Journal of Intelligent Manufacturing, 13, 367±377, 2002
# 2002 Kluwer Academic Publishers. Manufactured in The Netherlands.
A fuzzy AHP approach to the determination of
importance weights of customer requirements in
quality function deployment
C . K . K W O N G and H . B A I
Department of Manufacturing Engineering, The Hong Kong Polytechnic University, Hung Hom,
Kowloon, Hong Kong
E-mail: mfckkong@inet.polyu.edu.hk
Received March and accepted November 2001
Quality function deployment (QFD) is an important tool in product planning that could contribute to
increase in customer satisfaction and shorten product design and development time. During the QFD
process, determination of the importance weights of customer requirements is a crucial and essential
step. The analytic hierarchy process (AHP) has been used in weighting the importance. However,
due to the vagueness and uncertainty existing in the importance attributed to judgement of customer
requirements, the crisp pairwise comparison in the conventional AHP seems to be insuf®cient and
imprecise to capture the degree of importance of customer requirements. In this paper, fuzzy number
is introduced in the pairwise comparison of AHP. An AHP based on fuzzy scales is proposed to
determine the importance weights of customer requirements. The new approach can improve the
imprecise ranking of customer requirements which is based on the conventional AHP. Finally, an
example of bicycle splashguard design is used to illustrate the proposed approach.
Keywords: Fuzzy AHP, quality function deployment, customer requirements, importance weights,
product design
1. Introduction
Quality function deployment (QFD) is a management
tool that provides a visual connective process to help
teams focus on the customer requirements throughout
the total product design and development cycle. QFD
is now being used for de®ning new products, as well as
for diagnosing and improving existing products. The
basic concept of QFD is to translate the desires of
customers into appropriate product designs or engineering characteristics, and subsequently into parts
characteristics, process plans and production requirements. It has been well documented that the use of QFD
can reduce the product development time by 50%, and
start-up and engineering costs by 30% (Hauser and
Clausing, 1988).
As a customer-driven quality management tool, the
main characteristic of QFD is to recognize the ``voice
of customers'', and hence to generate a set of
customer requirements. Determination of the importance weights of customer requirements is a crucial
and essential process of QFD as it could largely affect
the target value setting of engineering characteristics
to be determined in the later stage. Various methods
have been applied in this process. The simplest
method in prioritizing customer requirements is based
on point scoring scale, such as 1±5 and 1±10 (Grif®n
and Hauser, 1993). However, a substantial degree of
human subjective judgement has to be involved in this
method. Lai et al. (1998) developed a group decisionmaking technique to determine the importance
weights of customer requirements, in which the
agreed criteria and individual criteria methods
combine voting and linear programming techniques
to aggregate individual preferences into group
consensus. Conjoint analysis method was attempted
368
to determine the relative importance of customer
requirements (Gustafsson and Gustafsson, 1994). The
methodology employs pairwise comparison of customer requirements to determine their relative
importance. Che et al. (1999) employed arti®cial
neural network to determine the importance weights
of customer requirements. To train the neural network, a competitive assessment of the company and
its competitors must be conducted in order to generate
the training data sets. Considering the vagueness and
imprecision in the importance assessment of customer
requirements, Chan et al. (1999) directly converted
the importance assessment from crisp values to fuzzy
numbers, and then the importance weights of customer
requirements were calculated by using the entropy
method. Prioritizing customer requirements could be
viewed as a complex multi-criteria decision-making
problem. The analytic hierarchy process (AHP), a
multi-criteria decision making technique, was used in
weighing customer requirements (Lu et al., 1994). The
integration of AHP with the determination of trade-off
weights for customer requirements has been proposed
(Akao, 1990; Aswad, 1989). Armacost et al. (1994)
adopted AHP to generate the importance weights
of customer requirements in a case study of
industrialized housing. Based on the customer
requirements and engineering requirements of a
product collected from the QFD planning matrix,
Zakarian and Kusiak (1999) applied AHP in the
determination of the importance measures for individual team members.
