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Identifying Controlling Features of
Engineering Design Iteration

Robert P. Smith
Steven D. Eppinger

Revised September 19S2
WP #3348-9 1-MS






Next revision: January 1993.
Please write to address below after that date for reprints.

Identifying Controlling Features of
En^neering Design Iteration

Robert P. Smith
Steven D. Eppinger

Revised September 1992
WP #3348-9 1-MS


This research is funded by General Motors and by the Leaders for Manufacturing
Progi-am, a partnership involving eleven major U.S. manufacturing firms and
M.I.T.'s schools of engineering and management. The authors are also grateful
to Dan Whitney, Marde Tyre, Karl Ulrich, and two anonymous reviewers from
Management Science who have provided helpful and insightful comments on
earlier versions of this paper.

Send correspondence to:

Prof. Steven D. Eppinger

M.I.T. Sloan School of Management

30 Wadsworth Street, E53-347

Cambridge, Mass. 02139

'OV 1 3 1992


Engineering design generally involves a very complex set of relationships among
a large number of coupled elements. It is this complex coupling that leads to
iteration among the various engineering tasks in a large project. The Design
Structure Matrix (DSM) is useful in identifying where iteration is necessary. The
Work Transformation Model developed in this paper is a powerful extension of the
DSM method which can predict slow and rapid iteration within a project, and
predict those features of the design problem which will require many iterations to
reach a technical solution. This model is applied to an automotive brake system
development process in order to illustrate the model's utility in describing the
main features of an actual design process.


The goal of this work is to develop a modeling framework which is useful
for describing engineering design iteration. The framework is applied to brake
system design to illustrate its utility in understanding the engineering design

Engineering design is the process whereby a technical solution is created to
solve a given problem. There have been several attempts to give formal structure
to the design process, such as those of Sub [1990], Pahl and Beitz [1988] and
Alexander [1964]. This stream of research characterizes good design practice in
general terms, but does not describe what makes some design problems more
difficult than others. We intend to further the development of design process
modeling by providing richness to the descriptions of design procedures and
strategies which will enable a design organization to identify the difficult portions
of their particular design problem. Strategies can then be developed to facilitate
the effective execution of these difficult aspects.

The Design Structure Matrix (DSM) serves as the basis for our formal
analysis and will be briefly reviewed in this section. (For a more detailed overview
of the DSM method the reader is referred to Steward [1981] and Eppinger et al.

[1991].) The work herein extends the analytical method, and demonstrates the
utility of our framework for the management of engineering projects.

The philosophy of the DSM method is that the design project is divided into
individual tasks, and the relationships among these tasks can be analyzed to
identify the underlying structure of the project. It has been suggested that
studying the relationships between individual design tasks can improve the
overall design process, and is a powerful way to analyze alternative design
strategies [von Hippel 1990]. Earlier work developed a modeling formalism which
shows how different aspects of a design problem are related [Alexander 1964].
Alexander describes a graphical technique where the functional needs of the
technology are nodes, and interactions between tlie needs are arcs. His idea is to
segment the graph into subsections which have relatively few interactions which
cross boundaries. These graph segmentations give rise to technical subsystems
which should separate the technical needs into independently solvable problems.

The DSM method is similar to Alexander's technique, but the nodes are
now specific design tasks and the arcs are directed and indicate information flows
between tasks. The nodes in the graph are arranged in a square matrix where
each row and its corresponding colunm are identified with one of the tasks.
Along each row, the marks indicate from which other tasks the given task
requires input. Reading down each column indicates which other tasks receive
its output. Diagonal elements do not convey any meaning at this point, since a
task cannot depend upon its own completion. For example, in Figure 1 (based on
a simplified view of camera body design), task C requires input from tasks B, D, E
and F, task B requires input only fi"om task A, and task A needs no input to begin.

A Set Specifications

B Design Concept

C Design Shutter Mechanism

D Design Viewfinder

E Design Camera Body

F Design Film Mechanism

G Design Lens Optics

H Design Lens Housing






:y)x X X


X X x(X



Figure 1. Sample Design Structure Matrix

The DSM can be used to identify orderings of tasks and to identify difficult
aspects of the design process. Some or all of the elements of the matrix can be
made sub-diagonal (such as those corresponding to tasks A, B, G, and H in
Figure 1) by reordering the tasks of the matrix using a partitioning algorithm
[Steward 1981, Gebala and Eppinger 1991]. An entirely sub-diagonal matrix
indicates that there exists a sequence where all tasks can be completed with all
input information available. Such a sequence may contain both tasks which must
be done in series, or tasks which may be done in parallel. The information in a
sub-diagonal design matrix is then similar to that expressed in a CPM (Critical
Path Method) or PERT (Program Evaluation and Review Technique) chart.

More typically, due to the complexity in engineering design, the matrix
cannot be reordered to have all matrix elements sub-diagonal (such as tasks C-F
in Figure 1.) In these cases there is a cyclic flow of information in the design
process and standard CPM/PERT techniques are not applicable because of the
presence of such cycles. Likewise, a sequential progression of the design tasks is
not possible. Tasks where neither a purely sequential nor a parallel ordering is
feasible are coupled in such a way that some alternative process for resolving the
design interactions (such as iteration or negotiation) must be used. For this

reason, iteration is a typical feature of engineering design projects [Hubka 1980].
The sub-matrix in Figiare 1 depicts a design problem defined such that the tasks
are sufficiently complex and interrelated so that iteration will be necessary to
complete the tasks.

