A common compatibility index of judgement matrices

This paper first presents the transformation formulas of a reciprocal judgement matrix and a complementary judgement matrix, and then points out the relationship between the perfectly consistent reciprocal judgement matrix and the perfectly consistent complementary judgement matrix. Based on the characteristics of reciprocal judgement matrix and complementary judgement matrix, a common compatibility index of judgement matrices (reciprocal judgement matrices and complementary judgement matrices) and the concept of reciprocal weight matrix and complementary weight matrix are given. The compatibility and consistency of judgement matrices are studied in detail, and a criterion to measure the compatibility of judgement matrices is also provided. These results can help to improve the reasonability and effectiveness of decision, and thus have great practical application prospect in group decision making. Finally, a numerical example is given.