Grey Relational Analysis: A Powerful Tool for Comprehensive Evaluation

Idea and Principles of Grey Comprehensive Evaluation Method

In cybernetics, colour depth is used as a metaphor for information certainty: black for unknown, white for fully known, and grey for partially known. Systems with completely unknown information are black systems, fully known systems are white systems, and partially known systems are grey systems. Grey systems, which lie between black and white, often exhibit unique properties worth exploring.

Grey systems are information‑poor, making traditional statistical methods ineffective. Grey system theory, proposed by Professor Deng Julong in 1982, addresses such sparse‑information systems and can work with very few observations. Its key feature is that it does not require large sample sizes or any specific probability distribution.

Social, economic, and other complex systems display hierarchical complexity, fuzzy structural relationships, dynamic randomness, and incomplete or uncertain data—characteristics that make them grey. Consequently, grey system theory has broad development prospects and has already been applied in many fields.

The focus of this article is on grey relational analysis, which forms the basis of a grey comprehensive evaluation method.

In abstract systems such as social or economic systems, many factors interact. Determining which factors are primary, secondary, or most influential is essential for factor analysis. For example, grain production is affected by fertilizer, pesticides, seeds, weather, electricity, soil, irrigation, technology, policy, etc. To improve yield and achieve balanced economic, social, and ecological benefits, factor correlation analysis is necessary.

Traditional regression analysis works well for few, linear factors but struggles with many, nonlinear factors. Grey system theory offers a new approach: relational degree analysis, which measures the similarity or difference of factor development trends.

Relational degree analysis first requires selecting appropriate data series that reflect system behavior. Once the series are defined, the relational degree formula yields a quantitative measure of how closely each evaluation object approaches an ideal reference, thereby ranking the objects. The object with the highest grey relational degree is considered the best.

This method does not depend on sample size or specific distribution, requires minimal computation, and can be performed manually even with ten variables. Its mathematical basis is non‑statistical, making it practical when data are scarce or statistical assumptions are unmet.

Model and Steps of the Grey Comprehensive Evaluation Method

Grey Relational Analysis

Grey relational analysis quantifies the strength of relationships between factors by comparing the geometric similarity of their time‑series curves. The closer two curves are, the higher their relational degree, indicating a stronger connection.

It treats relational degree as a relative ranking based on the similarity of curve shapes, effectively comparing the development trends of multiple series.

Consider three data series: regional total revenue (1997‑2003), investment attraction revenue, and agricultural revenue. By plotting these series, the shape of the investment curve closely matches the total revenue curve, while the agricultural curve diverges, indicating that investment attraction has the greatest impact on revenue.

The analysis proceeds by defining a reference series (the mother factor) and comparing each candidate series (child factors) to it. Differences at each time point are calculated, optionally normalised, and a distinguishing coefficient (commonly 0.5) is applied to reduce the influence of extreme values.

After computing point‑wise relational coefficients, they are averaged to obtain an absolute relational degree for each candidate series. The series with the highest absolute relational degree is deemed most similar to the reference and thus most influential.

(1) When the relational degree of series i is greater than series j, i is considered superior to j.

(2) When r_i < r_j, i is inferior to j.

(3) When r_i = r_j, they are considered equal.

(4) When r_i ≥ r_j, i is not worse than j.

(5) When r_i ≤ r_j, i is not better than j.

By ranking the influencing factors of the reference series according to these definitions, one obtains a prioritized list of factors.

Grey Comprehensive Evaluation Based on Grey Relational Analysis

The comprehensive evaluation aims to rank multiple alternatives. The model uses the following components:

Determine the optimal indicator set for each criterion, selecting either the maximum or minimum value depending on whether larger or smaller values are preferable.

Normalize indicator values to eliminate differing units and scales, using methods such as initial‑value scaling, mean‑value scaling, or other standard normalization techniques.

Compute the comprehensive evaluation results by applying grey relational analysis between each alternative’s indicator values and the optimal indicator values, typically using a distinguishing coefficient of 0.5.

The alternative with the highest overall relational degree is considered the best, allowing a clear ranking of all alternatives.

Overall, grey relational analysis provides a quantitative comparison of system trends by measuring the geometric closeness of multiple series to an ideal reference, enabling effective multi‑criteria decision making even with limited data.