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南 台 科 技 大 學
專題討論報告
指導老師：黃振勝
姓
名：黃逸帆
學 號：M98U0207
中華民國98年12月23日
報告內容
2016/7/13
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資料來源
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Topic：
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Finding the most efficient DMUs in DEA: An improved integrated
model
Author ：
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1-Gholam R. Amin*, 2-M. Toloo
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1 Postgraduate Engineering Center, Islamic Azad University of
South Tehran Branch, Tehran, Iran
2 Department of Mathematics, Islamic Azad University of Central
Tehran Branch, Tehran, Iran
The Source ：
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2016/7/13
Computers & Industrial Engineering 52 (2007) 71–77
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Abstract
Introduction
Integrated DEA model
An improved integrating DEA model
Conclusion
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Abstract
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This paper presents an improved integrated DEA model
in order to detect the most efficient DMUs.
The proposed integrated DEA model does not use the
trial and error method in the objective function.
Also, it is able to find the most efficient DMUs without
solving the model n times (one linear programming (LP)
for each DMU) and therefore allows the user to get faster
results.
It is shown that the improved integrated DEA model is
always feasible and capable to rank the most efficient
one.
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Introduction
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Data envelopment analysis (DEA) introduced by
Charnes, Cooper, and Rhodes (1978) and followed by
Banker, Charnes, and Cooper (1984)
New applications with more variables and more
complicated models are being introduced, (Emrouznejad,
Tavares, & Parker, 2007).’
Yang and Kuo (2003) introduced a hierarchical DEA
methodology for the facilities layout design problem.
DEA allows each DMU to specify its own weights so as
to obtain its maximum efficiency score, which may result
in a relatively high number of efficient DMUs and avoid
DEA to appear as a robust approach in determining the
most efficient unit (Doyle & Green, 1994).
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Integrated DEA model
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In the DEA literature the non-Archimedean e has
introduced for removing some difficulties. There are two
main approaches, the two phase method, Cooper et al.
(2006)
Amin and Toloo (2004) introduced a polynomial time
algorithm, Epsilon algorithm, to find a suitable nonArchimedean which assures to remove the weak efficient
DMUs from the list of efficient DMUs. The Epsilon
algorithm is applied for Table 1 and concludes the
assurance value e0 = 2.64852*10-5.
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Integrated DEA model
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Integrated DEA model
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Integrated DEA model
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An optimal value of model is ε=0.0714
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Integrated DEA model
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The strong efficient DMUs are DMU3, DMU5,
DMU2, DMU11, DMU6, DMU1, and DMU4. The
first column of Table 2 in Ertay et al.
Therefore the weak efficient units DMU5 and
DMU19 (Current) appeared as efficient one in
the first column of Table 6.
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An improved integrating DEA
model
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The Ertay et al. (2006) minimax DEA procedure
contains a constraint regarding to specific DMUs.
Therefore it needs to solve n LPs, one LP for
each DMU. We propose an improved version of
the presented model without the need to choose
the parameter k. The model proposes as:
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An improved integrating DEA
model
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An improved integrating DEA
model
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ε* is the maximum non-Archimedean. Note that in the
absence of the alternative efficient DMUs when the
appropriate parameter is selected, the minimax method
proposed by Ertay et al. (2006) reaches a single relative
efficient DMU.
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Cook, Kress, and Seiford (1996) suggested theuse of
maximum value for e, a non-Archimedean, in order to
improve the discrimination among all the DMUs. The
following model is an extended integrated version of their
model:
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An improved integrating DEA
model
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Obviously model is feasible. Let (μ*, ω*, ε*) be
an optimal solution of model. Define
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An improved integrating DEA
model
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First, we show that |J1|≧1, i. e. in any optimal
solution (μ*, ω*, ε*) of model at least one of the
first n constraints must be tight.
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An improved integrating DEA
model
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Lemma1. |J1|≧1.
Proof. On the contrary, assume that |J1| = 0.
Consider the dual of the model shown below:
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An improved integrating DEA
model
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The complementary slackness conditions imply
that δ* = 0, where (δ *, β*, γ *, η*) is an optimal
solution of the dual. So the constraints of the
dual conclude that β * = 0, γ * = 0 and η * = 0,
which contradicts to the last constraint of model.
