管
第
卷
理
第 期
學
報
民國九十一年 月
1 - 21
以模糊積分建構層級分析法之群體決策
整合模式
An AHP Group Decision Fusion Model by Fuzzy Integral
耿伯文 Victor B. Kreng
國立成功大學資訊管理所副教授
Associate Professor, Graduate School of Information Management
National Cheng-Kung University
吳昭儀 Chao-Yi Wu
南台科技大學資訊管理系講師
Lecturer, Department of Information Management
Southern Taiwan University of Technology
摘 要：「決策」是企業的重要活動之一，多屬性決策與群體決策是廣泛被使用的決策方法，文獻中已
有相當多的研究提出不同的決策模式。多屬性決策的優點是可以讓決策者從問題的多個維度來考慮，
並考量各決策因素的重要性之下，完整地評估所有方案的優劣。群體決策則是由多位專家來同時進行
決策過程，藉由彼此不同的考量，激盪出最佳的決策結果。本研究提出一以層級分析法與模糊積分為
基礎的決策整合模式。各決策者先利用層級分析法針對問題建構其階層架構，再以成對比較方式表達
個別意見。待彙齊眾人意見後，利用模糊積分進行非線性的意見整合，得到各方案的最後評比值，再
據以挑選最佳方案。文中並將此模式應用在一供應鏈策略的選擇問題上，以驗證模式的可行性。在實
證範例中可證明，本模式不需經由冗長的會議討論，可以在較短的時間達成群體決策。而且非線性的
整合函數也比數學平均法提供了較佳的整合效果，尤其是決策者有極端偏好之情況下。
關鍵詞：決策融合、模糊積分、層級分析法、群體決策
ABSTRACT： Multi-Criteria Decision Making (MCDM) is a widely used method in decision science. Group
Decision Making (GDM) helps to fuse the judgments of individual decision makers (DMs) through MCDM
to reach better decision. This study presents a decision fusion model based on AHP and fuzzy integral to
possess both advantages of MCDM and GDM. Each DM uses hierarchical AHP structure to separately obtain
the evaluations of alternatives. In addition, the evaluations of DMs can be integrated by fuzzy integral
through nonlinear combination. Supply chain management strategy is used to illustrate the proposed model,
which demonstrates the feasibility of this model. From the demonstrative example, the group decision process
demands shorter time and avoids tedious group meetings. The nonlinear fusion function performs better than
average method especially for extreme preference.
Keywords: Decision fusion, Fuzzy integral, AHP, Group decision-making
以模糊積分建構層級分析法之群體決策整合模式
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1. Introduction
Decision-making is one of the most crucial activities within an organization. Multi-Criteria
Decision Making (MCDM) and Group Decision Making (GDM) are the widely used approaches in
decision science, which have been proposed and applied by many different models, such as Analytic
Hierarchy Process (AHP) and Delphi technique. The main advantage of MCDM is to make Decision
Makers (DMs) be capable of evaluating the alternatives from a set of various criteria, prioritizing them,
and, then, identifying the best. With GDM, decision quality can be significantly improved by group
discussion and knowledge sharing.
Within the scope of GDM, researchers usually use consensus method and mathematical average
method to integrate opinions from various DMs. Consensus method attempts to get a consensus of
decision through the intensive discussions among DMs. However, such method still has some drawbacks,
which include difficult meeting arrangement for DMs and substantial time spent during discussion.
Delphi is one of the methods to resolve the first drawback of consensus method, but it takes even more
time to accomplish (Hwang and Lin, 1987). Mathematic average method uses the mathematic average
equations, which include arithmetic average, geometric average, and weighted average to integrate
various opinions of DMs. Although the above approaches can integrate different sources of DMs within
short period of time, they oversimplify the issue of decision fusion.
In a GDM problem, the DMs, who provide evidence (opinions) by a MCDM method, can be
treated as the information sources. And the fuzzy integral could be used as a tool to fuse the evidence.
