An Enhanced Analytical Hierarchical Process for Group Decision
Victor B. Krenga, * and Chao-Yi Wub
a. Graduate School of Information Management, National Cheng-Kung University, Tainan, Taiwan,
ROC 70101
b. Department of Information Management, Southern Taiwan University of Technology, Tainan Hsin,
Taiwan, ROC 710
Abstract
The Analytical Hierarchical Process (AHP) is a widely used multi-criteria decision making
approach and has been successfully applied in different areas. Nevertheless, AHP still has
limitations. Based on fuzzy set theory and fuzzy integral, an extension model of AHP is
proposed to resolve rank reverse problem, decision uncertainty, and group decision making.
Furthermore, for dealing with decision uncertainty, an innovative concept of assurance level is
introduced. Supply chain management strategy is used to illustrate the proposed model, which
demonstrates the feasibility of this model as well.
Keywords: Analytical Hierarchical Process; Group decision making; Fuzzy sets;
Fuzzy integral
1. Introduction
Analytic Hierarchy Process (AHP) method is one of the most popular Multi-Criteria
Decision Making (MCDM) approaches. Since proposed by Saaty (1980), there have been
many successful applications in different areas (Vargas, 1990). However, there are many
discussions about the limitations in literature (Bryson and Mobolurin, 1994; Ramanathan and
Ganesh, 1994; Tavana et al., 1996; Zahir, 1999), especially for the rank reversal problem
(Zahir, 1999) and application in group decision making (GDM) (Aczel and Saaty, 1983;
Ramanathan and Ganesh, 1994; Van Den Honert and Lootsma, 1996; Van Den Honert, 1998;
Forman and Peniwati, 1998;Zahir, 1999; Xu, 2000). Another issue in decision-making is
uncertainty, which includes vagueness and ambiguity (Klir and Floger, 1988). Vagueness is
associated with the difficulty of making sharp or precise distinctions of boundaries. On the
other hand, ambiguity concerns one-to-many relations, in which it is hard to differentiate from
two or more alternatives.
Many papers extended Saaty’s AHP to fuzzy AHP to propose useful approaches to
solve the rank reversal problem (Xu and Zhai, 1992; Mon et al., 1994; Mohanty and Singh,
1994; Lee et al., 2001; Weck et al., 1997). However, the decision uncertainty and the GDM
circumstance have not been consider.
Within the scope of GDM, there are two kinds of approaches to aggregate individual
preferences into the group consensus. One attempts to reach a consensus through intensive
discussions by spending a great deal of time among DMs (Bard and Sousk, 1990; Liberatore
et al., 1992; Madu and Kuei, 1995; Tavana et al., 1996), where Group Decision Support
System (GDSS) is a powerful tool via information technology. The other one, called the
decision aggregation model, is to directly synthesize individual judgments into one common
preference. Geometric mean approach (GMM) and weighted arithmetic mean approach
(WAMM) are the two most popular methods among available literatures (Aczel and Saaty,
1983; Ramanathan and Ganesh, 1994; Van Den Honert, 1998; Van Den Honert and Lootsma,
1996; Forman and Peniwati, 1998; Xu, 2000). However, these models seem to oversimplify
the issue of decisions aggregation and overlook the decision uncertainty.
The fuzzy integral (Sugeo, 1977) provides a nonlinear function to integrate evidences
from different information sources and has been successfully applied to fulfill various
objectives (Chen and Chiou, 1999; Pham and Yan, 1996; Tahani and Keller, 1990). In a GDM
problem, each DM can be treated as single information source to provide his/her judgment or
preference; and the fuzzy integral, can, then, be used to synthesize all the evidences.
Accordingly, this study proposes an extension model of AHP based on fuzzy set theory
and fuzzy integral to overcome rank reversal, decision uncertainty, and group decision
aggregation.
1
2. Fuzzy set theory and assurance level in AHP
TFN from the fuzzy set theory is used to derive the preferences of DMs for solving rank
reversal problem and improving the ambiguity. The intervals among membership functions of
the TFNs in this study are almost equal, the rank reversal problem can, then, be solved. In
addition, the ambiguity is improved by that the preferences of DMs are presented in TFN, not
a real number.
Besides, the assurance level is introduced in this study to enhance the AHP model. While
filling out the questionnaires, DMs are not always confident to all the pairwise comparisons,
which causes vagueness. Therefore, this study adds one more question to each pairwise
comparison to indicate the assurance level with linguistic status of high, medium, and low.
Such assurance level can, then, be used to reform the vagueness. The steps to apply TFN and
assurance level to Saaty’s AHP approach are described as follows:
(1) DMs make the pairwise comparisons according to the hierarchical structure of the
problem. As suggested by Saaty, DMs use the seventeen scales, 1/9, 1/8, …, 1, 2, …, 8, 9,
to demonstrate their preferences. In addition to the conventional AHP, DMs need to
indicate the assurance level to each pairwise comparison with linguistic status of high,
medium, and low.
(2) The pairwise comparisons by seventeen scales are transferred to TFNs. Let a~ijk be the
fuzzy pairwise comparison of criterion i over criterion j from DM k. According to the
fuzzy number converting method proposed by Chen and Hwang (1992), the membership
functions for the TFNs used in this study are listed as follows:
~
1 9 : (0, 0, 0.05),
~
1 3 : (0.35, 0.4, 0.45),
~
5 : (0.7, 0.75, 0.8),
~
1 8 :(0, 0.05, 0.1),
~
1 2 : (0.4, 0.45, 0.5),
~
6 : (0.775, 0.825, 0.875),
~
1 7 :(0.05, 0.1, 0.15),
~
1 : (0.45, 0.5, 0.55),
~
7 : (0.85, 0.9, 0.95),
~
1 6 : (0.125, 0.175, 0.225),
~
2 : (0.5, 0.55, 0.6),
~
8 : (0.9, 0.95, 1),
~
1 5 : (0.2, 0.25, 0.3)
~
3 : (0.55, 0.6, 0.65),
~
9 : (0.95, 1, 1),
~
1 4 : (0.275, 0.325, 0.375),
~
4 : (0.625, 0.675, 0.725).
~
(3) Develop the pairwise comparison matrices. Let A ck be the pairwise comparison matrix
from DM k based on criteria/object c, then
2
~
 1
~
a
~
~
A ck = ( a cijk ) =  c 21k
 
