Positive/negative effects of undesirable output on technology gap ratio under meta-frontier framework

In nonradial data envelopment analysis (DEA), the efficiency of meta-frontier may exceed that of group frontier, leading to technology gap ratio (TGR) greater than 1. Although existing methods have attempted to resolve this problem, they have severely limitations: (1) The influence of weak disposability within undesirable outputs on TGR remains unclear, particularly in distinguishing its effects on meta-frontier and group frontier; (2) Current methods fail to systematically account for undesirable outputs in projection processes, potentially distorting efficiency evaluation. According to these limitations, graphical method is used to analyze the influence of undesirable output on TGR. Then, in order not to interfere with undesirable output in projection process, new models are proposed based on the closest targets, which can control the efficiency of meta-frontier is not greater than that of group frontier. Subsequently, undesirable output possible sets are defined for different models, and thus developed to determine its minimum/maximum variation ranges under different frontiers. By determining variation range of undesirable output under variable returns to scale (VRS), the positive/negative effects on TGR are obtained. The advantages of our methods over existing methods are as follows: (1) Our methods ensure theoretical consistency in efficiency evaluation; (2) The potential impact of undesirable output on TGR is first studied; (3) Our methods offer actionable insights for efficiency improvement through undesirable output ranges. Finally, two examples are provided to illustrate the validity and effectiveness of our methods.