3.4 PER MILLION
Axiomatic Design and DFSS
by Douglas P. Mader
In the last few years, design for Six Sigma (DFSS) has gained much popularity due to the widespread application of Six Sigma principles outside the operations environment, particularly in new product development.
As organizations realized the structured use of quantitative tools has a large financial impact due to increased efficiency, they began to look for ways to apply the same principles in other areas of the organization. Since the examination of performance relative to functional requirements is a key activity in DFSS, axiomatic design (AD) has received much recent attention.
In his pioneering book The Principles of Axiomatic Design, Nam Suh formalizes several design methodologies and collectively refers to them as AD.1 AD is a structured approach to performance improvement for complex systems that is useful in the concept, preliminary and detail design stages of the systems engineering model.
Suh provides a structured and analytical method to link functional requirements for systems to product and process design parameters. AD further provides a quantitative method to assess the sensitivity of certain functional requirements to changes in design parameters.
Axiomatic Design Context
Suh presents what he considers the basic principles of design. He also attempts to provide a coherent structure for the examination and optimization of functional requirements (FRs) relative to design parameters (DPs). The precept is that customers will state a design objective in the functional domain, and the physical solution will be manifested by choices of DPs in the physical domain.
FRs are established in a solution neutral environment and reflect the desired performance of the system. DPs are physical or other variables that can be selected by the designer to achieve the desired FRs. DPs may also represent noise factors the designer must compensate for with regard to system performance.
An important consideration in the design process is that FRs and DPs can be decomposed into hierarchies. The design process is accomplished by linking these two domains via the hierarchy for the system that is selected in the concept design phase.
A typical decomposition of a design hierarchy includes the definition of requirements, selection of outputs and choice of inputs for the system, subsystem, subassembly, component and process levels (see Figure 1). FRs at the ith level of the hierarchy cannot be decomposed into the next level of the hierarchy without first developing a solution that satisfies the ith level functional requirements in the physical domain for all corresponding inputs. Suh proposes two main axioms that govern the iterative nature of the design process:
- Axiom one (independence axiom): The mapping of FRs to DPs should be one to one and independent. In engineering terms, this means we wish to map m FRs to m X’s such that a change to any X only affects the associated FR.
- Axiom two
(information axiom): This means we wish to minimize the information content of the
design. Among all the designs that could satisfy the FRs, the design with the least
information content is the best.
The intent of axiom one is to make the system easy to produce and use by ensuring a change to a single DP does not affect more than a single FR. This type of design is said to be uncoupled. If a change to a single DP affects all the FRs, then the design is difficult to use or produce and is said to be coupled.
By examining the hierarchy of FRs and DPs for a given system, the designer can attempt to minimize the sensitivity of the system to changes in the DPs by striving for an uncoupled design in which changes to individual DPs affect only subsets of FRs. For uncoupled designs, the designer can typically adjust the DPs to optimize the FRs by adjusting them in a particular order.
The outputs of any design process consist of information in the form of drawings, specifications, tolerances on the DPs and other relevant knowledge required to produce and use the physical system. The information content involved in the design is a direct measure of design complexity.
The intention of axiom two is to create as simple a solution as possible so the design can be conveyed and produced with minimal effort. Suh says the second axiom is difficult to quantify, so we should simply attempt to maximize the probability of meeting the functional requirements. The second axiom is completely consistent with DFSS principles because the maximization of process and design capability for key variables has always been a goal.
Transfer Function Modeling
Suh proposes the following design equation as the basis for examining the hierarchy of FRs and DPs. His design equation is simply a first order Taylor Series approximation in which we can attempt to predict the changes to the FRs if we know the changes to the DPs.
To create the design equation, we must have a differentiable function that relates the FRs to the DPs. Such functions are typically developed theoretically using knowledge of the system or empirically using designed experiments and linear regression techniques. Suh’s design equation is:
This equation represents a mapping of the DPs to the FRs by way of the m x n design matrix A. The Aij must be evaluated at the specific design point in the physical space. In nonlinear cases, the Aij vary with both the FRs and DPs. If m � n in the design equation then, by definition, the design is either coupled or we have redundant FRs.
While this design equation is interesting and may be useful for some systems, it has two inherent limitations:
1. It is often difficult to estimate all the Aij. A fundamental tenet of DFSS is the estimation of transfer functions relating X’s (DPs in AD) to Y’s (FRs in AD).