The fuzzy integral, proposed by Sugeo (1977), provides a nonlinear function to integrate evidence from
different information sources. In literature, the fuzzy integral has been used in many applications with
different objectives (Chen and Chiou, 1999; Pham and Yan, 1996; Tahani and Keller, 1990) where the
fuzzy integral is recognized as a fuzzy expectation, a maximal grade of agreement between two opposite
tendencies, a maximal grade of agreement between the objective evidence and the expectation, a
maximal degree of belief obtained from the fusion of several objective evidences where the importance
of multiple attributes are subject to fuzzy measure (Pham and Yan, 1996), and a maximal degree of
agreement reached by evaluating the objectiveness and importance of various information sources (Chen
and Chiou, 1999).
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AHP is one of the most popular MCDM methods. Since AHP method was proposed (Saaty, 1980),
there have been many discussions and applications in literature, which came to positive conclusions
mostly. The hierarchical structure of AHP is a useful mechanism to identify the scope of decision
problem, in which pairwise comparisons enable DMs to make precise decision between two alternatives
respect to the upper level factors. Therefore, this paper takes AHP as the MCDM method for further
discussion in GDM.
Accordingly, this study proposes a decision fusion model based on fuzzy integral to synthesize the
AHP weights from DMs, where the conceptual structure is shown in figure 1. And this decision fusion
model can be applied to other MCDM methods, in which the weights about alternatives are in [0,1].
Under the circumstances that the DMs are rational and objective, the decision fusion can be achieved by
fuzzy integral.
DM 1
DM 2
. . .
DM m
AHP
method
AHP
method
. . .
AHP
method
Judgment
of DM 1
Judgment
of DM 2
. . .
Judgment
of DM m
Fuzzy Integral
Fusion of
judgments
Figure 1 Conceptual structure of the decision fusion model
以模糊積分建構層級分析法之群體決策整合模式
4
2. Uncertainty in decision-making
In the process of decision-making, DMs usually have to face the issue of uncertainty. Klir and
Floger (1988) divided the uncertainties into two parts, which are vagueness and ambiguity. Vagueness is
associated with the difficulty of making sharp or precise distinctions of boundaries. On the other hand,
ambiguity is associated with one-to-many relations, in which it is hard to choose from two or more
alternatives. Within the domain of MCDM, when DMs conduct AHP pairwise comparison to evaluate
alternatives and reach the consensus, they normally have to deal with ambiguity and vagueness.
With the AHP method, DMs pair-wisely evaluate alternatives with 1-9 scale. Although there has
been some criticism concerning the use of a ratio versus interval scale, there is a large body of literature
in psychology, which readily accepts the use of a ratio scale in measuring the relative intensity of stimuli.
In administrative decision problems, especially strategy planning problems, DMs usually could not have
precise numerical data. By interval scale, DMs need not precisely describe the priorities among
alternatives and can make the judgment according to their expertise. This provides a solution for dealing
with vagueness in MCDM
Therefore, this paper focuses on the issue of ambiguity in GDM. Currently, the concept of fuzzy
measures provides remedy to deal with the ambiguity issues through mathematical framework.
Accordingly, the authors propose an innovative approach to integrate fuzzy measure with decision
fusion to overcome the obstacle of ambiguity.
3. Decision fusion by fuzzy integral
A fuzzy measure indicates a set of values to each alternative subset of the universal set, which
signify the degree of evidence or belief that a particular element belongs in the subset. The concept and
computational procedure of the fuzzy measure is briefly described in Appendix A.1.
Using the notation of fuzzy measure, Sugeno(1977) defined the concept of the fuzzy integral as a
non-linear function over measurable sets. The concept of the fuzzy integral is described in Appendix A.2,
and the calculation procedure of the fuzzy integral can be described as the following four steps:
管理學報，第
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Step 1: Use equation (A.2) to obtain a unique   (-1, ), and 0.
Step 2: Rearrange h(x) in as decreasing order.
Step 3: Use equation (A.5) and , which is determined in step 1, to calculate g(Ai), i=2,,n,
based on the new order obtained in step 2.
Step 4: Use equation (A.4) to calculate the fuzzy integral value.
i
The explications of h(xi) and g are the critical issues while implementing fuzzy integral. Tahani
and Keller (1990) applied fuzzy integral in automatic target recognition whose objective is to integrate
i
the results from different information sources to improve the rate of recognition. They explicate g as the
degree of importance of each information source and h(xi) as the objective evidence. Pham and Yan
(1996) applied fuzzy integral to enhance handwritten numerical recognition, which integrates
i
information from various classifiers. Their explications of h(xi) and g are similar to those by Tahani and
i
Keller, however, the g is obtained from data training. In addition, Chen and Chiou (1999) made an
application in financial credit rating. They modeled objective evidence as h(xi) and the degree of
i
importance from information sources as g . According to the available literature, h(xi) is represented by
i
the objective evidence and g is represented by the weights of information sources in applying fuzzy
integral to information fusion, which is proposed to extend to the process of decision fusion by the
author.
