Advanced Materials Research Vols 44-46 (2008) pp 421-428
© (2008) Trans Tech Publications, Switzerland
doi:10.4028/www.scientific.net/AMR.44-46.421
Online: 2008-06-12
Integration of Axiomatic Design and Design Structure Matrix for Product
Design
D.B. Tanga, G.J. Zhang and S. Dai
College of Mechanical and Electrical Engineering, Nanjing University of Aeronautics & Astronautics,
Nanjing 210016, China.
a
d.tang@nuaa.edu.cn
Keywords: Axiomatic Design, Design Structure Matrix, Integration, Design Theory
Abstract. Axiomatic design and design structure matrix (DSM) are two popular design methods at
the moment, while most related researches only apply the basic ideas of axiomatic design or DSM to
some cases. This paper first analyses the disadvantages of both axiomatic design and DSM. After
deep investigations, the paper then puts forward that there are strong complementarities between
axiomatic design and DSM. Based on the complementarities between axiomatic design and DSM, it
is proposed that axiomatic design’s design matrix (DM) can be transformed into corresponding DSM
for structural evaluation. In this way, axiomatic design and DSM can benefit from each other. The
logic of integration between axiomatic design and DSM is interpreted in this paper. A computer aided
conceptual design system has been developed to enable the integration of axiomatic design and DSM.
Introduction
Axiomatic design (AD) is a prescriptive engineering design theory and methodology that provides a
systematic and scientific basis for making design decisions. In addition to corollaries and theorems
derived from them, axioms give design teams a solid basis for formalizing design problems,
conceptualizing solution alternatives, eliminating bad design ideas during the conceptual stages,
choosing the best design among those proposed, and improving existing designs [1, 2]. One of the
findings to date is that one of the limitations of axiomatic design is its concentration on the
architectural design, at the expense of the system design context. As a result, certain factors and
constraints, such as cost, time, and physical integration are not taken into account directly by the
axiomatic model [3, 4]. As a structured modelling method, more recently the DSM (Design Structure
Matrix) model has been regarded as a good roadmap of system design [5]. DSM is a good tool
mapping information flow and its impact in product development processes; and it can represent
visually the network of interactions among development activities or design objectives and facilitates
analysis of these interactions [6-11]. The process of making DSM involves understanding, tracing and
capturing the interaction relations among system elements, which actually is a process of system level
knowledge creation and recording. System level knowledge can be defined as the knowledge
concerning the interactions among system elements. Knowing system interactions in the product
development process is critical for project management and decisions. System interactions mainly
refer to the inter-relationships existing between element types during the product development cycle.
Capturing inter-relationships between affiliated system elements is necessary if the product
engineering is to be handled efficiently and effectively. Currently DSM has been a common means to
represent the system interaction. Although DSM is powerful on the analysis of design interactions, it
is less effective in innovative design. One reason is that it is not possible to obtain a DSM for a new
product that has never been designed before. Another reason is that DSM fails to record explicitly the
reasons for these interactions.
There has been evidence of attempts to link axiomatic design and the DSM [3, 4]. This research
considers that axiomatic design and DSM can be linked in a complimentary manner, and captures and
models the underlying logic of integration of both. In Section 2, a brief review and analysis of
axiomatic design and DSM is presented. In Section 3, the logic of integration between axiomatic
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Materials and Product Technologies
design and DSM is interpreted. In Section 4, the related system development is given, and Section 5
concludes the paper.
