FuzzySTAR: Fuzzy set theory of axiomatic design review.
TitleFuzzySTAR: Fuzzy set theory of axiomatic design review
Author(s)Huang, GQ; Jiang, Z
Artificial Intelligence for Engineering Design, Analysis and
Manufacturing, 2002, v. 16 n. 4, p. 291-302
Artificial Intelligence for Engineering Design, Analysis and
Manufacturing. Copyright © Cambridge University Press.
FuzzySTAR: Fuzzy set theory of axiomatic design review
GEORGE Q. HUANG1and ZUHUA JIANG2
1University of Hong Kong, Hong Kong, People’s Republic of China
2School of Mechanical Engineering, Shanghai Jiao Tong University, Shanghai, People’s Republic of China
(Received September 1, 2001; Accepted May 26, 2002!
Product development involves multiple phases. Design review ~DR! is an essential activity formally conducted to
ensure a smooth transition from one phase to another. Such a formal DR is usually a multicriteria decision problem,
involving multiple disciplines. This paper proposes a systematic framework for DR using fuzzy set theory. This fuzzy
approach to DR is considered particularly relevant for several reasons. First, information available at early design
phases is often incomplete and imprecise. Second, the relationships between the product design parameters and the
review criteria cannot usually be exactly expressed by mathematical functions due to the enormous complexity. Third,
DR is frequently carried out using subjective expert judgments with some degree of uncertainty. The DR is defined as
the reverse mapping between the design parameter domain and design requirement ~review criterion! domain, as
compared with Suh’s theory of axiomatic design. Fuzzy sets are extensively introduced in the definitions of the
domains and the mapping process to deal with imprecision, uncertainty, and incompleteness. A simple case study is
used to demonstrate the resulting fuzzy set theory of axiomatic DR.
Keywords: Axiomatic Design; Design Review; Fuzzy Sets; Product Development
Axiomatic design was originally proposed by Suh in the
1980s, and it was later formulated as a generic theory of
axiomatic design ~TAD! as demonstrated systematically in
Suh ~1990!. Since then, the method has gained wide recog-
nition in both the research and industrial communities.Typ-
ical applications include mechanical products ~Park et al.,
1996; Cha & Cho, 1999!, software products ~Kim et al.,
and design for environment ~Wallace & Suh, 1993; Chen,
2001!. Recently, the method has been implemented as a
commercial software package ~Harutunian et al., 1996;
In TAD, Suh ~1990! defines design as the mapping pro-
cess between the functional requirements ~FRs! in the func-
tional domain and the design parameters ~DPs! in the
physical domain. Conceptually, the design process can be
interpreted as a process involving choosing the right set of
DPs to satisfy the given FRs. Mathematically, the mapping
is expressed as
design: $FR% r $DP%,
or, more specifically,
$FR% ? @A# ? $DP%,
where $FR% is the FR vector, $DP% is the DPvector, and @A#
is the design matrix.
The above is only a very brief summary of TAD. De-
tailed discussions on TAD should be sought from the Suh’s
original work ~1990, 1999!. More introductory discussions
on the method can be found at http:00www.axiomaticdesign.
com and http:00axiom.mit.edu!. Hintersteiner and Fried-
man ~1999! and Tate ~1999! demonstrate and examine
various key issues regarding how TAD as an abstract math-
ematical model can be put into practical applications.
Axiomatic design guides designers in using all their
existing design tools and software to arrive at a successful
new design or to diagnose and correct an existing design.
The latter application in diagnosis and correction has mo-
tivated the authors to explore the possibility of extending
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Artificial Intelligence for Engineering Design, Analysis and Manufacturing ~2002!, 16, 291–302. Printed in the USA.
Copyright © 2002 Cambridge University Press 0890-0604002 $12.50
the TAD into a framework for design review ~DR, Huang,
2002a!. The resulting framework is called STAR ~system-
atic theory for axiomatic DR!, which has also been imple-
mented as a prototype web application for setting up a DR
portal ~Huang, 2002b!.
