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1、Review of CCR modelAssume that there are H decision making units (DMU)Each DMU produces M outputs by using N inputs.The (relative) efficiency of j-th DMU is Subject to 第1页/共28页第一页，共29页。The objective is to obtain the most favorable input weights ( vn, n = 1, 2, N) and output weights (um, m = 1, 2, M)
2、, such that the efficiency measure for the j-th DMU is maximized. However, there are infinite numbers of solutions since it is a non-linear programming.If ( ) is an optimal solution, then ( ) is still an optimal solution. One can solve this problem by imposing the constraint第2页/共28页第二页，共29页。Subject
3、to 第3页/共28页第三页，共29页。The dual problem of the above linear program is expressed with a real variable (j) and non-negative variables (h, h = 1, 2, H) as follows:Subject to The advantage of the dual form is that it involves few constraints (in general, N + M H + 1). 第4页/共28页第四页，共29页。The above model has
4、to be solved H times, once for each DMU to obtain its efficiency measurement.The optimal value of j is the measure of technical efficiency of the j-th DMU.Although j is free, it can be show that its range is between 0 and 1.Efficient DMUs (with technical efficiency 1) construct the efficient frontie
5、r. The the j-th DMU is inefficient if its technical efficiency is less than 1 (j 0, D 0 . 第7页/共28页第七页，共29页。The benchmark of A is C , A = 0, B = 0, C 0, D = 0. A is not the efficient point since it can obtain the same amount of output by reducing the quantity AC of input x2/y. AC is called the non-ra
6、dial input slack 第8页/共28页第八页，共29页。Discover Non-radial Slacks The following two-phase LP problem could be used to discover the possible non-radial slacks, consisting of both input excesses and output shortfall (Cooper et al., 2000). Phase I: Find the optimal objective value from the dual form of the
7、CCR model.第9页/共28页第九页，共29页。Phase II: Using the knowledge of solves the following LP problem:Subject to第10页/共28页第十页，共29页。It can be shown that the following LP problem serves to unify the Phrase I and Phrase II procedures (Cooper et al., 2000): where is a small non-Archimedean quantity, usually being
8、106.第11页/共28页第十一页，共29页。4.1 IntroductionThe CCR model assume that the technology is constant returns to scale (CRS).Banker et al. (1984) extended the CCR model to account for variable returns to scale (VRS), called the BCC model. 第12页/共28页第十二页，共29页。CCRBCC第13页/共28页第十三页，共29页。For point P:第14页/共28页第十四页，共
9、29页。In reality, the scale of a DMU may be above or under the optimal scale. Hence, the inefficiency may result from the inappropriate scale as well as the mixture of inputs and outputs. Technical efficiency (TE): The efficiency value obtained from the CCR model.Pure technical efficiency (PTE): The e
10、fficiency value obtained from the BCC modelScale efficiency (SE): PE PTEThe CRS technical efficiency measure is decomposed into “pure technical efficiency” and “scale efficiency;” TE = PTE SE 第15页/共28页第十五页，共29页。4.1 IntroductionMathematically, the BCC model is easily modified from the CCR model by ad
11、ding the convexity constraint.Additive Model Treat nonzero SLACKS Characteristics of “translation invariance” But no *Slack-based measure of efficiency(SBM) a scalar both * and nonzero SLACKS第16页/共28页第十六页，共29页。BCC model VS CCR model.OBTCXYPQ.S.AR0 1 2345676543210D4.2 BCC models第17页/共28页第十七页，共29页。4.2
12、 BCC models (envelopment form)Subject to BCCo CCR中当e 1规模报酬(bo chou)递减阶段 当e=1规模报酬(bo chou)不变阶段第18页/共28页第十八页，共29页。DualmultiplierformU0相当于截距，前沿不需要(xyo)通过原点U00规模报酬递增第19页/共28页第十九页，共29页。Definition4.1BCC-Efficiency如果最优解满足B=1并且不存在冗余，则DMUO被称为BCC有效，否则为BCC非效率(xiol)对于BCC非效率(xiol)的DMU，如果*满足E0=j|j*0，我们定义其参考集合为E0D
13、efinition 4.2 参考参考(cnko)集合集合第20页/共28页第二十页，共29页。4.3加法加法(jif)模型模型CCR不需要(xyo)这个条件第21页/共28页第二十一页，共29页。4.3加法加法(jif)模型模型.OBTCXYPQ.S.AR0 1 2345676543210DS+S-.Additive效率(xio l)前沿与BCC相同只是评价方法不同第22页/共28页第二十二页，共29页。定义定义4.5如果改变最初投入如果改变最初投入(tur)或产出值所得到的或产出值所得到的新问题与原问题具有同样的最优解，则称其新问题与原问题具有同样的最优解，则称其为无差异变换。为无差异变换。
14、.OOBDXXYTRW.OOBDXXYTRW.第23页/共28页第二十三页，共29页。4.4基于冗余的效率基于冗余的效率(xiol)测度测度加法模型受到测量单位的影响，需要(xyo)寻找一种无量纲，单位不变性的测度变量基于冗余的效率测度具有如下两个性质单位不变性随着投入和产出冗余单调递减第24页/共28页第二十四页，共29页。SBM定义定义(dngy)具备(jbi)前面两个性质，而且当slack=0 =1 mixed inefficiency 第25页/共28页第二十五页，共29页。summary生产可能性集合的形状投入还是(hishi)产出导向？无差异转换投入产出指标的个数试验各种模型第26
15、页/共28页第二十六页，共29页。summary生产可能性集合的形状投入还是产出导向(doxin)？无差异转换投入产出指标的个数试验各种模型第27页/共28页第二十七页，共29页。感谢您的观看(gunkn)！第28页/共28页第二十八页，共29页。内容(nirng)总结Review of CCR model。第1页/共28页。U00规模(gum)报酬递增。如果最优解满足B=1并且不存在冗余，则DMUO被称为BCC有效，否则为BCC非效率。E0=j |j*0，我们定义其参考集合为E0。定义4.5 如果改变最初投入或产出值所得到的新问题与原问题具有同样的最优解，则称其为无差异变换。加法模型受到测量单位的影响，需要寻找一种无量纲，单位不变性的测度变量。感谢您的观看第二十九页，共29页。