In the conventional AHP, the pairwise comparisons
for each level with respect to the goal of customer
satisfaction are conducted using a nine-point scale.
Each pairwise comparison represents an estimate of
the priorities of the compared customer requirements.
The nine-point scale developed by Saaty (1980)
expresses preferences between options as equally,
moderately, strongly, very strongly, or extremely
preferred. These preferences are translated into
pairwise weights of 1, 3, 5, 7, and 9, respectively,
with 2, 4, 6, and 8 as intermediate values. The
pairwise comparison ratios are in crisp real numbers.
However, customer requirements always contain
ambiguity and multiplicity of meaning. The descriptions of customer requirements are usually linguistic
and vague. Furthermore, it is also recognized that
human assessment on qualitative attributes is always
subjective and thus imprecise. Therefore, conventional AHP seems inadequate to capture customer
Kwong and Bai
requirements explicitly and determine the importance
weights of customer requirements accurately.
In order to model this kind of uncertainty in human
preference, fuzzy sets could be incorporated with the
pairwise comparison as an extension of AHP. The
fuzzy AHP approach allows a more accurate
description of the decision-making process. The
earliest work in fuzzy AHP appeared in Van
Laarhoven and Pedrycz (1983), which compared
fuzzy ratios described by triangular membership
functions. Logarithmic least square was used to
derive the local fuzzy priorities. Later, using
geometric mean, Buckley (1985) determined fuzzy
priorities of comparison, whose membership functions were trapezoidal. By modifying the Van
Laarhoven and Pedrycz method, Boender et al.
(1989) presented a more robust approach to the
normalization of the local priorities.
In this paper, a fuzzy AHP approach to the
determination of the importance weights of customer
requirements for QFD is described. Firstly, the
linguistic assessment on customer requirements is
converted into triangular fuzzy numbers. These
triangular fuzzy numbers are used to build the
comparison matrices of AHP based on pairwise
comparison technique. The importance weights of
customer requirements can be calculated by applying
fuzzy AHP. At the end of this paper, an example of a
bicycle splashguard design is described to illustrate
the fuzzy AHP approach to the determination of the
importance weights of customer requirements.
2. Hierarchical structure for the development of
customer requirements
AHP is particularly useful for evaluating complex
multi-attribute alternatives involving subjective criteria. The essential steps in the application of AHP
contains (1) decomposing a general decision problem
in a hierarchical fashion into sub-problems that can be
easily comprehended and evaluated, (2) determining
the priorities of the elements at each level of the
decision hierarchy, and (3) synthesizing the priorities
to determine the overall priorities of the decision
alternatives.
To apply AHP in prioritizing customer requirements, all customer requirements have to be
structured into different hierarchical levels. Af®nity
diagram, tree diagram and cluster analysis can be used
369
Fuzzy AHP approach
Fig. 1. An example of a 3-level hierarchy for customer requirements.
for this purpose. Figure 1 shows an example of a
three-level hierarchy for customer requirements. In
the ®gure, the goal is ``customer satisfaction'', and
there are seven categories in the category level. All
customer requirements (attributes) are listed under
relevant categories, which form the lowest level of the
hierarchy. It is called attribute level. If there are a
large number of customer requirements, a four or
more levels structure should be required.
3. Fuzzy representation of pairwise comparison
The hierarchy of customer requirements need to be
established before performing the pairwise comparison of AHP. After constructing a hierarchy for
customer requirements, the decision maker is asked
to compare the elements at a given level on a pairwise
basis to estimate their relative importance in relation
to the element at the immediate proceeding level. In
the conventional AHP, the pairwise comparison is
made using a ratio scale. A frequently used scale is the
nine-point scale which shows the participants'
judgments or preferences among the options such as
equally, moderately, strongly, very strongly, or
extremely preferred. Even though the discrete scale
of 1±9 has the advantages of simplicity and easiness
for use, it does not take into account the uncertainty
associated with the mapping of one's perception (or
judgment) to a number.
