e.g., a sealed box with unknown contents.

Grey system – a system with partially known, partially unknown information, e.g., the economy.

Whitenization

In grey system theory, whitenization is the process of transforming a grey number (a value whose exact magnitude is unknown but bounded) into a definite, deterministic “white number.” It quantifies uncertainty by picking a representative crisp value from the possible range.

A simple representative value is chosen, often the midpoint of the interval: If ⊗ ∈ [a, b], the whitened value is typically ⊗̃ = (a + b)/2. More generally, it’s a convex combination: ⊗̃ = α a + ( 1 − α ) b, with α∈[0,1] reflecting optimism/pessimism.

References:
Ofosu-Adarkwa, J., Xie, N., & Javed, S. A. (2020). Forecasting CO2 emissions of China’s cement industry using a hybrid Verhulst-GM(1,N) model and emissions’ technical conversion. Renewable and Sustainable Energy Reviews, 130, 109945. https://doi.org/10.1016/j.rser.2020.109945