~
 a cn1k
where a~cijk = 1
a~c12k
~
1

a~cn 2 k
~
 1
~
 a c1nk   ~

 a~c 2 nk   1 a
=  c12k

   
~   ~

1   1
 a c1nk
a~c12k
~
1

~
1
a c 2 nk
 a~c1nk 

 a~c 2 nk 
,

 
~ 

1 

~
acijk
.
(4) In order to deal with the uncertainty of assurance, the fuzzy ranking method proposed by
Baldwin and Guilds (Chen and Hwang, 1992) is adopted to measure the fuzzy ranking
~
value of a TFN. Each a~cijk in the pairwise comparison matrix will be compared with 1
to derive its fuzzy ranking value. The calculation of fuzzy ranking value is as follows:
~
Supposed that a~cijk  {( x m ,  acijk ( x m ))} and 1  {(x1 , 1 ( x1 ))}, then, the fuzzy ranking
value of a~cijk , which is denoted by  O
cijk
O
cijk
 sup {min[  a ( x m ), 1 ( x1 ),  P ( x m , x1 )]} ,
cijk
m1
xm , x1
where  P
m1
The  O
cijk
, is defined as
( x m ) 0.5  ( x1 ) 0.5 ,

  x m  x1 ,
( x ) 2  ( x ) 2 ,
1
 m
(1)
assurance level is low,
assurance level is medium,
assurance level is high.
will be, further, transformed to fit in the interval (0,9), named  O
cijk
, to
preserve he same scale with Saaty’s AHP, and be recognized as the representative value
~ 
for the relative pairwise comparison. The matrix Ack will be defined as ( ̂ O
cijk
) and has
the form,
ˆ O
 1