In this research, triangular fuzzy numbers, ~1 to ~9,
are used to represent subjective pairwise comparisons
of customer requirements in order to capture the
vagueness. A fuzzy number is a special fuzzy set
F ˆ f…x; mF …x††; x [ Rg, where x takes its values on
the real line, R:
? 5 x 5 ‡ ? and mF …x† is a
continuous mapping from R to the closed interval
‰0; 1Š. A triangular fuzzy number denoted as
~ ˆ …a; b; c†, where a b c, has the following
M
triangular-type membership function:
8
0
x5a
>
>
<x a
a
xb
mM~ …x† ˆ bc xa
b
xc
>
>
:c b
0
x>c
Alternatively, by de®ning the interval of con®dence
level a, the triangular fuzzy number can be
characterized as:
Va [ ‰0; 1Š
~ a ˆ ‰aa ; ca Š ˆ ‰…b
M
a†a ‡ a;
c
b†a ‡ cŠ
Some main operations for positive fuzzy numbers
described by the interval of con®dence (Kaufmann,
1991) are:
~ a ˆ ‰maL ; maR Š;
VmL ; mR ; nL ; nR [ R ‡ ; M
N~a ˆ ‰naL ; naR Š; a [ ‰0; 1Š
~ N~ ˆ ‰maL ‡ naL ; maR ‡ naR Š
M+
~ N~ ˆ ‰maL naL ; maR naR Š
MY
~ N~ ˆ ‰maL naL ; maR naR Š
M6
~ N~ ˆ ‰maL =naR ; maR =naL Š
M
The triangular fuzzy numbers, ~1 to ~9, are utilized to
improve the conventional nine-point scaling scheme.
In order to take the imprecision of human qualitative
370
2
1
6 a~21
6
6 .
6 ..
~
Aˆ6
6 ..
6 .
6
4 a~…n 1†1
a~n1
a~12
1
..
.
..
.
a~13
a~23
..
.
..
.
a~…n 1†2
a~n2
..
.
a~…n 1†3
a~n3
Kwong and Bai
3
a~1n
a~1…n 1†
a~2n 7
a~2…n 1†
7
.. 7
..
. 7
.
7
.. 7
..
. 7
.
7
1
a~…n 1†n 5
a~n…n 1†
1
where
a~ij ˆ
Fig. 2. The membership functions of triangular fuzzy numbers
~
1; ~
3; ~
5; ~
7; ~
9.
1;
~
1; ~
3; ~
5; ~
7; ~
9 or ~
1
1
;~
3
1
;~
5
1
;~
7
1
;~
9
1
;
Step 3: Solving fuzzy eigenvalues. A fuzzy eigenvalue, ~l, is a fuzzy number solution to
~x ˆ ~l~
A~
x
assessments into consideration, the ®ve triangular
fuzzy numbers are de®ned with the corresponding
membership functions as shown in Fig. 2.
4. Algorithm of fuzzy AHP
Saaty's AHP method is known as an eigenvector
method. It indicates that the eigenvector corresponding to the largest eigenvalue of the pairwise
comparisons matrix provides the relative priorities of
the factors, and preserves ordinal preferences among
the alternatives (Saaty, 1980). This means that if an
alternative is preferred to another, its eigenvector
component is larger than that of the other. A vector of
weights obtained from the pairwise comparisons
matrix re¯ects the relative importance of the various
factors. In the fuzzy AHP triangular fuzzy numbers
are utilized to improve the scaling scheme in the
judgment matrices, and interval arithmetic is used to
solve the fuzzy eigenvector (Cheng and Mon, 1994).
The computational procedure of this methodology is
summarized as follows:
Step 1: Comparing the performance score. Triangular
fuzzy numbers …~
1; ~
3; ~
5; ~
7; ~
9† are used to indicate the
relative strength of each pair of elements in the same
hierarchy.
Step 2: Constructing the fuzzy comparison matrix.
