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
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List of Report
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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
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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.
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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.
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DEA methodology
Basic DEA model-(model 1)
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DEA methodology
Dual model-(model 2)
efficient
if
and all slacks are of zero in the DMUj DEA run
inefficient if
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DEA methodology
Decomposition of the normalizing equation-(model 3)
Decompose the normalizing equation(2)
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DEA methodology
Dual model-(model 4)
∵
efficient
if
inefficient if
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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.
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DEA methodology
Solve model 4
Specify the values of
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for
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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.
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Alternative targets of factors
If
is relatively inefficient, the reference
coordinates of
on the frontier are
,
According to constraints (11) and (12)
We have
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Alternative targets of factors
So for the inefficient
efficiency it must
to achieve Pareto
Decrease input r
increase output i
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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
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Alternative targets of factors
For the outputs, as shown in equations (14) and (16)
is the unit revenue of output i for the
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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.
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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
=
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， The improvement targets are the same.
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Illustration
For example
6
2
1
1、Using LINGO 6.0.
2、 =
.
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Illustration
Set of weights and the values of
and
1、Using CCR model.
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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.
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Reduction costs for inefficient DMUs
Total cost of DMUs
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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
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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.
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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.
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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.
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is eliminated in the objective
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Treatment of the exogenous inputs
BM model
Therefore, the proposed method can deal with the exogenous variables
as the BM model.
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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.
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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。
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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。
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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。
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~Thank you for your listening~
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