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專題討論 指導老師：黃振勝 學生：鄭富元 學號：M98U0202 中華民國98年12月16日 2009/12/16 Source of the Paper Topic：ANOTHER VIEW OF EFFICIENCY IMPROVEMENT IN DATA ENVELOPMENT ANALYSIS 中譯：資料包絡分析模式之效率改善方法 Source：Journal of the Chinese Institute of Industrial Engineers, Vol. 26, No. 2 (2009) Authors：Tien-Hui Chen Chiao-Pin Bao Shiow-Yun Chang 2009/12/16 2/32 List of Report 1. 2. 3. 4. 5. 6. 7. 8. 9. 2009/12/16 Abstract Introduction DEA Methodology ( Include decomposition) Alternative targets of factors An illustration Treatment of the exogenous inputs Conclusions My review of the Paper Q&A 3/32 Abstract This study modifies the original DEA model by decomposing the normalizing equation in order to for it to be associated with different dual variables. As a consequence, to improve efficiency the adjustment proportion of each input or output factor can be different. In essence, the proposed approach can not only set targets of factors for inefficient decision making units to eliminate inefficiency, but can also deal with the exogenous variables in a DEA context. 2009/12/16 4/32 Introduction 1、Data envelopment analysis (DEA) is a systematic programming approach for measuring relative efficiencies within a group of decision making units (DMUs), which utilize several inputs to produce a set of outputs. 2、If the efficiency score of a DMU is equal to one, then the DMU is classified as efficient; otherwise it is inefficient. 3、This study decomposes the normalizing equation in the original DEA model inorder for it to be associated with different dual variables to obtain improved inputs and outputs targets for inefficient DMUs. The proposed approach can not only set targets of factors for inefficient DMUs to achieve Pareto efficiency, but can also deal with the exogenous variables in a DEA context. 2009/12/16 5/32 DEA methodology Basic DEA model-(model 1) 2009/12/16 6/32 DEA methodology Dual model-(model 2) efficient if and all slacks are of zero in the DMUj DEA run inefficient if 2009/12/16 7/32 DEA methodology Decomposition of the normalizing equation-(model 3) Decompose the normalizing equation(2) 2009/12/16 8/32 DEA methodology Dual model-(model 4) ∵ efficient if inefficient if 2009/12/16 9/32 DEA methodology Dual model-(model 4) 1、However, implies that DMUj is inefficient because evidence (from the efficiencies of efficient DMUs) shows that DMUj could reduce its input r in the proportion without worsening any output, r = 1,2,…,m . 2、Therefore, an improvement possibility for the inefficient DMUj to eliminate input inefficiency is to decrease input r with the amount of . 3、Thus, it is not necessary to reduce all of the inputs in the same proportion as in the traditional model. 2009/12/16 10/32 DEA methodology Solve model 4 Specify the values of 2009/12/16 for 11/32 DEA methodology To continue 1、The contribution of input r to the current efficiency of . 2、The proposed procedure can ensure that the best score of by model (4) is the same as that of the traditional DEA model. 2009/12/16 12/32 Alternative targets of factors If is relatively inefficient, the reference coordinates of on the frontier are , According to constraints (11) and (12) We have 2009/12/16 13/32 Alternative targets of factors So for the inefficient efficiency it must to achieve Pareto Decrease input r increase output i 2009/12/16 14/32 Alternative targets of factors In equation (13), the larger the value of , the smaller the adjustment proportion of input r for the inefficient . The targets to eliminate inefficiency for using model (2) are 2009/12/16 15/32 Alternative targets of factors For the outputs, as shown in equations (14) and (16) is the unit revenue of output i for the 2009/12/16 16/32 Compare the total reduction in cost Using equations (13) and (15) are stated as and , respectively. is the unit cost of input r for the 1、Compare the total reduction in cost for performance improvement. 2、For an inefficient the total reduction costs in eliminating input inefficiency. 2009/12/16 17/32 Compare equations (17) and (18) Set the reduced targets for each input to eliminate the input wastage The reduced targets of input r is ， Otherwise Notably = 2009/12/16 ， The improvement targets are the same. 18/32 Illustration For example 6 2 1 1、Using LINGO 6.0. 2、 = . 2009/12/16 19/32 Illustration Set of weights and the values of and 1、Using CCR model. 2009/12/16 20/32 Illustration Improvement DMUE and DMUF =0.600*5=3 =0.385*5=1.925 Based on equations (13) and (15), the improvement input targets for and are presented in Table 3. 2009/12/16 21/32 Reduction costs for inefficient DMUs Total cost of DMUs 2009/12/16 22/32 Reduction costs for inefficient DMUs If , we find If the unit cost ratio fo input X1 to input X2 of DMUE is greater than 0.5, i.e. Then the proposed approach offers a larger reduction in cost with regard to eliminating input inefficiency; otherwise, the traditional method provides a better choice. ：input X1 to input X2 is greater than Provide a better targets to eliminate input inefficiency 2009/12/16 23/32 Reduction costs for inefficient DMUs In this illustration, there are only two inputs under the same respective output target so that the decision maker can determine which targets of inputs should be applied based on the cost ratio. For cases with more than two inputs, decision makers can apply Equations (17) and (18) to obtain and compare the total reduction costs for efficiency improvement in order to arrive at a better decision. 2009/12/16 24/32 Treatment of the exogenous inputs In many realistic situations, a few of the input factors are within the DMU’s control, i.e. they can be varied at the discretion of decision makers; however, some are uncontrollable. BM model A dual of BM model with m inputs and in which input m is an exogenous variable is shown as follow. 2009/12/16 25/32 Treatment of the exogenous inputs Because the exogenous variables are not possible to vary them at the discretion of management. 1、 2、 3、Then the dual variable function. 2009/12/16 is eliminated in the objective 26/32 Treatment of the exogenous inputs BM model Therefore, the proposed method can deal with the exogenous variables as the BM model. 2009/12/16 27/32 Conclusions 1、The characteristic of a DEA model is that it allows DMUs to select the best weights in calculating their efficiencies. 2、This study modifies the CCR model by decomposing the normalizing equation to provide another choice for efficiency improvement. 3、 If the efficiency score by the CCR model is one, then the DMUs being evaluated are already Pareto efficiency. 4、This paper provides another choice of efficiency improvement for inefficient business units. 2009/12/16 28/32 My review of the Paper 很多分析經營績效的模式並非為最 佳模式，且閱讀此篇文章後，讓後 學更能瞭解，若能多利用坊間或學 術上各式各樣的數理分析模式且加 以應用，藉此便可透過許多模式進 行多方面分析，以得到最適結果。 2009/12/16 29/32 Q&A Q：何謂柏拉圖效率( Pareto efficient )？ A：經濟學定義：是指在不損及他人的情 況下，使其他人福利提高的一種狀態 。而完全競爭市場的均衡就是柏拉圖 效率。 2009/12/16 30/32 Q&A Q：此篇文章的目的最主要在探討何種 問題？ A：此篇文章最主要在探討，為了改善 效率，則藉由給予(或調整)每一個 投入或產出的不同權重，來進行績 效衡量與改善。 2009/12/16 31/32 ~Thank you for your listening~ 2009/12/16 32/32