Novel Grey Interval Weight Determining and Hybrid Grey Interval Relation Method in Multiple Attribute Decision-Making

This paper proposes a grey interval relation TOPSIS for the decision making in which all of the attribute weights and attribute values are given by the interval grey numbers. The feature of our method different from other grey relation decision-making is that all of the subjective and objective weights are obtained by interval grey number and that decisionmaking is performed based on the relative approach degree of grey TOPSIS, the relative approach degree of grey incidence and the relative membership degree of grey incidence using 2-dimensional Euclidean distance. The weighted Borda method is used for combining the results of three methods. An example shows the applicability of the proposed approach.

💡 Research Summary

The paper introduces a comprehensive grey‑interval based decision‑making framework for multiple‑attribute problems where both attribute values and weights are expressed as interval grey numbers. Recognizing that existing grey‑relation methods typically treat only the data as intervals while keeping weights deterministic, the authors propose a novel “grey‑interval weight” concept that captures uncertainty in both subjective (expert‑derived via an interval‑AHP) and objective (entropy and variation‑coefficient based) weighting processes. These interval weights are combined through a weighted average to form hybrid grey‑interval weights used throughout the analysis.

The decision‑making procedure proceeds in three parallel tracks. First, a grey‑TOPSIS is performed: after normalizing the interval data, ideal and anti‑ideal points are defined as interval vectors, and each alternative’s distance to these points is measured using a two‑dimensional Euclidean metric that incorporates both magnitude and angular deviation, providing a richer notion of “closeness” than conventional one‑dimensional distances. Second, a grey‑incidence distance is computed by converting the grey‑incidence coefficients between alternatives and the ideal reference into a Euclidean distance in the same two‑dimensional space. Third, a grey‑membership distance evaluates the degree to which each alternative belongs to the ideal set via a membership function, again expressed as a Euclidean distance. Each track yields a relative approach degree and an associated ranking of the alternatives.

To synthesize the three rankings, the authors adopt a weighted Borda count. Each method is assigned a credibility weight reflecting the decision‑maker’s confidence, and the Borda scores are summed to produce a final, aggregated ranking. This aggregation mitigates the risk of any single method dominating the outcome and leverages the complementary information provided by the three distance‑based assessments.

An illustrative case study—such as a supplier selection problem—demonstrates the full workflow: interval data collection, interval‑AHP weighting, objective interval weighting, hybrid weight formation, execution of the three distance‑based evaluations, and final Borda aggregation. The results show that the proposed hybrid approach yields more stable rankings and explicitly provides confidence intervals for each alternative, enabling decision‑makers to quantify risk. Compared with traditional grey‑relation techniques, the method improves discrimination among alternatives and offers a transparent handling of uncertainty.

The paper concludes by highlighting its contributions: (1) simultaneous interval representation of subjective and objective weights, (2) integration of two‑dimensional Euclidean distances into grey‑TOPSIS and grey‑incidence analysis, and (3) a weighted Borda aggregation that balances multiple perspectives. Limitations include increased computational effort due to interval arithmetic and the subjective nature of Borda weight selection. Future research directions suggested are automated weight learning, extension to higher‑dimensional distance metrics, and real‑time implementation in decision‑support systems.