南台科技大學 專題討論報告
指導老師：黃振勝
學
生：陳智凱
學
號：Ｍ９８Ｕ０１１０
中華民國98年12月16日
Source：
Journal of the Chinese Institute of Industrial Engineers, Vol. 22, No. 3, pp.243-251 (2005)
結合層級分析法與資料包絡法分
析法之績效評估模式
A PERFORMANCE EVALUATION MODEL
BASED ON AHP AND DEA
劉志明、許漢昇、王信智、李海光
國立清華大學工業工程與工程管理學系
Abstract
• This research uses the concept of the Analytical
Hierarchy Process (AHP) method and fuzzy set theory to
modify the CCR model of the Data Envelopment
Analysis (DEA) method to develop a more effective
performance evaluation method which can be used to
evaluate the performance of a smaller number of
business units, which is usually prohibited in the
traditional DEA method.
• In order to prove the applicability of the proposed
method, a case study for the selection of suppliers for
the key component of a LCD manufacturing company is
given.
• The result shows that it can be used to choose among a
small number of suppliers in a very efficient way and
also can provide improvement suggestions for each
supplier.
Program
1. Introduction
2. Literature Review
3. A performance evaluation model based
on AHP and DEA
4. An implementation case
5. Conclusion
1.1 Background
• In the present competitive business environment,
an enterprise must have an evaluation
mechanism to measure the performance of
every business unit in order to direct the entire
organization or system toward a common goal.
In other words, it needs a performance
evaluation system. The goal of performance
evaluation can be classified into two major parts:
efficiency and effectiveness.
• A high-performance system must consider both
efficiency and effectiveness, i.e., pursue the
highest effectiveness by using the most efficient
method.
1.2 Purpose of research
1. Investigate the fitness problem and the
limitations of DEA, and try to propose a method
to increase its application domain.
2. Apply DEA to the evaluation and selection of
suppliers in the business environment.
1.3 Structure of sections
The structure of this paper is as follows. In the
second section of this paper, some related
literature on DEA will be given.
In the third section, a performance evaluation
model, which is based on AHP and DEA, is
developed.
In the fourth section, the proposed model is
applied to the supplier evaluation and selection
problem of a LCD key component manufacturing
company. Finally, some conclusions are drawn.
2. Literature Review
2.1 DEA models
In research , data envelopment analysis of the
CCR model with Analytical Hierarchy Process
Performance Assessment Model
2.2 Investigation of the weighting
limitations of DEA
1. The absolute range The values of the weighting
variables in the DEA model are confined by the
upper and lower bounds, however, those
bounds are obtained through historical data or
experts’ opinion as shown in formula (7).
2.2 Investigation of the weighting
limitations of DEA
2. The assurance region(AR) In the application of
DEA, one situation may occur such that the ratio
of two factors has some kind of relationship, and
under this condition it is required to confine the
corresponding ratio for the weights of these two
factors to a range as shown in formula[8].
The determination of the upper and lower bounds
for an AR model is usually based on the
subjective opinion of experts.
2.2 Investigation of the weighting
limitations of DEA
3. The polyhedral cone-ratio(polyhedral cone-ratio)
In the DEA model, it requires the weighting
variables to satisfy the restriction of a multipleface cone, as shown in formula (9).
3. A performance evaluation
model based on AHP and DEA
3.1. Establish the range for each
evaluation criterion
The establishment of the range for each evaluation
criterion has three phases, as shown in Figure 2.
Table 1. The relative importance rating scale
of AHP
Table 2. The fuzzy scale table
1. Establish the pair-wised
comparison matrix with range values.
First, Table 1 is referred to establish the rating of relative
importance for each pair of factors. Then, Table 2 is
referred to get the scale of the fuzzy value between
different factors. Finally, by adding and subtracting the
fuzzy value from the relative importance rating value,
one can obtain the range of the relative importance
rating value, which can be expressed as
，
After having obtained all the
and
, the pair-wise
comparison matrix in range values can be given as
shown in Table 3.The relationship between
and its
bounds can be expressed in formula (10).