As those who are familiar with DFSS will likely attest, much effort is involved in determining which transfer functions to estimate and which to approach subjectively. We use screening experiments to select the vital few DPs, steepest ascent or other optimization procedures to zero in on the optimal condition for the design and second order or robust designs to estimate transfer functions for the key DPs and FRs.
Once these steps have been taken, the design can be further refined using nonlinear optimization, and tolerances on the DPs can be established for the key DPs and FRs using Monte Carlo simulation. At no point in DFSS do we attempt to estimate every possible transfer function.
2. If the transfer functions are known via experimental design techniques, Monte Carlo simulation is easier to use than calculus to estimate performance to the FRs.
Axiomatic Design as a DFSS Tool
What clever people have done to make use of Suh’s design equation to examine the hierarchy of FRs and DPs without calculus is to simply accept the elements in the matrix A do not have to be known or estimated. Sometimes it is sufficient to simply use 1s and 0s to evaluate the linkage of FRs to DPs.
For example, if an FR changes as a result of a perturbation to some DP, then the corresponding cell in A must have a 1 or a 0. This approach will yield a table that should remind many in the DFSS community of the XY matrix, a common tool used to subjectively assess the relationship between system inputs and outputs.
As a simple example, let’s consider an LCD projector. If we consider a design that involves only one lens with a single adjustment for that lens, then optics dictates we cannot adjust image size and focus at the same time. The resulting design would be completely coupled:
However, if we use a double lens system that has two adjustments, we can set image size and focus independently without having to move the projector relative to the screen. The resulting design would be uncoupled:
To generalize the above approach for any design scenario, several sets of parameters should be considered: the system inputs (X’s), which are labeled as DPs in the AD notation, the specifications for the X’s, the system outputs (Y’s), the specifications for the Y’s and the relationship of the X’s and Y’s to the FRs. We will introduce two changes to Suh’s structure:
- The FRs will represent the internal or external customer requirements.
- Measurable system outputs or Y’s will be an intermediate quantity between the X’s and the FRs.
The actual value of Y accumulated for many observations will allow us to estimate distributional parameters and thereby the probability of conformance to specification for the FR.
Therefore, a more comprehensive approach might be to examine the hierarchy by decomposing the relationships without having to estimate all the transfer functions. The following 13-step requirements decomposition process can be used along with XY matrix analysis in most cases:
- Determine the functional re-quirements (FRs).
- Brainstorm a list of feasible design concepts.
- Rank the design concepts against the FRs.
- Brainstorm all measurable system outputs (Y’s) for the best concept.
- Examine the mapping of Y’s to FRs, and determine if the level of coupling is acceptable.
- Postulate target and specification limits for each Y.
- For each FR-Y pair, ensure the target and specification limits provide an acceptable probability of conformance for the FR.
- Brainstorm all system inputs (X’s) for the concept under consideration.
- Examine the mapping of X’s to Y’s, and determine if the level of coupling is acceptable.
- Examine the mapping of X’s to FRs, and determine if the level of coupling is acceptable. (Note this step is similar to Suh’s approach.)
- Postulate target and specification limits for each X.
- For each Y-X pair, ensure the target and specification limits for the X provide an acceptable probability of conformance for the Y.
- For each FR-X pair, ensure the target and specification limits for the X provide acceptable probability of conformance for the FR.
Figure 2 is a visual representation
of these steps.
This procedure helps address the coupling of X’s, Y’s and FRs in the design of systems. It also provides an overall structure for the typical DFSS project in that 13 steps can be expanded to sync with any DFSS methodology, including define, measure, analyze, design and verify (DMADV) and identify, design, optimize and validate (IDOV). The Six Sigma tools can also be employed within the same structure in a manner that facilitates rapid execution of DFSS projects.
- Nam P. Suh, The Principles of Axiomatic Design, Oxford University Press, 1990.
DOUGLAS P. MADER is an international speaker, seminar leader and certified master instructor for Six Sigma and design for Six Sigma. He is the founder and president of SigmaPro Inc., a consulting firm in Fort Collins, CO, that specializes in integrated deployment of Six Sigma, design for Six Sigma and lean systems. Mader earned a doctorate in mechanical engineering from Colorado State University and is a Senior Member of ASQ and the Institute for Industrial Engineers.