By using triangular fuzzy numbers, via pairwise
Ä …aij † is
comparison, the fuzzy judgment matrix A
constructed as shown below:
iˆj
i=j
1†
where A~ is a n 6 n fuzzy matrix containing fuzzy
numbers a~ij and x~ is a non-zero n 6 1 fuzzy vector
containing fuzzy numbers x~i .
To perform fuzzy multiplications and additions
using the interval arithmetic and a-cut, Equation 1 is
equivalent to
‰aai1l xa1l ; aai1u xa1u Š+ +‰aainl xanl ; aainu xanu Š ˆ ‰lxail ; lxaiu Š
where
x1 ; . . . ; x~n †;
A~ ˆ ‰~
aij Š; x~t ˆ …~
a
a
a
a
a~ij ˆ aijl ; aiju ; x~i ˆ ‰xail ; xaiu Š; ~la ˆ ‰lal ; lau Š …2†
for 0 5 a 1 and all i; j, where i ˆ 1; 2; . . . n; j ˆ 1; 2;
. . . ; n:
Ä is
Degree of satisfaction for the judgment matrix A
estimated by the index of optimism m. The larger
value of the index m indicates the higher degree of
optimism. The index of optimism is a linear convex
combination (Lee, 1999) de®ned as:
a^aij ˆ maaiju ‡ …1
m†aaijl ;
Vm [ ‰0; 1Š
3†
While a is ®xed, the following matrix can be obtained
after setting the index of optimism, m, in order to
estimate the degree of satisfaction.
2
3
1 a^a12 a^a1n
6 a^a21 1 a^a2n 7
6
7
A~ ˆ 6 .
4†
.. 7
..
..
4 ..
. 5
.
.
a^an1 a^an2 1
The eigenvector is calculated by ®xing the m value and
identifying the maximal eigenvalue.
371
Fuzzy AHP approach
Step 4: Determining the total weights. By synthesizing the priorities over all levels, the overall
importance weights of customer requirements are
obtained by varying a value.
CR9
CR10
CR11
S3: FCM3 ˆ
CR12
CR13
CR14
5. An illustrative example
The design of a removable mountain bicycle
splashguard (Ullman, 1992) is used as an example
to illustrate the fuzzy AHP approach to the determination of the importance weights of customer
requirements.
5.1. Developing hierarchical structure
of customer requirements for bicycle
splashguard design
There are 19 customer requirements to be considered
in the design of a bicycle splashguard. They are
classi®ed into three main categories and seven
subcategories. A four-level hierarchy of customer
requirements for the splashguard design was constructed as shown in Fig. 3.
CR16
S5: FCM4 ˆ
CR17
CR9
2
1
CR10
6~
1
61
6 1
6~
5
6
6 ~
6 3
6 1
4~
1
~
3
CR11
~
5
~
3
~
3 1 1
~
~
5
9
~
1
3
~
~
3
7
~
1
1
CR12 CR13 CR14
3
~
~
~ 1
1
3 1
3
~ 1
~
5
1
3 17
7
7
~
7 17
3 1 ~
9 1 ~
7
~
~
1
5
1 7
7
7
~
~
5 1
1
3 15
~
~
1 1
3
1
CR
17
" 16 CR#
~
1
1
~
1 1 1
2 S1 S12 S3 1 3
~
~
S1 1 3
1
6~
~ 7
C1: FCM5 ˆ S2 4 3
5
1
1
~ 1