~   ˆ Oc 21k
Ack =
 

 ˆ Ocn1k
c12k
1

ˆ O
cn 2 k
(5) The values of ̂ O
cijk
 ˆ O

c1nk

 ˆ O
c 2 nk 
, where ̂ O  1
̂ O .
cijk

 
cjik


1 

can be treated as the preference of criterion i over criterion j from
~ 
DM k. Therefore, the preferences for criterion i in pairwise comparison matrix Ack ,
3
denoted as wcik, can be obtained by eigenvector, where wik is grouped into matrix, Wck,
according to each DM.
Wck = the preference matrix from DM k based on criteria/objective c,
= (wcik) = (wc1k, wc2k, …, wcnk).
3. Aggregating individual pairwise comparison matrices by fuzzy integral
In this study, the fuzzy integral, which is one of the fuzzy measure approaches, is
employed to fuse DMs’ preferences about each pairwise comparison matrix, not the final
evaluations to alternatives. Fuzzy measure assigns the value to each alternative subset from
the universal set, which signify the degree of evidence or belief that a particular element
belongs to the subset. Using the notation of fuzzy measure, Sugeno (1977) defined the
concept of fuzzy integral as a non-linear function over measurable sets. The necessary
equations are listed as follows.
g(AB) = g(A) + g(B) + g(A)g(B), for some >-1.
+1 =
(2)
n
 (1  g i ) .
(3)
i 1
n
e = max [min( h( x i ), g( Ai ))] , where Ai={x1,x2, … , xi}.
(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.
(5)
Both the explications of h(xi) and gi are the critical issues while implementing fuzzy
integral. According to the available literature, h(xi) is represented by the objective evidence;
and gi is represented by the weights of information sources while applying fuzzy integral to
information fusion (Tahani and Keller, 1990; Pham and Yan, 1996; Chen and Chiou, 1999).In
this study, the fuzzy integral is utilized to integrate the judgments of DMs to each pairwise
comparison, where h(xi) is interpreted as the DM’s judgment to a pairwise comparison and gi
as the assurance level of each DM as well. The details about h(xi) and gi will be discussed as
follows.
3.1 Explications of h(xi) for decision fusion
With respect to decision fusion, each DM can be regarded as an individual information
source. The preference matrix of DM i, Wck, which is obtained by the method proposed in
section 2, are treated as the evidence from information source k; moreover, the boundary of
wik, 0 wik 1, is identical to the constraint of h(xi). This study, therefore, considers wcik as
h(xck)i to reflect the expertise/judgment of DM k,
where h(xck)i = wcik = the preference to criterion i from DM k.
3.2 Explications of gi for decision fusion
4
In decision aggregation issue, gi is introduced as the assurance level of DM i to one
pairwise comparison matrix, instead of using the degree of importance from single
information source. The target of decision aggregation in this study is the pairwise
comparison matrix, not the final evaluations to alternatives; thus, each pairwise comparison
matrix has an assurance level. Accordingly, each DM can signify his different assurance levels
to judgments about various criteria. This let DMs’ assurances to different criteria are
completely considered.
~
The assurance levels of DM k to pairwise comparison matrix c ( A ck ), g ck , are computed
by the following steps.
~
Step 1: Indicate the assurance level to each pairwise comparison in A ck by each DM with
linguistic status of high, medium, and low.
Step 2: Transfer the linguistic status into the assurance level, ccijk, with the scale from 0 to 1.
In this study, 1 is used to represent the high assurance level, 0.5 as the medium
assurance level, and 0 as the low assurance level,
where ccijk: the assurance level for the pairwise comparison of criterion i over criterion
~
j in A ck .
Step 3: Group these assurance levels of DM k into the assurance matrices, Cck,
where Cck = the assurance matrix from DM k based on one criterion/objective,
 1
c
= (ccijk)=  c 21k
 

 ccn1k
cc12k
1

ccn 2 k
 cc1nk 
 cc 2 nk 
.
1
 


1 
Step 4: Compute the maximum eigenvalue,  ack , of each assurance matrix,
where  ack : the maximum eigenvalues of the assurance matrix Cck.
Since the interval of  ack changes with the size of matrix, the  ack has, therefore, to

be transferred to a relative value,  ack , by equation (6).

