Abstract: To solve the problems of the traditional data envelopment analysis (DEA) and directional distance function (DDF), scholars have proposed crossing DDF methods, but those methods are still defective in that evaluation results are inaccurate for the slacks of indicators being neglected; evaluation benchmarks (reference points) are inconsistent among DMUs; there may be multiple optimal solutions; and the efficiency of decision-making units cannot be fully ranked. In order to solve the above defects, this paper gives a cross efficiency evaluation method through DDF projections according to the indicator ratios between two DMUs. Firstly, this method gives the definition of relative efficiency based on reference DMUs, i.e., the ratio of comprehensive input ratio and comprehensive output ratio of DMUs, and makes all indicators' information of DMUs to be taken into account in efficiency evaluation. Secondly, based on the DDF method, the second objective function model that minimizes the sum of the slacks of inputs and outputs is constructed to obtain unique projection and unique evaluation result for DMUs. Finally, taking the DDF projections as references, the crossing efficiency of DMUs is evaluated according to the definition of relative efficiency, which ensures that evaluation reference points are consistent among the DMUs, evaluation results are not affected by the length of direction vectors and have strong consensus, and all DMUs are fully ranked according to their efficiencies. Two numerical examples are used to show that to a certain extent, the proposed method is scientific and feasible.

Key words: DEA, DDF, indicator ratio, projection, crossing-evaluation