S3 ~
1 1
1
S4
C2: FCM6 ˆ
S5
" S4 S5 #
~
1
3
1
~
3
1
S6
C3: FCM7 ˆ
S7
" S6 S7 #
~
1
1
1
~
1
1
2 C1 C2 C3 3
~
~
C1
1
5
7
7
6~ 1
~
G: FCM8 ˆ C2 4 5
1
15
C3 ~
7 1 ~
1 1 1
5.3. Computing importance weights of customer
requirements
5.2. Constructing fuzzy comparison matrices
Triangular fuzzy numbers, ~
1-~
9, are used to express the
preference in the pairwise comparisons. By using
geometric means of the pairwise comparisons, the
fuzzy comparison matrices (FCM) for each level can
be obtained as follows:
CR1
CR2
CR3
S1: FCM1 ˆ
CR4
CR5
CR6
CR1
2
1
CR2
6~
1
63
6 1
6~
65
6~
67 1
6
4 ~
5
~
7 1
~
3
~
1
~
3
~
3
1
~
1
8
"CR7 CR#
~
CR7
1
1
S2: FCM2 ˆ
CR8 ~
1 1 1
CR3
1
1
1
CR4
~
7
~
3
~
3
~
3 1 1
~
1
3
1
~
3
1
~
5
~
1
1
CR5
~
5
~
1
~
1
~
3 1
1
~
3 1
CR6
3
~
7
~7
3
7
7
~
37
7
17
7
7
~
35
1
The lower limit and upper limit of the fuzzy numbers
with respect to the a can be de®ned as follows by
applying Equation 2:
~
1a ˆ ‰1; 3 2aŠ; ~
3a ˆ ‰1 ‡ 2a; 5 2aŠ;
1
1
~
3a 1 ˆ
;
;
5 2a 1 ‡ 2a
1
1
1
~
~
5a ˆ
;
;
5a ˆ ‰3 ‡ 2a; 7 2aŠ;
7 2a 3 ‡ 2a
1
1
~
~
;
;
7a 1 ˆ
7a ˆ ‰5 ‡ 2a; 9 2aŠ;
9 2a 5 ‡ 2a
1
1
~
9a ˆ ‰7 ‡ 2a; 11 2aŠ; ~
9a 1 ˆ
;
5†
11 2a 7 ‡ 2a
For example, let a ˆ 0.5 and m ˆ 0.5, and substitute
above expression into the fuzzy comparison matrices,
FCM1 to FCM8 , all the a-cuts fuzzy comparison
matrices can be obtained as follows:
372
Kwong and Bai
Fig. 3. A hierarchy of customer requirements for bicycle splashguard design.
2
1
6 ‰1=4; 1=2Š
6
6
6 ‰1=6; 1=4Š
S1: FCMa1 ˆ 6
6 ‰1=8; 1=6Š
6
6
4 ‰4; 6Š
‰1=8; 1=6Š
S2: FCMa2 ˆ
1
‰1=2; 1Š
‰1; 2Š
1
‰1; 2Š
‰1=4; 1=2Š
‰1=2; 1Š
‰1=4; 1=2Š
‰1; 2Š
1
3
‰6; 8Š
‰2; 4Š 7
7
7
1
‰2; 4Š
‰1; 2Š
‰2; 4Š 7
7
‰1=4; 1=2Š
1
‰1=4; 1=2Š
1 7
7
7
1
‰2; 4Š
1
‰2; 4Š 5
‰1=4; 1=2Š
1
‰1=4; 1=2Š
1
‰4; 6Š
‰1; 2Š
‰6; 8Š
‰2; 4Š
‰4; 6Š
‰1; 2Š
373
Fuzzy AHP approach
2
1
6 ‰1=2; 1Š
6
6
6 ‰1=6; 1=4Š
a
S3: FCM3 ˆ 6
6 ‰2; 4Š
6
6
4 ‰1=2; 1Š
‰1; 2Š
1
1
‰8; 10Š
‰2; 4Š
1
‰1=6; 1=4Š
‰4; 6Š
1
‰2; 4Š
‰6; 8Š
‰1=2; 1Š
‰2; 4Š
‰1; 2Š
‰1=2; 1Š
1
C2:
C3:
FCMa7
ˆ
‰2; 4Š
‰1=4; 1=2Š
1
1
ˆ
‰1=2; 1Š
2
‰1=2; 1Š
1
‰1; 2Š
1
1
6
G: FCMa8 ˆ 4 ‰1=6; 1=4Š
‰1=8; 1=6Š
1
‰4; 6Š
1
‰1=2; 1Š
‰6; 8Š
1
Let FCM0:5
5 ˆ A. Eigenvalues of the matrix A can be
calculated as follows by solving the characteristic
equation of A, det…A lI† ˆ 0.
l2 ˆ
0:081;
3
7
‰1; 2Š 5
Equation 3 and MATLAB package (Harman, 1997)
are used to calculate eigenvectors for all comparison
matrices, from which the importance weights of
individual customer requirements can be obtained.