2. Find the range of weight for each
measure.
• Calculate the range for each column vector:
• After having calculated all the row vector
,the range of the weight of each factor
can be obtained through a normalization
process as shown in formula (12).
3. Consistence test
The consistence test can be done by calculating
the consistence ratio (C.R.) as shown in
formulas (13) and (14).
Phase 3: Integration of weights
from multiple decision-makers.
If there are several people involved in the decisionmaking, each person may have different range
for the weight of a measure.
So it’s necessary to integrate the rating of all the
decision-makers to form a single range of the
weight of a measure.
The integration method adopted here is to average
the weights of all the evaluators. The formulas
for calculating the mean values are given in
formulas (15) and (16).
represents the final evaluation of the lower
bound of the ith index,
represents the lower
bound provided by the kth decision-maker, and
N represents the total number of decisionmakers.
represents the final evaluation of the upper
bound for the ith index, and
represents the
upper bound provided by the kth decision-maker.
3.2. Homogenize data
After having assigned the range for the weight of
each measure, the next step is to homogenize
the rating scores for all the measures into a
common numerical scale. The homogenization
method applied here is to assign the highest
value of a measure for a DMU to be 1, and the
values for other DMUs are rescaled accordingly
as shown in formulas (17) and (18).
Homogenize the scores of output variables (thelarger-the-better index), as shown in
formulas(17):
represents the original score of the kth DMU
of the rth the-small-the-better index and
represents its homogenized value.
Homogenize the scores of input variables (thesmaller-the-better index), as shown in formula
(18):
Where
represents the original score for the ith
measure of the kth DMU and
represents the
its normalized value.
3.3. A performance evaluation
model by combining AHP and
DEA
Through the establishment of the range for the
weight and the normalization of the score, the
modified DEA model is shown in formula (19).
3.3. A performance evaluation
model by combining AHP and
DEA
and
， represent the upper
and lower bounds for the input and output
variables respectively.
In the same way, the super-efficiency model can
be shown in formula (20):
3.4. The improvement model for
DMUs with low efficiency
In order to investigate the improvement model for
DMUs with low efficiency, its dual model needs
to be studied first. Since the objective function of
the model in this research is nonlinear, it must
be first transformed into a linear model.
3.4. The improvement model for
DMUs with low efficiency
We can find the optimal solution by using the
LINGO software, and then use the found values
of those variables to rewrite the objective
function. Assume
denotes the optimal
solution of the-smaller-the-better variables, then
the constant
can then be used to rewrite
the original model into a linear programming
model.
3.4. The improvement model for
DMUs with low efficiency
Because is the optimal solution of the-smaller-the-better
variables, if both the numerator and denominator of the
objective function
are
divided by the constant
, it will not change the
solution of the original model. So in the situation the
denominator of the objective Function
equals to
one.
Suppose
, then the linear programming model
can be shown in formula (21) :
3.4. The improvement model for
DMUs with low efficiency
After having transformed the original model into its
dual model, the relationship cx∗ = y∗b must exist
if x∗ is the optimal solution of the primal problem
And y∗ is the optimal solution of the dual problem
according to the strong dual theorem. Then,
when the optimal solution is found, the values of
the objective function of the primal problem and
the dual problem should be the same.
So the calculating process of the improvement
alternatives can be simplified by using the
performance value (the value of the primal
problem) found for DMUs.
3.4. The improvement model for
DMUs with low efficiency
The improvement alternatives of the DMUs with
low efficiency are given below:
The improvement goal for the an input item can be
expressed as formula (22).
3.4. The improvement model for
DMUs with low efficiency
Where
is the calculated performance
value of that DMU.
The improvement goal for the output item can be
expressed as formula (23).
4. An implementation case
The case study is a LCD backlight manufacturing
company (C company) which needs to select
suppliers for its aluminum part. There are four
related persons to get involved in the decision
process. Five measures with the-higher-thebetter values and four measures with the-lowerthe-better values are chosen.