In essence, STAR is defined as the reverse mapping from
the DP domain to the FR domain, that is,
review: $DP% r $FR%.
Here the FR domain should be generalized as the DR crite-
ria, which is the DR criterion domain ~RC domain!. The
RC domain can be established from the functional ~FRs!
and other types of design requirements ~DRs!, that is,
$RC% ? $DR%.
The establishment of STAR contributes to the scarce lit-
erature on DR. The ad hoc DR practice ~Ichida, 1989! can
now be guided in a systematic way. Thus, such systematic
DR practice is more likely to meet the requirements im-
posed by the ISO 9000 quality standard under which DR is
mandatory ~Schoonmaker, 1996!.An extra benefit of a sys-
tematic DR framework is that computerized decision sup-
port systems with the latest internet and web technologies
can be developed for professional applications in industry.
This is a significant addition to the product data manage-
ment technology, where DR has traditionally not been em-
phasized as much as engineering change management.
However, DR is never a clear-cut exercise. Vague and
uncertain descriptions of DPs and DR criteria are often
involved ~Li &Azarm, 2000!.This has made the mathemat-
itative analysis is widely needed with linguistic and vague
descriptions of the complex DR system ~Cheng, 1999!. Fur-
thermore, the mapping results from STAR do not address
the ultimate question that DR must answer: is the design
good or bad and which parts are good, which parts are bad,
and to what extent?
Whereas Helander and Lin ~2000! incorporated fuzzy set
theory into Suh’s TAD, this paper aims to resolve the ques-
to form FuzzySTAR ~fuzzy set theory for axiomatic DR!.
Section 2 details the extension process from STAR to
FuzzySTAR and associated research issues. Section 3
presents a case study to illustrate how the proposed
FuzzySTAR can be applied in an industrial environment.
Observations and implications are discussed in Section 4 to
conclude the paper.
FuzzySTAR is an extension of a previous version of the
STAR by incorporating fuzzy set theory. STAR is derived
from Suh’s TAD. Naturally, the definition and description
of FuzzySTAR share some common features and conven-
tions with TAD and STAR. Most noticeably, the constructs
of domains, mappings, and axioms are all used in Fuzzy-
STAR. Figure 1 shows the overview of FuzzySTAR. Its
major steps, including the key concepts and associated
mathematics, will be discussed in detail in the rest of this
Fig. 1. An overview and the steps of FuzzySTAR.
G.Q. Huang and Z. Jiang
2.1. Step 1: Define the design objects $DO% domain
As in TAD and STAR, the product design domain must be
defined first. Let $DO% be defined as a collection ~domain
or vector! of DOs that make up the product under review.
Although the mathematical definition here seems to imply
that the $DO% domain is a flat structure, the hierarchical
tree representation is commonly used in practice. The lev-
els from the top to the bottom in the DO hierarchy reflect
the evolution of the design process from early stages to
more detailed stages, on the one hand, and the shifting foci
or concerns of the DR process, on the other.
A DO is defined as a triplet: DP, target value ~TV!, and
design weight ~DW!. The first element in the DO triplet is
a DP that describes and defines a product DO collectively
with other DPs. The collection of all the DPs in the $DO%
domain forms another DP domain $DP%.
The second element of the DO triplet, TV, is the value ~or
range of values! of the corresponding DP. The collection of
all the TVs in the $DO% domain is the new value domain
$TV%. The TVs of DPs can be defined as either crisp num-
bers or a region ~range! of crisp numbers. At this stage of
the work reported in this paper, only crisp values are con-
sidered and region values are avoided for simplicity.
The third element in the DO triplet, DW, is the weight or
rating based on a certain aspect ~e.g., the cost of changing
the TV of the corresponding DP!. The aspect of “changing
a DP’s value” is mentioned here in regard to modular de-
sign, in which some DPs cannot be changed if the DO is a
module. In addition, the subsequent action following the
DR is usually design revision or change. The collection of
all the DWs forms the set of weights on DPwhen taking TV
as its value.