For example, FCM0:5
5 can be obtained as shown below
after applying Equation 3.
2
3
1:000 3:000 1:000
0:5
FCM5 ˆ 4 0:375 1:000 0:375 5
1:000 3:000 1:000
l1 ˆ 3:081;
1
3
‰1=2; 1Š
7
‰1; 2Š 5
1
‰1=4; 1=2Š
6
C1: FCMa5 ˆ 4 ‰2; 4Š
1
‰1=10; 1=8Š ‰1=4; 1=2Š
3
‰1=4; 1=2Š
‰1=4; 1=2Š 7
7
7
‰1=8; 1=6Š 7
7
‰1; 2Š 7
7
7
‰1=4; 1=2Š 5
2
FCMa6
‰1; 2Š
1
‰4; 6Š
1
1
‰1; 2Š
‰1=4; 1=2Š
‰1=6; 1=4Š
‰1=4; 1=2Š
‰2; 4Š
S5: FCMa4 ˆ
‰4; 6Š
‰2; 4Š
l3 ˆ 0:000
As the value of l1 is the largest, the corresponding
eigenvectors of A can be calculated as follows by
substituting the l1 into the equation, AX ˆ lX.
X1 ˆ …0:6852; 0:2469; 0:6852†
T
After normalization, the importance weights of the
sub-categories of the customer requirements, S1 ; S2 ,
and S3 , can be determined as shown below.
C1: ‰WS1 ; WS2 ; WS3 Š ˆ ‰0:4237; 0:1527; 0:4237Š
Following the similar calculation, the importance
weights of C1 to C3 ; S4 to S7 and CR1 to CR19 can be
determined as shown below.
374
Kwong and Bai
Table 1. Importance weights of customer requirements for bicycle splashguard design
Category
Subcategory
Attribute
Functional performance (0.7387)
Attach/Detach (0.3130)
Easy to attach (0.1278)
Easy to detach (0.0575)
Fast to attach (0.0423)
Fast to detach (0.0172)
Can attach when bike is dirty (0.0423)
Can detach when bike is dirty (0.0260)
Not mar (0.0661)
Not catch water, etc. (0.0467)
Not rattle (0.0489)
Not wobble (0.0331)
Not bend (0.0118)
Long life (0.1042)
Lightweight (0.0331)
Not release accidentally (0.0819)
Most bikes (0.0635)
With lights and generator (0.0526)
With brakes (0.0372)
Streamlined (0.0447)
Popular colour (0.0633)
Interface with bike (0.1128)
Structural integrity (0.3130)
Spatial constraints (0.1533)
Fit (0.0635)
Not interfere (0.0898)
Appearance (0.1080)
Shape (0.0447)
Colour (0.0633)
S1: ‰WCR1 ; WCR2 ; WCR3 ; WCR4 ; WCR5 ; WCR6 Š ˆ
C3: ‰WS6 ; WS7 Š ˆ ‰0:4142; 0:5858Š
‰0:4082; 0:1386; 0:1351; 0:0551; 0:1351; 0:0829Š
S2: ‰WCR1 ; WCR8 Š ˆ ‰0:5858; 0:4142Š
G: ‰WC1 ; WC2 ; WC3 Š ˆ ‰0:7387; 0:1533; 0:1080Š
S3: ‰WCR9 ; WCR10 ; WCR11 ; WCR12 ; WCR13 ; WCR14 Š ˆ
‰0:1564; 0:1058; 0:0376; 0:3330; 0:1058; 0:2616Š
S5: ‰WCR16 ; WCR17 Š ˆ ‰0:5858; 0:4142
Based on the above results, the total importance
weights of individual customer requirements can be
calculated by using the following equations, and the
results are shown in Table 1.
