變動規模報酬下以虛擬差額方法評量超高效與超低效

Abstract

有許多文獻探討解決資料包絡分析(data envelopment analysis, DEA)在變動規模報酬(variable returns-to-scale, VRS)條件下進行超高效率(Super-efficiency)分析時，如被評量的決策者有極端高的產出項以輻射型(radial-based)投入導向(super-BCC-I)、或是有極端低的投入項以產出導向(super-BCC-O)，數學規劃模型求解時將因不可行解而受阻的問題。但是以非輻射模型(non-radial-based models)分析超高效率時，不可行解的困難仍待解決。(Liu & Huang, 2015)提出之虛擬差額模型(virtual-gap-measurement, VGM)屬於非輻射型模型之差額變數為基(slack-based)，(Liu & Laio, 2015) 利用VGM析解固定規模報酬(constant-return-to-scale, CRS)條件之超高效率， 。本研究延伸此議題至在VRS條件下之超高效率，可成功克服計算時不可行解的問題，得到虛擬技術差額(virtual-technical-inefficiency-gap) ( )與虛擬規模差額(virtual-scale-inefficiency-gap) ( )。本研究將此方法延伸至在VRS條件下，被評量的決策者有極端低的產出項或是有極端高的投入項時，分析最低效率(worst-practice frontier, WPF-VGM-VRS), 與超低效率(Hypo-efficiency, Hypo-VGM-VRS), 。

Many data envelopment analysis (DEA) literatures introduced the methods to measure variable-return-to-scale (VRS) super-efficiency of a DMU (decision-making unit) against to its peers. The linear programming of input-oriented radial-based models (Super-BBC-I) will have infeasible solutions if the DMU has any output is lower than the other DMUs. Similar problems would appear in the case of output-oriented models (Super-BBC-O) and the DMU has any input is higher than the other DMUs. Radial-based models have drawbacks as it limits either one orientation and the DMU may not project on the frontier at certain circumstances. Non-radial-based DEA models are measuring the slack of inputs and outputs at same time and the DMU projects on the frontier. (Liu & Huang, 2015) introduced non-radial-based models, virtual-gap-measurement (VGM) to solve DEA problems. (Liu & Laio, 2015) employed VGM to measure the technical super-efficiency under the condition of constant-return-to scale (Super-VGM-CRS), . In the current paper, we established a procedure to measure the super-efficiency under the condition of VRS (Super-VGM-VRS), . The main approaches are extended to two other measurements: worst-practice-frontier analysis (WPF-VGM-VRS), and hypo efficiency analysis (Hypo-VGM-VRS), . For the cases the DMU under measurement has any input is lower and any output is higher than the other DMUs, the problems could be solved successfully. The three models, super-, WPR- and Hypo-VGM-VRSs would have the optimal values of virtual-gap-of-scale ( ). It would be proportioned to inputs and outputs to have accurate efficiency score.