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. 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
Topic：ANOTHER VIEW OF EFFICIENCY
IMPROVEMENT IN DATA
Source：Journal of the Chinese Institute of Industrial
Engineers, Vol. 26, No. 2 (2009)
This study modifies the original DEA model by
decomposing the normalizing equation in order
to for it to be associated with different dual
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
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.
Basic DEA model-(model 1)
Dual model-(model 2)
and all slacks are of zero in the DMUj DEA run
Decomposition of the normalizing equation-(model 3)
Decompose the normalizing equation(2)
Dual model-(model 4)
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
3、Thus, it is not necessary to reduce all of the inputs in the same
proportion as in the traditional model.
Solve model 4
Specify the values of for
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.
If is relatively inefficient, the reference
coordinates of on the frontier are
According to constraints (11) and (12)
So for the inefficient to achieve Pareto
efficiency it must
Decrease input r
increase output i
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
For the outputs, as shown in equations (14) and (16)
is the unit revenue of output i for the
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.
Set the reduced targets for each input to eliminate the input wastage
The reduced targets of input r is
The improvement targets are the same.
1、Using LINGO 6.0.
Set of weights and the values of and
1、Using CCR model.
Improvement DMUE and DMUF
Based on equations (13) and (15), the improvement input targets for
and are presented in Table 3.
Total cost of 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
：input X1 to input X2 is greater than
Provide a better targets to eliminate input inefficiency
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.
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.
A dual of BM model
with m inputs and in
which input m is an
is shown as follow.
Because the exogenous variables are not possible to vary them at
the discretion of management.
3、Then the dual variable is eliminated in the objective
Therefore, the proposed method can deal with the exogenous variables
as the BM model.
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.
Q：何謂柏拉圖效率( Pareto efficient )？