Analysis of Axiomatic Design and DSM
The underlying hypothesis of axiomatic design [1, 2] is that there exist fundamental principles that
govern good design practice. The main distinguishable components of the axiomatic design theory are
domains, hierarchies, and design axioms. The axiomatic design method proceeds from a high level of
abstraction to detailed design elements (i.e., from meta-product systems to assemblies to components
to features to parameters). This activity of definition and detailing produces a prescriptive model of
design hierarchy for the design entity in each of three domains: functional, physical, and process. The
decisions made at higher levels affect the statement of the design definition at lower levels. That is,
the design team goes through a process whereby they zigzag between domains to decompose the
design problem (see Fig.1). At a given level of design hierarchy, there exists a set of FRs (function
requirements) defined as the minimum set of requirements needed at that level. The definition of FRs
depends on the solutions with DPs (design parameters), namely, defining acceptable FRs may involve
several iterations. Before zigzagging FRs, the corresponding hierarchical level DPs (even PVs) shall
be selected. Once a corresponding DP can satisfy an FR (and/or a PV can satisfy a DP), a focused FR
can be decomposed into a set of lower-level requirements. The design team will develop different
solutions for each DP to satisfy an FR and select the best alternative (option) at each hierarchical level
according to Axiom 1 “Maintain the independence of the functional requirements” and Axiom 2
“Minimise the information content of the design”. The axiom 1 states that the DPs and FRs are related
such that a specific DP can be adjusted to satisfy its corresponding FR without affecting other FRs.
The axiom 2 states that the independent (uncoupled) design alternatives that minimize the
information content are the best.
Customer Requirements
CRs
Customer Domain
Function Requirements
Design Parameters
DPs
FRs
Functional Domain
Physical Domain
Process Variables
PVs
Process Domain
Fig. 1: Domain and hierarchy in Axiomatic Design
Fig. 2: DP candidates for one FR (based on [16])
Guided by the axioms, the design team conceives, selects, and optimizes the “best,” even at the
conceptual stages. Once a set of FRs has been formulated and feasible sets of DPs have been
synthesized at a given hierarchical level, the two design axioms are applied to evaluate the design
concepts proposed. Axiomatic design method’s best practice is always to satisfy the axioms in
sequence (i.e., the independence axiom, then the information axiom). Application of the
independence axiom may be described in terms of the design matrixes (DM) [1]. A DM prescribes the
relationships between the FR array and the corresponding DP array at the same hierarchical level,
which can be described as
{FR}=[DM]{DP} ,
 A11
A
[ DM ] =  21
 M

 Am1
A12
A22
M
Am 2
A1n 
A2 n  , A = ∂FR /∂DP .

ij
i
j
O
M 

L Amn 
L
L
(1)
(2)
Advanced Materials Research Vols. 44-46
423
Dependent on the type of resulting design matrix DM, three types of designs exist: uncoupled,
decoupled and coupled. Regarding the uncoupled design, each FR is satisfied by exactly one DP, and
the resulting DM is diagonal. Axiom 1 is fully satisfied with the uncoupled design. The decoupled
design occurs when the DM is lower triangular, which means that a sequence exists, where by
adjusting DPs in a certain order, the FRs can be satisfied. The DM of a coupled design contains mostly
non-zero elements and thus the FRs cannot be satisfied independently.
Axiomatic design has been proved to be an excellent design method by real applications [12-15],
but it still has some shortages which are followed:
(1) Suh considers that in the case of a product that is new and innovative, the FRs should be defined
in a solution-neutral environment without considering any physical solution in mind. This,
however, can rarely happen in practice, particularly in complex product environments, where
economic considerations dictate maximum possible utilisation of mature designs and existing
knowledge [4].
(2) Axiomatic Design guides the designer finding suitable DPs to meet the needs of FRs. But how can
the designer know the interactions amongst the design parameters, including geometry, spatial
layout, interfaces (e.g. logical and physical connectivity) which will decide the quality of design?
Meanwhile, for a FR, there may be more than one corresponding DPs (Fig. 2), and several
candidate solutions may all satisfy the Function Independency Axiom. The final solution has to be
decided based on the interactions among DPs.
The idea of using Design Structure Matrix (DSM) to represent the system interaction is not new.
DSM method was proposed for system modelling and analysis more than 30 years ago. Steward [5]
firstly introduced design structure matrix (DSM) and developed some algorithms for manipulating the
matrix as tools for systems design and analysis. However, it is not until recently that DSM methods
started to attract attention for managing the complexity of large engineering systems and complex
product development processes [6-11]. A design structure matrix is a matrix representation of a
system or a project. System components are listed in the first row and the first column of the matrix.