TWS1 ˆ WC1 ? WS1 ;
TWS2 ˆ WC1 ? WS2 ;
TWS4 ˆ WC2 ? WS4 ;
TWS5 ˆ WC2 ? WS5 ;
TWS7 ˆ WC3 ? WS7 ;
TWCR1 ˆ WC1 ? WS1 ? WCR1 ;
TWCR3 ˆ WC1 ? WS1 ? WCR3 ;
TWCR5 ˆ WC1 ? WS1 ? WCR5 ;
TWCR6 ˆ WC1 ? WS1 ? WCR6 ;
TWCR7 ˆ WC1 ? WS2 ? WCR7 ;
TWCR9 ˆ WC1 ? WS3 ? WCR9 ;
TWCR10 ˆ WC1 ? WS3 ? WCR10 ;
TWCR12 ˆ WC1 ? WS3 ? WCR12 ;
TWCR13 ˆ WC1 ? WS3 ? WCR13 ;
TWCR15 ˆ WC2 ? WS4 ;
TWCR16 ˆ WC1 ? WS5 ? WCR16 ;
TWCR18 ˆ WC3 ? WS6 ;
TWCR19 ˆ WC3 ? WS7
6. Discussions
In the pairwise comparisons of AHP, triangular fuzzy
numbers were introduced to improve the scaling
scheme of Saaty's method. The central value of a
TWS3 ˆ WC1 ? WS3 ;
TWS6 ˆ WC3 ? WS6 ;
TWCR2 ˆ WC1 ? WS1 ? WCR2 ;
TWCR4 ˆ WC1 ? WS1 ? WCR4 ;
TWCR8 ˆ WC1 ? WS2 ? WCR8 ;
TWCR11 ˆ WC1 ? WS3 ? WCR11 ;
TWCR14 ˆ WC1 ? WS3 ? WCR14 ;
TWCR17 ˆ WC2 ? WS5 ? WCR17 ;
fuzzy number is the corresponding real crisp number.
The spread of the number is the estimation from the
real crisp number. Equation 3 de®nes how the
estimated number, a^ij , reacts to the real crisp
number by adjusting the index of optimism, m. The
375
Fuzzy AHP approach
m indicates the degree of optimism, which could be
determined by design team. If the real crisp number is
overestimated …m 4 0.5†, the value of a^ij is higher than
the central value. If it is underestimated …m 5 0.5†, the
value of a^ij is lower than the central value.
By setting m value as 0.05, 0.5, and 0.95
respectively (re¯ecting the pessimistic, the moderate
and the optimistic situations), three graphs as shown
in Appendix A, B, and C were generated by using
MATALAB package with the a varying from 0 to 1.
From the graphs, mutual comparisons can be
performed from the most uncertain situation …a ˆ 0†
to the most certain situation …a ˆ 1†, from which
relative importance of the customer requirements
CR1 *CR19 † can be realized. For example, from the
three graphs, the importance weight of customer
requirement CR7 is less than the one of the customer
requirement CR15 under the most uncertain comparison …a ˆ 0† and highly optimistic situation
m ˆ 0.95†. For the pessimistic situation …m ˆ 0.05†,
the importance weight of customer requirement CR7
is larger than the one of CR15 . For the moderate
situation …m ˆ 0.50†, the importance weights of
the customer requirements CR7 and CR15 are very
close.
7. Conclusions
In this paper, a fuzzy AHP approach to the
determination of the importance weights of customer
requirements in QFD was presented. In the methodology, triangular fuzzy numbers were introduced
into the conventional AHP in order to improve the
imprecise ranking of customer requirements. Using
fuzzy AHP in the determination of importance
weights of customer requirements has the following
two major advantages:
(1) Fuzzy numbers are preferable to extend the
range of a crisp comparison matrix of the conventional AHP method, as human judgement in the
comparisons of customer requirements is fuzzy in
nature.
(2) Adoption of fuzzy numbers can allow design
team number of QFD to have freedom of estimation
regarding the overall customer satisfaction goal and
actual situations. Judgement can go from optimistic to
pessimistic.
