基于样本评价的广义SBM方法及其有效性
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摘要
基于样本评价的广义DEA方法可以通过不同参考集对决策单元的有效性进行评价,但是现有的模型基本都是径向模型,鉴于此提出基于样本生产可能集的广义SBM模型。首先给出决策单元广义SBM有效的定义及其与多目标规划Pareto有效之间的关系,然后改进广义SBM模型,给出决策单元不属于样本生产可能集的判断条件和决策单元效率值的定义,讨论决策单元在样本生产可能集中的投影性质,最后通过算例分析决策单元的广义SBM有效性排序问题。结果表明,广义SBM方法能给出较为完整的有效性排序,且随着移动因子d的增大,可以进一步预测有效决策单元的改进方向。
By using sample units, the generalized DEA method can evaluate the efficiency of decision making units through different reference sets, but the existing models are radial models. In view of this, a generalized SBM model based on the sample production possible set is proposed. First, the definition of generalized SBM efficiency is presented, the relationship between the generalized SBM efficiency and Pareto efficiency of multi-objective programming are explored. Then, the improved generalized SBM is proposed, the judgement condition of decision making unit does not belong to the sample production possible set, and the efficiency score of decision making unit are given. The projection property of decision making unit in the sample production possible set is discussed. Finally, the ranking of generalized SBM efficiency is analyzed by an example. The results show that the generalized SBM method can provide relatively complete efficiency sequencing, and with the increase of moving factor d, the improvement direction of efficient decision making unit can be further predicted.
By using sample units, the generalized DEA method can evaluate the efficiency of decision making units through different reference sets, but the existing models are radial models. In view of this, a generalized SBM model based on the sample production possible set is proposed. First, the definition of generalized SBM efficiency is presented, the relationship between the generalized SBM efficiency and Pareto efficiency of multi-objective programming are explored. Then, the improved generalized SBM is proposed, the judgement condition of decision making unit does not belong to the sample production possible set, and the efficiency score of decision making unit are given. The projection property of decision making unit in the sample production possible set is discussed. Finally, the ranking of generalized SBM efficiency is analyzed by an example. The results show that the generalized SBM method can provide relatively complete efficiency sequencing, and with the increase of moving factor d, the improvement direction of efficient decision making unit can be further predicted.
引文
[1] Charnes A,Cooper W W,Rhodes E.Measuring the efficiency of decision making units[J].European Journal of Operational Research,1978,2(6):429～444.
[2] 马占新.数据包络分析模型与方法[M].北京:科学出版社,2010.
[3] Emrouznejad A,Yang G L.A survey and analysis of the first 40 years of scholarly literature in DEA:1978-2016[J].Socio-Economic Planning Sciences,2018,61(ST):4～8.
[4] Banker R D,Charnes A,Cooper W W.Some models for estimating technical and scale in efficiencies in data envelopment analysis[J].Management Science,1984,30(9):1078～1092.
[5] F?re R,Grosskopf S.A nonparametric cost approach to scale efficiency[J].The Scandinavian Journal of Economics,1985,87(4):594～604.
[6] Seiford L M,Thrall R M.Recent developments in DEA:The mathematical programming approach to frontier analysis[J].Journal of Econometrics,1990,46(1～2):7～38.
[7] Aida K,Cooper W W,Pastor J T.Evaluating water supply services in Japan with RAM:A range- adjusted measure of inefficiency[J].Omega,1998,26(2):207～232.
[8] Tone K.A slacks-based measure of efficiency in data envelopment analysis[J].European Journal of Operational Research,2001,130(3):498～509.
[9] Cooper W W,Pastor J T,Borras F.BAM:A bounded adjusted measure of efficiency for use with bounded additive models[J].Journal of Productivity Analysis,2011,35(2):85～94.
[10] Chen J X,Deng M.A cross-dependence based ranking system for efficient and inefficient units in DEA[J].Expert Systems with Applications,2011,38(8):9648～9655.
[11] Sun J,Wu J,Guo D.Performance ranking of units considering ideal and anti-ideal DMU with common weights[J].Applied Mathematical Modelling,2013,37(9):6301～6310.
[12] Andersen P,Petersen N C.A procedure for ranking efficient units in data envelopment analysis[J].Management Science,1993,39(10):1261～1264.
[13] Wu J,Chu J,Sun J,Zhu Q.DEA cross-efficiency evaluation based on Pareto improvement[J].European Journal of Operational Research,2016,248(2):571～579.
[14] Tracy D L,Chen B.A generalized model for weight restrictions in data envelopment analysis[J].Journal of the Operational Research Society,2005,56(4):390～396.
[15] Li X B,Reeves G R.A multiple criteria approach to data envelopment analysis[J].European Journal of Operational Research,1999,115(3):507～517.
[16] Shen W F,Zhang D Q,Liu W B,Yang G L.Increasing discrimination of DEA evaluation by utilizing distances to anti-efficient frontiers [J].Computers & Operations Research,2016,75:163～173.
[17] 马占新.一种基于样本前沿面的综合评价方法[J].内蒙古大学学报(自然科学版),2002,33(6):606～610.
[18] 马占新.样本数据包络面的研究与应用[J].系统工程理论与实践,2003,23(12):32～37+58.