Off-diagonal cells indicate the interactions (i.e. dependency, information flow) between system
elements. To construct a DSM, one assigns the individual system elements of a domain to the rows
and columns of a square matrix. Then, going down the list, if element b is an input to element c, one
puts a mark in the column of b and the row of c. One continues until all (known) hierarchical and
interdependent relationships have been accounted for. Marks below the diagonal represent forward
information transfer which affects the later element. Marks above the diagonal represent information
feed back or iteration (see Fig.3).
It is not enough to know only the form of DSM, it is also necessary to know the meaning of its
marks and the knowledge inside. In a DSM, the off-diagonal marks are not simply placeholders.
Corresponding to each is a specific real issue, involving at least two system elements. The matrix
understates the depth of the designers' knowledge for them; each of the marks stands for a specific
problem or question.
DSM has the advantages at recording the interaction relationship between existing product
elements, and the DSM-based method facilitates for minimizing interaction iterations. However, for a
new product that has never been designed before, it is difficult to make the DSM. At the conceptual
design stage, it is also difficult to make the DSM.
After deep analysis, it has already been found that axiomatic design and DSM could be
complementary to each other with the linking of DM and DSM [3,4]. The reasoning behind these
attempts to combine the DM with the DSM was to obtain the design information flow at an early stage
of the design, and thus allowing the use of the DSM “at the time when the most important decisions
about the system and design are made” [3](Dong and Whitney, 2001). A DSM cannot be created for a
system which has never been designed before. It is reasonable to conclude from this that the DSM
needs pre-existing system knowledge. In combining the two matrices, therefore, the concentration has
tended to be on deriving corresponding DSM from the DM, which could lead to the integration of
Axiomatic Design and DSM. The logic of integration between AD and DSM, however is still an open
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question. In this research, the integration of AD and DSM includes following issues: (1) using
existing DSM-based knowledge to accelerate axiomatic design; (2) Conversion of DM to DSM to get
the interactions among DPs at the conceptual design stage; (3) Using derived DSM to evaluate the
design result of axiomatic design; and (4)Using derived DSM to conduct project planning at early
design stage. Meanwhile the intrinsic logic of integration between AD and DSM is uncovered by
formal representations.
a
b
c
d
e
f
h
X
a
b
g
Feed
Backward
X
c
X
X
a solution principle
d
e
Feed
Forward
f
X
FR1
X
FR2
…
X
g
h
X
FRm
X
Fig. 3: The general DSM form
DP1
X
X
=
DP2
…
…
X
X
DPn
Fig. 4: DM construction guided by solution principles
Integration Logic between Axiomatic Design and DSM
The complementary manner between AD and DSM has been found by researchers, whist the
conducted research is by no means concluding and profound. The entire exploration of integration
between AD and DSM is still in pre-theory stage. One of the major reasons is the lack of a good
combination of precise mathematical representation languages and laws governing AD and DSM.
The aim of this section is two folds: (1) to establish a basic mathematical formal representation
scheme to define the predicates involved in the axiomatic design and DSM; (2) investigate the logic
of integration between AD and DSM with the mathematical representation of design objects.
A designer associates the customer requirement (CR) with function requirements (FRs) which are
engineering language, and tackles the customer requirements specified in this way. Relation defined
in Eq. (3) expresses an assertion of such a reasoning step.
∃FR ⊆ Ω : λspecify ( FR , CR )
(3)
Where Ω refers to the function definition domain, and λspecify represents that FR can fully specify
CR or not.
How to find solutions to realize the function requirements is the main aim of conceptual design.
Axiomatic Design has given Axiom 1 to judge the design matrix (DM) and evaluate the solutions
candidates. The process of DM construction is actually knowledge-based. During the mapping
process from FR to DP, the designer in the first place considers that the FR can be realized though
what solution principle (SP), and then conceives the corresponding structure candidates guided by the
specified solution principle. The solution principle could be physical principle (such as Newtonian
law of action and reaction), chemical principle, magnetic principle, etc. In other words, the diagonal
elements of Axiomatic Design’s DM are associated with various solution principles (see Fig. 4).
A specified solution principle (SP) is selected from a domain theory Θ which is a generic,
problem-independent knowledge, possibly applicable to the different problems. For example, physics
is a generic domain theory applicable to a design of an elevator as well as a spacecraft. A SP is
‘instantiated’ for a particular conceptual design, thus creating a usable theory for solving a particular
problem. With the specified SP, the mapping from FR to DP can be formulated in Eqs. (4), (5) and (6).