In this study, the fuzzy integral is utilized to integrate the decisions of DMs during decision fusion,
i
where h(xi) is interpreted as the maximal degree of agreement among the DMs’ judgments and g as
i
the associated degree of decision information provided by various DM. The details about h(xi) and g
will be discussed as following.
3.1 Explications of h(xi) for decision fusion
With respect to decision fusion, each DM can be regarded as an individual information source that
uses AHP model to obtain the preferences, wij, about all alternatives. Since the boundary of wij, 0 wij 1,
is compatible to the constraint of h(xi), this study, therefore, considers wij as h(xi) to reflect the
expertise/judgment of DM.
i
3.2 Explications of g for decision fusion
i
In information fusion, the weights of information sources, g , are difficult to be evaluated since
there is no objective standards existed. In addition, the judgments from all of the DMs are normally
considered to be equally contributed. Accordingly, it is necessary to develop an appropriate definition
以模糊積分建構層級分析法之群體決策整合模式
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for the weights of information sources.
Reviewing the process of decision fusion under the condition that DMs are all rational and
objective, another factor, other than the expertise of DMs, to be considered is the degree of decision
i
information of each DM. Such degree of decision information, represented as g , can be used as a
modified factor in decision fusion. The properties required by degree of decision information are
addressed as follows:
(1) When the degree of preference among alternatives by a DM is not obvious, which means any
alternative can serve as the final decision for this DM without any serious objection, the judgment
of such DM has been considered with less contribution.
(2) On the other hand, while there is a significant gap of preference among alternatives, which means
that this DM deeply believes that one alternative is superior to the others, the opinion of such DM
has been valued with more contribution to the final decision.
Accordingly, this study proposes a gap index, which possesses the above properties for the degree
of decision information, to reflect different contribution during the process of decision fusion and
explain that in next section.
i
3.3 The index of g
Shannon entropy, H, is a very popular concept in information theory, which is defined as follows
(Klir and Floger, 1988),
n
H(p1 , p 2 ,..., p n )    p i  log2 p i ,
(1)
i 1
where pi is a probability distribution on a finite set.
Shannon entropy represents the degree of uncertainty existing in one set of data. In other words, it
can be interpreted as the maximum amount of information contained in data. Since the boundary of H is
i
[0, log2n], which is not compatible to the boundary of g , the normalized Shannon entropy (Klir and
Floger, 1988),
Ĥ , is, therefore, introduced to compensate such issue.
Ĥ(p1 , p 2 ,..., p n ) 
H ( p1 , p 2,, p n ) , where 0  Ĥ(p1 , p 2 ,..., p n )  1.
log2 n
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In addition to Shannon entropy, Onicescu (1966) proposed the concept of information energy, E,
which is defined as follows,
n
2
E(p1 , p 2 ,  , p n )   p i , where 0  pi  1.
(2)
i 1
Information energy possesses complementary characteristics to Shannon entropy, in which Bhatia (1997)
extends it to a general form as follows:
n

E(p1 , p2 , , p n )   [ pi /(  1)]
i 1
, > 0 and   1,
(3)
where  is the order of information energy.
Based on the equation (3), the boundary of information energy is [0,1] while  is equal to 2, which
i
is identical to the boundary of g (it is easy to show that information energy reaches minimum while
evaluations are all equal and maximum while one of evaluations is 1 and all the others are 0). However,
in the case that  is equal to 3, the information energy will fall in [0, 0.5]. And when  is larger, the
upper bound of information energy will become smaller (the lower bound is still equal to 0). Therefore,
when  is greater than or equal to three, the information energy will become too small to be an index of
g , which will distort the function of g in decision fusion. Accordingly, the order  has been set to be
i
i
two in the following discussions.