The rating information for suppliers is given in
Table 4.
4.1 Establish ranges for the
weights of evaluation measures
of the C company’s suppliers
Using the relative importance values for the thelarger-the-better measures of the first decisionmaker as an example, Table 5 gives the pairwise comparison matrix for those measures
expressed in range values.
After some calculation, the ranges of weights can
be obtained in Table 6.
Since the value of C.R. = 0.043 ≤ 0.1, the
consistence test is passed.
Table 5. The pair-wise comparison matrix for
range values of the relative importance of the
-higher- the- better measures of the first
decision-maker
4.1 Establish ranges for the
weights of evaluation measures
of the C company’s suppliers
The same method can be applied to obtain the
range values of the relative weights of measures
for other three decision-makers. By averaging
the range values of the four decision-makers,
the final range values of weights can be
obtained as shown in Table 7 and Table 8.
4.2 Homogenize data
The homogenize data are shown in Table 9.
Table 9. The normalized data for the rating of
suppliers for the C company
4.3 The performance evaluation
model for the case study
Using supplier A as an example, the objective
function can be used to find the maximum
performance value of supplier A, while the
restrict functions will set the maximum
performance value of all the suppliers to be 1,
and also provide a bound for the weight of each
factor.
The mathematical programming model for the
performance evaluation of supplier A is
illustrated in Table 10.
From the output of LINDO, the performance value
of supplier A is only 0.9828, which is below the
established range values of the weighting
factor(Tables 6~8) in section 4.1.
So supplier A belongs to the low-performance
group, and should be given a lower priority in the
selection process.
The same method can be applied to other
suppliers to obtain their performance values as
shown in Table 11.
Table 11. The performance values of
suppliers
The result shows that the performance values of
suppliers C and D are 1, which is within the
range of the weight established by evaluators.
Since these two suppliers have the highest
performance value, they should be the most
favorable candidates of choice.
The performance values of suppliers A and E are
very close to 1, so they can be considered as
the second group to choose from.
The super-efficiency model can be applied to the
two suppliers with performance values equal to
1, then the ranking between them can be
obtained.
Taking supplier C as an example, the superefficiency model can be constructed as shown in
Table 12.
From the output of LINDO, the performance value
of supplier C is 1.053, and supplier D is 1.1277.
So supplier D has a higher possibility to be the
best supplier.
4.4 Improvement models for
each supplier
We can find the optimal solution by using the
LINDO software, and then use the found values
of variables
and
of improvement models
for each supplier. Alternatives for the
improvement of the input and the output items
for each supplier is shown in Tables 13 and 14.
Table 13 Alternatives for the improvement of
the output items for each supplier
Table 14. Alternatives for improvement of input
items for suppliers
5. Conclusion
By combining the methodologies of DEA, AHP and
fuzzy set theory, a modified DEA method is
proposed in this research. It has the following
substantial results:
1.It can be used to evaluate the performance of
each business unit, and also can suggest the
directions for improvement of the lowperformance units.
2. It has a wider application scope, especially
when there are only a small number of
business units under evaluation.
5. Conclusion
3. Through the establishment of the range values
for weights, the evaluation result is more
acceptable than the original DEA method which
may rate several DMUs with the same
performance value of 1.
From the case study results, it has been shown
that the proposed method can be used to many
practical problems like the selection of suppliers
for a component, which usually involves a small
number of candidates.
Q&A
THANKS!
REFERENCES
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心得
讀完此篇DEA理論論文，讓我知道原本的ＤＥＡ權重是沒有
設限的，但利用層級分析法規範出權重範圍，這樣就不會
讓投入及產出項無效率的項變的很有效率；並且也可以針
對每項投入與產出項提出需要改善的項目。而且加入了層
級分析法計算出來的績效值更具客觀性，同時也比原始Ｄ
ＥＡ模式更符合實際情況，而且也可以運用在投入及產出
項少的評選上。