∃SP ⊆ Θ : λsatisfy ( SP , FR )
(4)
Advanced Materials Research Vols. 44-46
(SP : FR → DP) ⇔ (SP~DM)
DM ij =
425
(5)
FRi
∀i ∈ [1,L , m ] , ∀j ∈ [1,L , n ]
DPj
(6)
In Eq. (4), the relation λsatisfy would be binary, because it associates a problem solving model with
the current explicit function requirement FR. Eq. (5) denotes that the mapping from FR to DP is
supported by SP, and the construction of DM is guided by SP.
Based on the constructed DM, axiomatic design can evaluate the solution quality according to the
Independence Axiom from the function point of view, but cannot guarantee the design technical
feasibility from the physical synthesis point of view. Clearly, a system or product is made of a number
of modules. Each module is pieced from a set of DPs, which is a key design decision to be taken for
physical design synthesis. A design parameter set forming a design module may have elements that
belong to different design mappings in several hierarchical levels. Feasibility thinking should guard
this synthesis activity [16]. Design feasibility here has two aspects: the first is that the set of design
parameters constituting the module should be able to deliver the FRs hosted in the same the module.
Second, the relations or interactions among design parameters shall be feasible for modular design
and the module itself should be feasible as to good properties (such as durability, safety,
manufacturability, assembly, cost, etc.), which means that the constituent DPs module can be
synthesized in a manner that meets the required properties. To check the design technical feasibility,
the relations among DPs would be the main focus to be considered and DSM subsequently is capable
to deal with the description of interactions between different DPs (see Eq. (7)).
DSM ij =
DPi
∀i , j ∈ [1,L , n ]
DPj
(7)
It has been noted that for a new product design or conceptual design stage, it is difficult to make a
DSM directly as the interaction relation among DPs is vague. Dong and Whiley [3] and Marin and
Barker [4] have addressed the conversion of DM to DSM, which can be defined by
T : DM → DSM or DSM=T(DM)
(8)
The conversion operator T includes following operations: (i) In each row of the DM chose the
dominant entry (X0 in DM) as output variable, (ii) Construct a composite matrix to describe the
equation relationship between DPs and FRs , (iii) Permute the composite matrix by exchanging rows
and columns so that all dominant entries appear on the main diagonal and get the derived DSM. A
simple example in Fig. 5 can illustrate such a conversion process.
Construct DM
DP1
FR1
X
FR2
X
DP2
Choose dominant DP for each FR
DP3
X
X
FR3
X
DP1
FR1
FR2
X
DP2
X0
X
X0
FR3
DP3
X
X
X0
DP3=f(FR1,DP1)
DP1=f(FR2,DP2)
DP2=f(FR3,DP3)
Get derived DSM
DP1 DP2
DP1
X
X
DP2
DP3
X
Construct CM(Composite Matrix)
DP3
X
FR2
DP1
X
FR3
DP2
X
FR1
DP3
DP1 DP2
DP3
X0
X
X0
X
X0
Fig. 5: Conversion of DM to DSM
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Materials and Product Technologies
When a DSM has been derived, the physical structure could be built with the synthesis of DPs in
preparation for concept selection. Meanwhile, based on the derived DSM, the product property (PP)
can be predicted from related property knowledge. Denoting the property knowledge K p and the
property knowledge domain Γ , we have
∃K p ⊆ Γ : DSM → PP .
(9)
With the derived DSM and predicted PP, the product structure solution can be evaluated whether it
can meet (i.e. are close enough to) the design constraints (DC) such as component compatibility,
manufacturability, cost, safety, etc., which can be described as
λconflict ( PP,DC) ⇒ ¬λspecify ( FR , CR ) ∨ ¬λsatisfy ( SP , FR ) .
(10)
If there are any conflicts, the former design content needs to be improved or refined. It is necessary
to distinguish the function requirements FR from the design constraints DC. Function requirements
would be those statements demanding the explicit presence of a particular feature, whereas design
constraints are typically the conditions that must not be explicitly violated by a design solution.