While analyzing the relation among
Ĥ , E, and the degree of decision information, four different
scenarios are investigated, which include (1) preference among alternatives is equal; (2) evaluation of
one alternative is significantly higher than the others; (3) evaluation of one alternative is significantly
lower with the others are equal; (4) evaluation of one alternative is significantly lower with another one
is much higher.
Suppose there are four alternatives in a decision problem, the values of
Ĥ and E are calculated in
these scenarios respectively from the decision problem of table 1. The observations of this case study are
addressed as follows:
(1) In Scenario 1,
Ĥ has the maximum value and E has the minimum.
(2) In Scenario 2, when the preference difference among alternatives is obvious,
Ĥ becomes smaller
以模糊積分建構層級分析法之群體決策整合模式
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and E becomes larger. In the extreme case, when the evaluation of alternative A closes to 1 and
others close to 0, which means DM almost 100% insist in selecting alternative A,
Ĥ reaches the
minimum and E reaches the maximum.
(3) In Scenario 3,
Ĥ becomes large and E becomes small.
(4) In Scenario 4, when the evaluation difference among alternatives is limited,
Ĥ becomes smaller
and E becomes larger.
By comparing the above findings with the properties required by the degree of decision
information, it is obvious that information energy can serve as an appropriate index for this study. Since,
i
the boundary of E fits in the constraint of g , the information energy is, therefore, considered as the index
i
of g in the decision fusion model of this study. According to equation (2), the degree of decision
information for DM j can be described as follows:
n
n
n
2
2
E j   p ij   w ij   h( x ij ) 2 , j = 1, 2, , m.
i 1
i 1
(4)
i 1
Table 1
Evaluations, Shannon entropy, and information energy of different scenarios
Scenarios
Evaluations of alternatives by
a
H
Ĥ b
E
c
AHP model
1. Preference among alternatives is equal
d
0.25
0.25
0.25
2.000
1.000 0.250
0.9
0.03
0.03
0.04
0.626
0.313 0.813
3. Evaluation of one alternative is significantly 0.05
0.31
0.31
0.32
1.790
0.895 0.297
0.65
0.15
0.15
1.441
0.721 0.470
2. Evaluation of one alternative is significantly
0.25
higher than the others
lower with the others are equal
4. Evaluation of one alternative is significantly 0.05
lower with another one is much higher
a. H: Shannon entropy
b. Ĥ : Normalized Shannon entropy
c. E : Information energy
d. The values of H, Ĥ , E are the same even though evaluations of alternatives are in different
order.
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4. A decision fusion model based on AHP and fuzzy
integral
The process of decision fusion model based on AHP and fuzzy integral can be summarized as
follows:
(1) Construct a hierarchical AHP structure for the decision problem.
(2) Find individual preference, h(xij), of each DM according to the approach of
AHP.
(i) Collect the judgments of DMs by the questionnaire of pairwise comparison.
(ii) Obtain the individual alternative preference of each DM from the eigenvalue. Suppose there
are n DMs and m alternatives, then the preference for the alternative i by DM j is represented
as h(xij)=wij, where i=1,2,…,m, j=1,2,…,n.
(3) Calculate information energy of each DM.
(i) Information energy for DM j, Ej, is calculated by equation (4).
j
(ii) Using Ej as g to represent the degree of information provided by each DM.
(4) Calculate g(Aj)
j
(i) Within each alternative i, rearrange g in the decreasing order of h(xij).
(ii) Use equation (A.2) to find .
(iii) Calculate g(Aj) by equation (A.5).
(5) Fuse all decisions from DMs by fuzzy integral
By equation (A.4), the various preferences from DMs can be integrated into a single preference of
alternatives.
(6) Make decision.
Choose the alternative with the highest value as the best decision.
5. An illustrative example
In this section, the authors will use the following example to illustrate the feasibility and flexibility
of the proposed decision fusion model and, then, discuss the results of such example under different
scenarios.
以模糊積分建構層級分析法之群體決策整合模式
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5.1 Scope of the example
The problem of identifying a Supply Chain Management (SCM) strategy is used here to illustrate
this model. To integrate and perform both logistics and manufacturing activities effectively is the main
objective of SCM. According to Pagh and Cooper (1998), postponement and speculation strategies offer
opportunities to achieve a timely and cost-effective delivery by rearranging the conventional production
and logistics structures. In addition, they identify four strategies, which are full speculation strategy,
manufacturing postponement strategy, logistics postponement strategy, and full postponement strategy,
and designed Profile Analysis to help selecting the most appropriate SCM strategy.