The design of a bicycle splashguard was used as an
example to illustrate the application of fuzzy AHP
method in the determination of the importance
weights of customer requirements for bicycle
splashguard design. The overall results show that the
combination of fuzzy decision making with AHP
could become a useful tool for implementing QFD in
future research.
Acknowledgment
The work described in this paper was supported by a
grant from The Hong Kong Polytechnic University,
Hong Kong (Project no. A-PC06).
Appendix A
376
Kwong and Bai
Appendix B
References
Appendix C
Akao, Y. (1990) Quality Function Deployment: Integrating
Customer Requirements into Product Design,
Productivity Press, Cambridge, MA.
Armacost, R. T., Componation, P. J., Mullens, M. A. and
Swart, W. W. (1994) An AHP framework for
prioritizing custom requirements in QFD: An industrialized housing application. IIE Transactions, 26(4),
72±79.
Aswad, A. (1989) Quality function deployment: A systems
approach, in Proceedings of the 1989 IIE Integrated
Systems Conference, Institute of Industrial Engineers,
Atlanta, GA, pp. 27±32.
Boender, C. G. E., de Grann, J. G., Lootsma, F. A. (1989)
Multicriteria decision analysis with fuzzy pairwise
comparison. Fuzzy Sets and Systems, 29, 133±143.
Buckley, J. J. (1985) Fuzzy hierarchical analysis. Fuzzy Sets
and Systems, 17, 233±247.
Chan, L. K., Kao, H. P., Ng, A. and Wu, M. L. (1999) Rating
the importance of customer needs in quality function
deployment by fuzzy and entropy methods.
International Journal of Production Research,
37(11), 2499±2518.
Che, A., Lin, Z. H. and Chen, K. N. (1999) Capturing weight
of voice of the customer using arti®cial neural network
in quality function deployment. Journal of Xi'an
Jiaotong University, 33(5), 75±78.
Cheng, C. H. and Mon, D. L. (1994) Evaluating weapon
system by analytical hierarchy process based on fuzzy
scales. Fuzzy Sets and Systems, 63, 1±10.
Grif®n, K. and Hauser, J. R. (1993) The voice of the
customer. Marketing Science, 12(1), 1±27.
Gustafsson, A. and Gustafsson, N. (1994) Exceeding
customer expectations. The Sixth Symposium on
Quality Function Deployment, Novi, Michigan.
Harman, T. L. (1997) Advanced Engineering Mathematics
Using MATALAB, PWS Pub. Co., Boston.
Hauser, J. R. and Clausing, D. (1988) The house of quality.
Harvard Business Review, 66, 63±73.
Lai, Y. J., Ho, E. S. S. A. and Chang, S. I. (1998) Identifying customer preferences in quality function deployment using group decision-making techniques, in
Usher, J., Roy, U. and Parsaei, H. (ed.), Integrated
Product and Process Development, Wiley, New York,
pp. 1±28.
Lee, A. R. (1999) Application of Modi®ed Fuzzy AHP
Method to Analyze Bolting Sequence of Structural
Joints, UMI Dissertation Services, A Bell & Howell
Company.
Lu, M. H., Madu, C. N., Kuei, C. and Dena Winokur (1994)
Integrating QFD, AHP and benchmarking in strategic
marketing. Journal of Business & Industrial
Marketing, 9(1), 41±50.
Fuzzy AHP approach
Saaty, T. L. (1980) The Analytic Hierarchy Process,
McGraw-Hill, New York.
Ullman and David G. (1992) The Mechanical Design
Process, McGraw-Hill, New York.
Van Laarhoven, P. J. M. and Pedrycz, W. (1983) A fuzzy
377
extension of Saaty's priority theory. Fuzzy Sets and
Systems, 11, 229±241.
Zakarian, A. and Kusiak, A. (1999) Forming teams: An
analytical approach. IIE Transactions on Design and
Manufacturing, 31(1), 85±97.