The DSM-based resulting design may affect the functionality of the systems, for example,
grouping design parameters or integrating components together may couple functions. If that is the
case, function requirements may need to change or more design parameters may need to be added.
System Development
The framework of the AD-DSM integration design system is shown in Fig. 6. The top-level
components within the system are FR definition, DM construction, DSM deriving, DSM-based
structural synthesis and design technical feasibility check. The DM construction is guided by the
repository of solution principles, allowing the user to consult the FR-DP mapping knowledge.
The process of solution principle consulting for FR-DP napping is as follows: (1) Designer inputs
a desired function requirement in the form of verb attributes; (2) The design system searches the
domain of solution principles for possible solutions according to the given verb and attributes; (3) The
design system outputs related solution principles, and designer could visit the introduction of each
corresponding SP which refers to a possible conceptual design solution. Fig. 7 shows a screenshot of
solution principle consulting for FR-DP mapping. The axiomatic design suggests the zigzag process,
while it is designer’s job to perform the identification of sustainable DPs for specified FRs.
Meanwhile, the designer can have the chance of selecting the optimum SP from a set of solution
alternatives, which can potentially lead to the emergence of products of high-quality, and in redesign,
designers are able to substitute better SPs for those old unsatisfactory ones. With the consulting aid of
solution principles, the designer will be helped to identify most suitable means to carry the functions
and generate most desirable design concepts and construct the design matrix (DM). Each diagonal
element in DM may hint at a SP.
Solution Principles
SP domain Θ
Customer Requirements CR
FRs Definition
SP Repository
DM Construction
Derived DSM
Technical requirements
DSM synthesis
Design constraints
Product performances
……
Past DSMs
Design Cases
Technical feasibility check
Knowledge Repository Γ
Fig. 6: AD-DSM integration design system
Advanced Materials Research Vols. 44-46
427
As stated before in this paper, DM serves to define the manner in which the DPs will satisfy the
FRs. However, the effects of decisions relating to the product such as cost, capacity, and physical
integration, are not dealt with particularly well. The architecture of the product is defined not only by
the decomposition of the complete product into elemental DPs or components, but also by the
interactions between these DPs or components. The interactions may include well-specified
interfaces and undesired or incidental interactions. As a result, a DSM structure needs to be generated
to accommodate these issues and find undesired interactions. Therefore the conversation or derivation
of a DM to a DSM shall be conducted. The derived DSM may need to be re-engineered (so called
RDSM) to find feedback loops of interactions which could reveal the physical or capacity problems.
With result of RDSM, the original DM may be backtracked and modified to a refined one (called
RDM). The developed system can support a designer to perform such conversion process (see Fig. 8).
It is noted that a product has various kinds of DSM according to different domains such as physical
connection, material, cost, layout, etc. The derived DSM from DM at the conceptual design phase is
not totally equal to real final DSMs of designed product, and it could act as a base to deduce real final
DSMs of different domains.
Fig. 7: Solution principles for FR-DP mapping
Fig. 8: Interaction of DM and DSM
Conclusions
Conceptual design problems are usually complex and involve various functional requirements and a
large number of potential means as solutions. Axiomatic Design suggests the zigzag mapping process
to decompose the function requirement and provides two axioms to judge the quality of the function
decomposing. Like many tools, axiomatic design has been applied in a variety of areas, as researchers
and practitioners have sought to leverage its advantages. Along the way, however, its practicability
has been challenged. Under the axiomatic design context, the designer is not able to master the
interactions amongst the design parameters, including geometry, spatial layout, interfaces (e.g. logical
and physical connectivity) which will decide the quality of structural design. DSM has been proved
that it can enhance axiomatic design on such issue. Therefore, this paper has investigated the logic of
integration between axiomatic design and DSM. A computer aided conceptual design system has
been developed to realize the proposed integration model of axiomatic design and DSM. Further
practical use cases to evaluate and validate the proposed idea and the developed prototype form the
scope of our future work.
Acknowledgment
This research was supported by NSFC (Nature and Science Foundation of China) research grants
under projects no. 50505017 and no. 50775111.
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