In this study, the framework of SCM strategy selection is based on the study of Pagh and Cooper.
Four domain experts (DMs) are selected to choose the most appropriate strategy for a designated
corporation. The details of decision process about this example are described as follows.
Step 1: Analyze the decision problem and construct the hierarchical AHP structure, which is shown in
figure 2.
Best strategy of SCM
Market
Product
Manufacturing
Cooperation
因素
Demand
uncertainty
Logistic
capability
Product Life-cycle
type
stage
Full Speculation
Strategy
Manufacturing
Postponement
Strategy
Production Modulization
technology
Logistics
Postponement
Strategy
Information
sharing
Power
Full
Postponement
Strategy
Figure 2 The hierarchical structure for decision of SCM strategy
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Table 2
The evaluations of alternatives for all DMs in the decision of Supply Chain Management strategy
Full Speculation
Manufacturing
Logistics
Full
Postponement
Postponement
Postponement
Strategy
Strategy
Strategy
Strategy
E
DM 1
0.511
0.261
0.181
0.046
0.3641
DM 2
0.603
0.187
0.163
0.047
0.4274
DM 3
0.344
0.269
0.239
0.147
0.2694
DM 4
0.477
0.229
0.185
0.109
0.3261
Step 2: Establish the questionnaire in the form of pairwise comparison. Gather all questionnaires and,
then, separately calculate the preference according to AHP, which are listed in table 2.
j
Step 3: Calculate the information energy, Ej, of each DM, which is used as g in the proposed model. The
results are listed in the last column of table 2.
Step 4: According to equation (A.2), the  can be obtained as –0.6255. Then, rearrange g in
j
descending order of h(xij) and calculate g(Aj) from equation (A.5), which are listed in the fourth
and fifth columns of table 3 respectively.
Step 5: Use equation (A.4) of fuzzy integral to integrate the judgments of DMs and obtain a set of final
evaluations of alternatives, which are listed in the sixth column of table 3.
Step 6: Select the alternative with the highest value (0.5110), strategy I, as the best decision.
5.2 Discussions
In order to illustrate the flexibility of the decision fusion model, the authors discuss the following
different scenarios during group decision making. Other than the four scenarios mentioned above,
another case, a total conflict situation, is added here as scenario 5 in table 4.
In scenario 1 to 4, the evaluations of DM 2 are substituted with the data in table 1. In scenario 5,
the four evaluations at the last row of table 1 are reassigned in totally different sequence for four DMs to
represent the total conflict situation. The results of decision fusion for the five scenarios are listed in
table 5 and discussed as follows.
以模糊積分建構層級分析法之群體決策整合模式
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Table 3
The process of the decision fusion in the decision of Supply Chain Management strategy
i
Alt.
DM
1
2
3
4
g
i
(sort on g )
g(Ai)
h(xi)
2
0.4274
0.4274
0.6030
1
0.3641
0.6941
0.5110
4
0.3261
0.8786
0.4770
3
0.2694
1.0000
0.3440
3
0.2694
0.2694
0.2690
1
0.3641
0.5722
0.2610
4
0.3261
0.7816
0.2290
2
0.4274
1.0000
0.1870
3
0.2694
0.2694
0.2390
4
0.3261
0.5406
0.1850
1
0.3641
0.7816
0.1810
2
0.4274
1.0000
0.1630
3
0.2694
0.2694
0.1470
4
0.3261
0.5406
0.1090
2
0.4274
0.8234
0.0470
1
0.3641
1.0000
0.0460
e
Decision
0.5110
0.5110
0.2690
0.2390
0.1470
Table 4
The evaluations of alternatives from DM 2 in the different scenarios
Alt. 1
Alt. 2
Alt. 3
Alt. 4
0.603
0.187
0.163
0.047
a
0.25
0.25
0.25
0.25
b
0.9
0.03
0.03
0.04
c
0.05
0.31
0.31
0.32
d
0.05
0.65
0.15
0.15
e
0.7
0.19
0.08
0.03
Original
Scenario 1
Scenario 2
Scenario 3
Scenario 4
Scenario 5
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(1) In scenario 1, information energy reaches its minimum value of 0.25. Actually, when there are four
alternatives, the value of E will always be above 0.25. With the number of alternatives increases,
the lower bound of E will decrease, which is shown in figure 3. In this case, the evaluations of the
DM can rarely influence the final decision. Such phenomenon meets the requirement that if one
DM doesn’t show clear preference among alternatives, his/her judgment has been treated with less
decision information and will be nearly ignored.