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An improved integrating DEA
model
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Lemma 2. 0 < ε* <∞
Proof. Using model , a similar manner applied in
Lemma 1 completes the proof.
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Conclusion
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This paper started with the motivation for determining the most
efficient DMUs in DEA and developed a new integrated DEA model.
The merits of the proposed formulation compared with DEA-based
approaches that have previously been used for finding the most
efficient DMUs can be listed as follows. First, this formulation allows
the computation of the efficiency scores of all DMUs by a single
formulation, i.e. all DMUs are evaluated by a common set of weights.
Second, it identifies the most efficient units by using fewer
formulations and without the need to solve n LPs, one LP for each of
DMUs. Further, the proposed integrated DEA formulation does not
use the trial and error method that has been applied for finding the
most efficient DMUs.
Finally to illustrate the model capability it is applied to a real data set
consisting of the 19 FLDs.
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參考文獻
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Amin, Gholam R., & Toloo, M. (2004). A polynomial-time
algorithm for finding Epsilon in DEA models. Computers
and Operations Research, 31(5), 803–805.
Banker, R. D., Charnes, A., & Cooper, W. W. (1984).
Some models for estimating technical and scale
inefficiency in data envelopment analysis. Management
Science, 30, 1078–1092.
Charnes, A., Cooper, W. W., & Rhodes, E. (1978).
Measuring the efficiency of decision-making units.
European Journal of Operational Research, 2, 429–444.
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參考文獻
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Cook, W. D., Kress, M., & Seiford, L. M. (1996). Data
Envelopment Analysis in the presence of both
quantitative and qualitative factors. Journal of
Operational Research Society, 47, 945–953.
Cooper, W. W., Seiford, L. M., & Tone, K. (2006).
Introduction to Data Envelopment Analysis and its uses
with DEA-Solver software and references, Springer.
Doyle, J., & Green, R. (1994). Efficiency and cross
efficiency in DEA: derivations, meanings and uses.
Journal of the Operational Research Society, 45(5),
567–578.
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參考文獻
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Emrouznejad, A., Tavares, G., & Parker, B. (2007). A
bibliography of data envelopment analysis (1978–2003).
Socio-Economic Planning Sciences, in press.
Ertay, T., Ruan, D., & Tuzkaya, U. R. (2006). Integrating
data envelopment analysis and analytic hierarchy for the
facility layout design in manufacturing systems.
Information Sciences, 176, 237–262.
Li, X. B., & Reeves, G. R. (1999). A multiple criteria
approach to data envelopment analysis. European
Journal of Operational Research, 115, 507–517.
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參考文獻
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Mehrabian, S., Jahanshahloo, G. R., Alirezaee, M. R., &
Amin, Gholam R. (2000). An assurance interval for the
non-Archimedean epsilon in DEA models. Operations
Research, 48(2), 344–347.
Yang, T., & Kuo, C. A. (2003). A hierarchical AHP/DEA
methodology for the facilities layout design problem.
European Journal of Operational Research, 147, 128–
136.
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心得及討論
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Impression：
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一般的資料包絡法都只是評測出最有效率的幾個評測
單位為何，卻沒有針對這些被評測為有效率的評測單
位再做其中最有效率的分析，因此本篇針對此一問題
提出了所謂改良式的整合資料包絡法，此一方法最主
要是套入了一個non-Archimedean的概念，藉由此一方
法的參數ε達到最大，藉此找到傳統資料包絡法所評測
出的幾個最有效的評測單位中最有效率的評測單位為
何。因而能夠找出最有效率的DMU中最有效率的DMU
為何，總而言之此一改良式整合資料包絡法不但能強
化傳統資料包絡法的缺點，更能有效找出最有效率的
評測單位。
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心得及討論
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Question ：
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剛剛在報告中所提及的non-Archimedean的作用是什麼
呢？
Answer ：
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2016/7/13
non-Archimedean就是所謂的非阿基米德數，是由
Cook, Kress, and Seiford 三人在1996年所提出的一個
資料包絡分析法的新概念，並利用線性規劃模式使
non-Archimedean的參數ε達到最大，藉以求出資料包
絡分析法中最有效率的評測單位。
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