The DWs are normally expressed as subjective crisp num-
bers ~see, e.g., Khoo & Ho, 1996; Moskowitz & Kim, 1997!.
Very often, these weights are typically uncertain and impre-
cise. Therefore, they have also been expressed as fuzzy
numbers ~Masud & Dean, 1993; Wang, 1999!. Typical lin-
guistic terms for describing grades of the cost of changing
the value of a DP are difficult, possible, and easy.
DWs are usually established based on past technical ex-
periences or expert judgments. It is outside the scope of this
work to discuss how this can be accomplished. Several tech-
niques are available to obtain these values, whether crisp or
fuzzy. One of the most widely used methods is the analyt-
ical hierarchy process ~AHP; e.g., Park & Kim, 1998!.
Weights from AHP can be converted into fuzzy numbers
based on the concept of the “fuzzy line segment” proposed
by Carnahan et al. ~1994!.
2.2. Step 2: Define the $RC% domain
Let $RC% define a collection ~domain or vector! of RCs
used for reviewing the $DO%. Just like the $DO% domain,
the $RC% domain is commonly represented hierarchically,
as a tree in practice. The higher levels are called “view-
points” of DR.The RC at the bottom level are those directly
used for evaluating the DOs. That is, the relationships be-
tween the DPs and the RC at the bottom level can be di-
rectly established and analyzed. The results are propagated
or aggregated upward in the hierarchy as discussed in a
A RC is also expressed as a triplet: evaluation criterion
~EC!, design capability ~DC!, and functional weight ~FW!.
The first element in the $RC% domain is an EC. This is a
description of an objective that DOs intend to achieve. The
collection of all the RCs in the $RC% domain forms another
EC domain $EC%.
The second element of the RC triplet, DC, is the value
~or range of values! of the corresponding EC. This value or
range of values describes the capability of the DOs to meet
the intended DRs. The collection of all the ECs in the $RC%
domain is the new DC domain $DC%.
The third element in the RC triplet, FW, is the weight or
rating that the DOs collectively achieve against this RC ~or
EC!. The collection of all the FWs forms the set of weights
on RC. FWs can be either crisp values or fuzzy numbers,
depending on the type of number used in evaluating the
EC–DP relationships. If the relationships are described in
fuzzy numbers, FWs are naturally described in fuzzy num-
bers. Typical linguistic terms for describing grades of an
EC are excellent, good, satisfactory, barely satisfactory, and
unsatisfactory or simply good, fair, and unsatisfactory.
Finally, there are two ways of obtaining the FW of a RC.
One method is to derive the FW from fuzzy relationships
between DPs and the EC through certain aggregation algo-
rithms. This method is called the direct fuzzy mapping
method because the FW is obtained directly from the mem-
bership functions established between the EC across the
TVs of the DPs.
The other method is to derive the FWfrom a membership
is called indirect or tandem fuzzy mapping method. In the
tandem approach to DR, it is necessary to define the rela-
tionships between the DPs and RCs mathematically so that
the absolute values of RCs can be estimated as output vari-
ables with respect to DPs as input variables. The complex-
ity of the DR problem usually makes such mathematical
ping method does not have this problem.
Both methods are necessary in practical applications. For
example, the case study on the bus inner configuration de-
sign to be discussed in a later section uses the first method.