 ack = [  ack -min(  ack )]/[max(  ack )-min(  ack )].
(6)
The maximum of  ack , max(  ack ), occurs when all the elements from assurance matrix
are 1, except for those in the diagonal. And the  ack reaches its minimum, min(  ack ),
5
while all the elements of assurance matrix are 0, except for those in the diagonal.

Step 5: Normalize those  ack s from all DMs based on criteria/objective c to demonstrate the

relative strengths of assurances among DMs. Then, the normalized  ack , g ck , is
~
regarded as the assurance level to pairwise comparison matrix A ck .
g ck = ack /

a
cj
.
(7)
all DMs
4. An extension model of AHP based on fuzzy set theory and fuzzy integral
The steps of the AHP-based decision aggregation model based on fuzzy set theory and
fuzzy integral are summarized as follows:
(1) Construct a hierarchical AHP structure for the decision problem.
(2) Collect the judgments from DMs by the AHP type questionnaires of pairwise comparisons.
In addition, DMs have to fill out the assurance level for each comparison with high,
medium, and low. In order to limit the amount of questions been answered, DMs only
need to reply to those with the high or low assurance.
(3) Derive preference matrixes and assurance levels of all DMs for all pairwise comparison
matrices.
~
The following steps are for the A ck , and will repeat for all pairwise comparison matrices.
(3.1) Find individual preference matrices, h(xck)i, of DMs by the steps in section 2.
(3.2) Calculate the assurance levels of DMs, g ck , by the steps listed in section 3.2.
(3.3) Calculate g(Aci) about criteria i.
(i) Use equation (3) to find c.
(ii) Rearrange g ck in the decreasing order of h(xck)i for criteria i.
(iii)Calculate g(Aci) by equation (5).
(3.4) Fuse the evaluations from DMs to one criterion/objective by fuzzy integral. Based
on equation (4), the various preferences from DMs can be integrated into a single
preference of each criterion.
(4) Synthesize the weights of all criteria through the AHP hierarchical structure to obtain the
final weights of all alternatives.
(5) Choose the alternative with the highest value as the decision.
5. An illustrative example
The problem to identify a Supply Chain Management (SCM) strategy is used, here, to
illustrate this model. To integrate and perform both logistics and manufacturing activities
6
effectively is the main objective of SCM. According to Pagh and Cooper (1998),
postponement and speculation strategies both offer opportunities to achieve a timely and
cost-effective delivery by rearranging the conventional production and logistics structures. In
addition, four strategies have been further developed accordingly, which are Full Speculation
Strategy (FSS), Manufacturing Postponement Ptrategy (MPS), Logistics Postponement
Strategy (LPS), and Full Postponement Strategy (FPS).
In this study, the framework of SCM strategy selection is also based on the study of Pagh
and Cooper (1998), which is shown in figure 1. Since there are numerous computations for
such decision, only those data, which are related to the criteria in the first level of hierarchical
AHP structure, are listed for illustration.
Best strategy of SCM
Market
Manufacturing
因素
Product
Cooperation
Demand
Logistic Product Life-cycle Production Modulization Information Power
technology
uncertainty capability type
stage
sharing
Full
Speculation
Strategy
Figure 1
Manufacturing
Postponement
Strategy
Logistics
Postponement
Strategy
Full
Postponement
Strategy
The hierarchical structure for decision of SCM strategy.
The pairwise comparisons from DM 1, including those relate to the criteria in the first level of
hierarchical AHP structure, are listed in table 2.
~ 
~
Accord to the data in table 2, the matrices, Ac1 and Ac1 , are as follows:
 ~
1
 ~
1 5
~
Ac1 =  ~
1 / 6
 ~
1 / 3
~ 
~ ~
5 6
3
~ ~
~ 
1 1 / 4 1 / 3
,
~ ~
~ 
4 1
4 
~ 
~ ~
3 1/ 4 1 
6.457 6.436 4.881
 1
2.377
1
2.679 3.242
~ 
Ac1 = 
.
1.551 5.309
1
5.633


