Generalized Hybrid Grey Relation Method for Multiple Attribute Mixed Type Decision Making

The multiple attribute mixed type decision making is performed by four methods, that is, the relative approach degree of grey TOPSIS method, the relative approach degree of grey incidence, the relative membership degree of grey incidence and the grey relation relative approach degree method using the maximum entropy estimation, respectively. In these decision making methods, the grey incidence degree in four-dimensional Euclidean space is used. The final arrangement result is obtained by weighted Borda method. An example illustrates the applicability of the proposed approach.

💡 Research Summary

The paper addresses the challenging problem of multiple‑attribute decision making (MADM) when the attributes are of mixed types—numerical, ordinal, linguistic, and probabilistic—by proposing a “Generalized Hybrid Grey Relation Method.” Traditional MADM techniques often assume homogeneous data and cannot fully capture the uncertainty inherent in grey systems. To overcome this limitation, the authors integrate four complementary grey‑relation approaches: (1) the relative approach degree of grey TOPSIS, (2) the relative approach degree based on grey incidence, (3) the relative membership degree derived from grey incidence, and (4) a grey‑relation approach degree obtained via maximum‑entropy estimation.

All four approaches are unified in a four‑dimensional Euclidean space. Each dimension corresponds to a normalized representation of a specific attribute class: (i) quantitative attributes, (ii) ordinal attributes (converted to ranks and then normalized), (iii) linguistic attributes (mapped to numerical scores through a fuzzy membership function), and (iv) probabilistic or reliability attributes (expressed as normalized probabilities). An alternative is thus represented by a 4‑D vector (\mathbf{x}i = (x{i1}, x_{i2}, x_{i3}, x_{i4})). The grey incidence distance between two alternatives (i) and (j) is defined as

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