The case study on fuel pump design uses the second method
~Huang & Jiang, 2002b!. The methods can also be applied
2.3. Step 3: Evaluate the impacts of DPs on ECs
This step is the most fundamental step in FuzzySTAR. It
creates the review workspace and forms the basis for car-
Fuzzy set theory of axiomatic design review
rying out fuzzy analysis. This step includes the following
1. Create DR space (matrix): the STAR mapping creates
the 2-dimensional review space ~or matrix!, which is
the so-called universe of discourse used in fuzzy
2. Determine for each EC if a particular DP is related to
this EC: If yes, a checkmark is placed in the corre-
sponding cell. Otherwise, the cell is left blank, indi-
cating that this DP does not affect the EC. This EC is
said to “not care” about the DP. In the case of “an
uncaring” EC–DP relationship, no further consider-
ation is necessary.
3. Evaluate the $DC% for each RC across the $DO% do-
main: This activity is only necessary in the tandem
mapping method and is not included in the direct map-
ping method. In the tandem mapping method, the DC
must be fuzzificated just like the fuzzification of the
impacts in the direct mapping method in point 4.
4. Fuzzificate the impact of the DP on the EC. As ad-
vocated by several researchers ~Cheng, 1999; Li &
Azarm, 2000; Wang, 2001!, this paper also advocates
the approach of capturing the DP–RC relationships
through fuzzy sets. Typical linguistic terms for de-
scribing the grades of the relationship between a DP
and an EC are strong, medium, and weak or excellent,
good, satisfactory, unsatisfactory, and poor.
The activity of fuzzification is worth further discussions
in two different situations. In the first situation of tandem
fuzzy mapping, the RCs obtained are crisp numbers. For
example, $RC% ?$~cost, $150, ?!, ~weight, 20 kg, ?!%. Now
the remaining question is if this cost of $150 and the weight
of 20 kg reflect a good design or bad design. To answer this
question, it is then necessary to fuzzificate these crisp num-
bers into fuzzy measures FWs.That is, $RC% ?$~cost, $150,
~0.3, 0.4, 0.1!!, ~weight, 20 kg, ~0.1, 0.2, 0.5!!%. These fuzzy
numbers will be defuzzificated later to establish the grades
excellent, fair, or poor.
In the case of direct fuzzy mapping, the activity of fuzzi-
fication is conducted to each cell in the review matrix. For
each cell of the DP–RC matrix, that is, each pair ~EC, DP!
where EC ? $EC% and DP ? $DP%, a fuzzy set is estab-
lished to represent the impact of DPon EC by searching the
fuzzy knowledge base that is a collection of membership
functions. If a corresponding membership function is found,
then the fuzzy number is evaluated from the membership
function for the cell or pair ~EC, DP!.
The membership functions play the important role in the
mapping function from $DO% to $RC%. For each pair of DOi
and RCj, a set of membership functions should be estab-
lished corresponding to the five grades of fuzzy measures:
excellent, good, satisfactory, unsatisfactory, and poor.
There are a variety of membership functions, such as
ezoidal function. Figure 2 shows one example. The limita-
tion to the minimum ideal value can be described as the
upper step function or S-membership function. The limita-
tion to an ideal range can be described as trapezoidal func-
tion, P-membership function, or normal distribution
function; and the limitation to the maximum ideal value can
be described as the complementation of a step function or
2.4. Step 4: Aggregate with fuzzy operators
Individual fuzzy impacts of DPs on ECs are processed to
obtained intermediate and final results in the following four
1. the aggregation from individual DPs to one EC,
2. the upward aggregation along the RC hierarchy,
3. the aggregation from individual ECs to one DP, and
4. the upward aggregation along the DO hierarchy.
The first occasion is the same as the third. Because the
aggregation takes place along rows or columns in a matrix,
it can be called a matrix ~flat! aggregation. The second
occasion is the same as the fourth. Because it takes place
upward on a hierarchical tree, it is therefore called a hier-
Let us consider the matrix flat aggregation first. The fol-
lowing equation is proposed for the flat aggregation in a
cell_weighti, j? column_weightj
for aggregating along the n columns in the matrix and
cell_weighti, j? row_weighti
for aggregating along the m rows in the matrix.
Fig. 2. Typical membership functions.
G.Q. Huang and Z. Jiang