(2) In scenario 2, if other DMs do not show obvious preference about other alternatives, then, the
judgment of DM 2 will dominate the decision fusion. This is because the larger difference among
evaluations of a DM represents more confidence of such DM in distinguishing fitness of
alternatives. Therefore, when other DMs are not capable of making discriminations, the decision of
DM 2 will dominate the decision fusion process.
Another finding reached by this scenario is that the highest evaluation, 0.9, is modified by the
information energy, 0.8130. This is a crucial feature reached by the proposed model, which means
this model is capable of decreasing the influence of extreme preference.
Information Energy
1
0.8
0.6
0.4
0.2
0
1
5
9
13
17
21
25
29
33
37
41
45
49
Amount of Alternatives
Figure 3 Values of Information energy with respect to various amount
of alternatives
以模糊積分建構層級分析法之群體決策整合模式
14
(3) In scenario 3, the process of decision fusion is similar to scenario 1. However, since the information
energy is larger than the minimum, the judgment of DM 2 can influence the final decision more
than others.
(4) In scenario 4, the process of decision fusion is similar to scenario 2. And actually scenario 2 can be
thought as an extreme situation of this scenario. The highest evaluation in this scenario is 0.65,
which is evaluation of alternative 2 by DM 2. However, while checking the results of decision
fusion in table 4, the final choice is alternative 1, not alternative 2. This is because the judgment of
DM 2 is modified by the information energy, 0.47.
(5) In scenario 5, when the judgments of all DMs are totally conflict, the fusion values of alternatives
are all equal. Accordingly, there is no conclusion reached and required further negotiation.
Table 5
Results of decision fusion in various scenarios
Alt. 1
Alt. 2
Alt. 3
Alt. 4
Decision
a
0.477
0.269
0.250
0.250
Alt. 1
b
0.813
0.269
0.239
0.147
Alt. 1
c
0.477
0.297
0.297
0.297
Alt. 1
d
0.477
0.470
0.239
0.150
Alt. 1
e
0.533
0.533
0.533
0.533
?
Scenario 1
Scenario 2
Scenario 3
Scenario 4
Scenario 5
a.
Preference among alternatives is equal.
b. Evaluation of one alternative is significantly higher than the others.
c.
Evaluation of one alternative is significantly lower with the others are equal.
d. Evaluation of one alternative is significantly lower with another one is much higher.
e.
Total conflict situation.
6. Conclusions
How to integrate judgments from numerous DMs is one of the most important issues in GDM. In
this study, a systematic methodology for decision fusion using AHP and fuzzy integral is proposed.
DMs use AHP model to construct a hierarchical structure of decision problem and, then, evaluate
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alternatives by pairwise comparison. After the evaluations are completed, the judgments of DMs are
integrated by fuzzy integral to obtain a set of final fusion among alternatives. The final decision is, then,
achieved based on the one with highest value of decision fusion.
This proposed decision fusion approach possesses the following features, which overcome the
obstacles of current methods in GDM.
(1) Compared with other consensus methods, the model proposed here has the following advantages: (a)
DMs evaluate alternatives separately, which is not necessary to round up all DMs for meetings; (b)
DMs use AHP to conduct evaluation and, then, fuzzy integral to integrate judgments, which
demands shorter process time and avoids tedious discussion.
(2) Compared with average method, the proposed model based on fuzzy integral provides a nonlinear
function to fuse judgments from various DMs instead of using simplified mathematic average
equation.
(3) The decision fusion model is able to modify extreme preference by information energy to reduce
the unnecessary dominance from single DM.
(4) Finally, the decision fusion model proposed here could be easily extended to other MCDM models,
which offer the similar evaluations with AHP.
7. Reference
1.
Bhatia, P.K. (1997), “On measures of information energy,” Information Science, 97(3-4),
pp.233-240.
2.
Chen, L.H. and Chiou, T.W. (1999), “A fuzzy credit-rating approach for commercial loans: A
Taiwan case,” Omega, 27(4), pp.407-419.