[19] 马生昀,马占新.基于样本前沿面移动的广义DEA有效性排序[J].系统工程学报,2014,29(4):443～457.
[20] 马占新,吕喜明.带有偏好锥的样本数据包络分析方法研究[J].系统工程与电子技术,2007,29(8):1275～1281.
[21] 马占新,马生昀.基于C2W模型的广义数据包络分析方法研究[J].系统工程与电子技术,2009,31(2):366～372.
[22] 马占新,马生昀.基于C2WY模型的广义数据包络分析方法[J].系统工程学报,2011,26(2):251～261.
[23] 马占新,马生昀.基于样本评价的广义数据包络分析方法[J].数学的实践与认识,2011,41(21):155～171.
[24] Tone K.A slacks-based measure of super-efficiency in data envelopment analysis[J].European Journal of Operational Research,2002,143(1):32～41.
[25] Du J,Liang L,Zhu J.A slacks-based measure of super-efficiency in data envelopmentanalysis:A comment[J].European Journal of Operational Research,2010,204(3):694～697.
[26] Fang H H,Li H S,Hwang,S N,Chung,C C.A slacks-based measure of super-efficiency in data envelopment analysis:An alternative approach[J].Omega,2013,41(4):731～734.
[2] 马占新.数据包络分析模型与方法[M].北京:科学出版社,2010.
[3] Emrouznejad A,Yang G L.A survey and analysis of the first 40 years of scholarly literature in DEA:1978-2016[J].Socio-Economic Planning Sciences,2018,61(ST):4～8.
[4] Banker R D,Charnes A,Cooper W W.Some models for estimating technical and scale in efficiencies in data envelopment analysis[J].Management Science,1984,30(9):1078～1092.
[5] F?re R,Grosskopf S.A nonparametric cost approach to scale efficiency[J].The Scandinavian Journal of Economics,1985,87(4):594～604.
[6] Seiford L M,Thrall R M.Recent developments in DEA:The mathematical programming approach to frontier analysis[J].Journal of Econometrics,1990,46(1～2):7～38.
[7] Aida K,Cooper W W,Pastor J T.Evaluating water supply services in Japan with RAM:A range- adjusted measure of inefficiency[J].Omega,1998,26(2):207～232.
[8] Tone K.A slacks-based measure of efficiency in data envelopment analysis[J].European Journal of Operational Research,2001,130(3):498～509.
[9] Cooper W W,Pastor J T,Borras F.BAM:A bounded adjusted measure of efficiency for use with bounded additive models[J].Journal of Productivity Analysis,2011,35(2):85～94.
[10] Chen J X,Deng M.A cross-dependence based ranking system for efficient and inefficient units in DEA[J].Expert Systems with Applications,2011,38(8):9648～9655.
[11] Sun J,Wu J,Guo D.Performance ranking of units considering ideal and anti-ideal DMU with common weights[J].Applied Mathematical Modelling,2013,37(9):6301～6310.
[12] Andersen P,Petersen N C.A procedure for ranking efficient units in data envelopment analysis[J].Management Science,1993,39(10):1261～1264.
[13] Wu J,Chu J,Sun J,Zhu Q.DEA cross-efficiency evaluation based on Pareto improvement[J].European Journal of Operational Research,2016,248(2):571～579.
[14] Tracy D L,Chen B.A generalized model for weight restrictions in data envelopment analysis[J].Journal of the Operational Research Society,2005,56(4):390～396.
[15] Li X B,Reeves G R.A multiple criteria approach to data envelopment analysis[J].European Journal of Operational Research,1999,115(3):507～517.
[16] Shen W F,Zhang D Q,Liu W B,Yang G L.Increasing discrimination of DEA evaluation by utilizing distances to anti-efficient frontiers [J].Computers & Operations Research,2016,75:163～173.
[17] 马占新.一种基于样本前沿面的综合评价方法[J].内蒙古大学学报(自然科学版),2002,33(6):606～610.
[18] 马占新.样本数据包络面的研究与应用[J].系统工程理论与实践,2003,23(12):32～37+58.
[19] 马生昀,马占新.基于样本前沿面移动的广义DEA有效性排序[J].系统工程学报,2014,29(4):443～457.
[20] 马占新,吕喜明.带有偏好锥的样本数据包络分析方法研究[J].系统工程与电子技术,2007,29(8):1275～1281.
[21] 马占新,马生昀.基于C2W模型的广义数据包络分析方法研究[J].系统工程与电子技术,2009,31(2):366～372.
[22] 马占新,马生昀.基于C2WY模型的广义数据包络分析方法[J].系统工程学报,2011,26(2):251～261.
[23] 马占新,马生昀.基于样本评价的广义数据包络分析方法[J].数学的实践与认识,2011,41(21):155～171.
[24] Tone K.A slacks-based measure of super-efficiency in data envelopment analysis[J].European Journal of Operational Research,2002,143(1):32～41.
[25] Du J,Liang L,Zhu J.A slacks-based measure of super-efficiency in data envelopmentanalysis:A comment[J].European Journal of Operational Research,2010,204(3):694～697.
[26] Fang H H,Li H S,Hwang,S N,Chung,C C.A slacks-based measure of super-efficiency in data envelopment analysis:An alternative approach[J].Omega,2013,41(4):731～734.