1 
3.242 4.881 2.770
The assurances matrix of DM 1 to the first level is
7
1 0 .5
1

1
1 0 .5
Cc1= 
 0 .5 0 .5 1

1
1
 1
1

1
.
1

1
Thus, the DM 1s’ assurance level is derived as g c1 = 0.845.
In addition, the decision aggregation process is shown in table 3. Then, FSS with the highest
value (0.4050) is selected as the decision.
Table 2
Examples of transformation from pairwise comparisons to the relative fuzzy ranking values.
Target of comparison
Assurance
Fuzzy Ranking
level
Value (  Oijk )
High
Medium
6.457
6.436
High
4.881
Medium
2.679
~
13
High
3.242
~
4
High
5.633
Product vs. Market
~
15
High
2.377
Manufacturing vs. Market
~
16
Medium
1.551
Cooperation vs. Market
~
13
High
3.242
Medium
5.309
High
4.881
High
2.770
Market vs. Product
Market vs. Manufacturing
Market vs. Cooperation
Product vs. Manufacturing
Product vs. Cooperation
Manufacturing vs. Cooperation
Manufacturing vs. Product
Cooperation vs. Product
Cooperation vs. Manufacturing
TFN
( a~ijk )
~
5
~
6
~
3
~
14
~
4
~
3
~
14
8
Table 3
The process of the decision fusion.
Criterion
Marketing
Production
Manufacturing
Cooperation
DM
gi
g(Ai)
h(xi)
min[g(Ai),h(xi)]
e
1
0.845
0.845
0.335
0.335
0.335
4
0.582
0.938
0.314
0.314
2
0.683
0.984
0.306
0.306
3
0.757
1.000
0.203
0.203
3
0.757
0.757
0.264
0.264
4
0.582
0.901
0.249
0.249
2
0.683
0.972
0.207
0.207
1
0.845
1.000
0.189
0.189
2
0.683
0.683
0.300
0.300
3
0.757
0.926
0.271
0.271
1
0.845
0.993
0.245
0.245
4
0.582
1.000
0.208
0.208
3
0.757
0.757
0.263
0.263
1
0.845
0.966
0.232
0.232
4
0.582
0.989
0.229
0.229
2
0.683
1.000
0.187
0.187
0.264
0.300
0.263
6. Conclusion
This study is proposed to overcome rank reversal problem, decision uncertainty, and
group decision of AHP. A systematic methodology by using fuzzy set theory and fuzzy
integral is demonstrated. In addition, the assurance level on each pairwise comparison is
introduced to improve the quality of decision. The proposed model possesses the following
features.
(1) The pairwise comparisons have been converted to TFNs, and, then, coupled with
assurance levels to further transform to become crisp fuzzy ranking values. Such
transformation not only improves the rank reversal problem, but also deals with ambiguity
and vagueness during decision-making process.
(2) The aggregation of the proposed model occurs on individual pairwise comparison, not
final preferences of alternatives. This makes DMs’ assurances, which are likely different
for various criteria, be entirely deliberated in decision fusion.
(3) While comparing with the average methods, such as GMM and WAMM, the proposed
aggregation model based on fuzzy integral provides a nonlinear function to fuse
9
preferences from various DMs instead of using simplified mathematic average equation.
Accordingly, such model can offer decision with better quality.
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