3.
Cheng, C.H., Yang, K.L., and Hwang, C.L. (1999), “Evaluating attack helicopters by AHP based
on linguistic variable weight,” European Journal of Operational Research, 116(2), pp.423-435.
4.
Hwang, C.L., and Lin, M.J. (1987), Group decision-making under multiple criteria, New York:
Springer-Verlag.
5.
Klir, G.J., and Folger, T.A. (1988), Fuzzy Sets, Uncertainty, and Information, Englewood Cliffs:
Prentice-Hall.
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6.
Klir, G.J., and Yuan, B. (1995), Fuzzy Sets and Fuzzy Logic – Theory and
7.
Onicescu, O. (1966), “Energie Informationnelle,” C. R. Academics Science Paris Ser. A, 263,
pp.841-842.
8.
Pagh, J.D., and Cooper, M.C. (1998), “Supply chain postponement and speculation strategies:
How to choose the right strategy,” Journal of Business Logistics, 19(2), pp.13-33.
9.
Pham, T.D., and Yan, H. (1996), “Information fusion by fuzzy integral,” Proceeding 1996
Australian New Zealand Conference on Intelligent Information Systems, pp.18-20.
10. Satty, T. L. (1980), The Analytic Hierarchy Process: Planning, Priority Setting, Resource
Allocation, New York: McGraw-Hill.
11. Sugeo, M. (1977), “Fuzzy measures and fuzzy integrals,” In Gupta, M.M., Saridis, G.N., and
Gaines, B.R. (ed.) Fuzzy Automata and Decision Processes, New York: North-Holland.
12. Tahani, H., and Keller, J.M. (1990), “Information fusion in computer vision using the fuzzy
integral,” IEEE Transactions on Systems, Man, and Cybernetics, 20(3), pp.733-741.
Appendix A
A.1 A brief introduction of the fuzzy measure
Definition 1: Given a universal set X and a nonempty family  of subsets of X, a
fuzzy measure on (X, ) is a function
g :   [0,1],
that satisfies the following requirements: (Klir and Yuan, 1995)
(1) g()=0 and g(X)=1 (boundary requirements);
(2) for all A, B  , if AB, then g(A)g(B) (Monotonicity);
(3) for any increasing sequence A1A2… in ¸if
i 
i 1
 , then
i 1

lim g ( Ai )  g ( Ai )

 Ai
(continuity from below)
(4) for any decreasing sequence A1A2… in ¸if

 Ai
i 1
 , then
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
lim g ( A i )  g ( A i ) (continuity from above)
i 
i 1
Based on the above characteristics of fuzzy measure, the union from two
disjoint subsets cannot be directly obtained from the component measures. In light
of this, Sugeno (1977) introduced the so-called “g-fuzzy measures” to augment
the following property: for all A, B  X and AB=,
g(AB) = g(A) + g(B) + g(A)g(B), for some >-1.
By such property, the following equation can be deduced.
n
+1 =  (1  g i ) .
(A.1)
(A.2)
i 1
If all the g are known, the  can, then, be calculated by equation (A.2).
i
A.2 A brief introduction of the fuzzy integral
Definition 2: Let (X,) be a measurable space and h: X[0,1] be a measurable function. The fuzzy integral over AX of the function h
with respect to a fuzzy measure g is defined by
h( x ), g( A  E))]
A h( x )  g()  sup[min( min
XE
E X
 sup [min( , g( A  F ))] ,
(A.3)
[ 0,1]
where F={x: h(x)  }.
The calculation of the fuzzy integral, when X is a finite set, can be obtained in
the following (Sugeo, 1977).
Let X={x1,x2, … , xn} be a finite set and h: X  [0,1] be a function. Suppose
h(x1) h(x2) … h(xn), a fuzzy integral, e, with respect to a fuzzy measure g over
以模糊積分建構層級分析法之群體決策整合模式
18
X is addressed as follows:
n
e = max [min( h( x i ), g( Ai ))] , where Ai={x1,x2, … , xi}.
(A.4)
i1
The values of g(Ai) can be recursively determined as
g(A1) = g({x1})=g1,
g(Ai) = gi+ g(Ai-1)+gig(Ai-1) for 1 < i  n.
(A.5)