There is an established set of models which allow looping within a PERT
modeling framework. This set of models is known as GERT, for General
Evaluation and Review Technique. Direct analysis of any but a simple GERT
network rapidly becomes unwieldy, so simulation is typically used to evaluate a
project. (Taylor and Moore [1980] discuss the application of GERT to R&D
projects.) It is the intention of this modeling effort to provide an analytically
tractable model of the design iteration process, even for large projects. It is hoped
that by preserving tractability it will be possible to observe the relationship between
the structiu-e of the problem and the development time of the project. Because
GERT relies on simulation for large projects, it is difficult to discern this

For our purposes, we assume that the tasks and interrelationships of a
design problem are known and unchangeable during the course of the project.
This assumption is reasonable for a firm is working on a design project in an
area in which they have a significant degree of familiarity (the example of brake
system design at General Motors, which serves as the basis for the application
described in this paper, fits this category). The assumption is less true for a
completely new or rapidly evolving technology.

There is evidence that some companies who are faced with the same design
problem choose differing design strategies, which implies a different underlying
design matrix. For example, to what extent they choose to work on tasks in series
or parallel affects development time significantly [Clark and Fujimoto 1991].

Development time is an important measure in engineering design
management. We believe that complex iteration is a major source of extended
development time. While the Design Structure Method is a useful tool to identify
the coupled blocks in which the complex iteration occurs, this work is intended to
characterize how such iteration occurs.

If we include task durations in the DSM, we can use this description to
estimate the total duration of the project. Series tasks can be evaluated by
summing their individual times, and parallel tasks can be evaluated by finding
the maximum of those task times. For the project characterized by the DSM in
Figure 1, if the task time are a, b, c, ... , h, the time of the camera design project
would be

a + b + max{ f(c,d.e,f) , g+h }
where f(-) is a function, undefined as yet, corresponding to the development time
for the coupled block.

The model presented in this paper illustrates how iteration time can be
evaluated for such a coupled block of tasks, and shows that the critical features
controlling the iteration can be identified. Each critical feature is a group of
parameters of the design solution which are strongly dependent on each other;
they may require many iterations to converge, as a set, to conform to design
constraints. We illustrate these concepts using a brake system design example.
The critical features in brake system design are important determinants of
product qvudity, and we believe that critical features which are strongly related to
both time and quality are typical of engineering design.

Our Approach

We believe that it is possible to lessen development time by analyzing and
restructuring the design process. We have developed extensions to the DSM

framework which have allowed us to suggest ways that the design process can be
restructured. A previous interpretation of the quantitative DSM developed a
probabilistic model of engineering design which predicts development time for a
sequential design iteration process [Smith and Eppinger 1991], but that model has
proven difficult to apply to actual design projects. The model presented in this
paper is a different interpretation of quantitative information in the DSM, known
as the Work Transformation Matrix (WTM), and is described below.

Our field work is based on extended exposure to the brake system design
engineering organization the brake system design division of General Motors.
Our observations include informal discussions with systems and component
engineers, internal documentation, and interviews with engineers and their
managers. We have found the brake system to provide a good subject for modeling
of the design process because of the nature of the design problem. Brake system
design is stable in that the technology and the market are mature and the form of
the base product is not undergoing radical change. The brake system design
engineers have considerable experience with brake system design. These factors
suggest that the data contained within the brake system DSM is not changing
rapidly, and the knowledge which is represented within the DSM is well

Ours is a descriptive model, not an optimization model. The description
developed below can be used by the design manager to analyze the design problem,
and to estimate how long the design process will take, and what aspects of the
design problem contribute to iteration time.

The novelty in this model is in the application of matrix mathematics to
analyze development time of an iterative design process. The model relies on
standard linear algebra results. The interpretation of the relationship between
the matrix mathematics and development time is novel.

Design Iteration Model Development

For the purposes of our analysis, we assume that each task creates a
deterministic amount of rework for other tasks. Rework is the work which is
necessary because the task originally was attempted with imperfect information
(assumptions). The rework adapts the original solution to account for the
modified information. Rework is measured in percent of the time that it takes to
complete the task in the original iteration.

We use a transformed version of a fully coupled Design Structure Matrix
which we call the Work Transformation Matrix (WTM). There are two types of
information in a Work Transformation Matrix. The off-diagonal elements
represent strength-of-dependence measures (defined in next section). (See Figure
2a.) The diagonal elements in the WTM represent the time that it takes to
complete each task during the first iteration stage. (See Figure 2b.) It is assumed
that there will be multiple iteration stages, and that the time for each stage is a
function of the amount of time spent working in the previous stage. We wish to
find the sum of the times of all stages.





(a) Strength of Dependence Measures (b) Task Times

Figure 2. Woi^ Transformation Matrix

The derivation below is divided into three sections. In the first section we
describe the assumptions underlying the model. In the second section we
describe why the eigenvalues and eigenvectors of the WTM are relevant to our
analysis of development time. In the third section we describe how the
eigenstructure of the matrix is interpreted. Following the derivation of the model,


we illustrate the analytical process using a simple example. Finally, we present
and analyze the WTM which describes brake system design.

A: Work Transformation Model Assxunptions

The assumptions in this model are:

• All tasks are done in every stage - fully parallel iteration

• Rework is created based on a linear rule - as a % of work done in
previous iteration stage

• The parameters in the matrix describing work transformation behavior
do not vary with time

These assumptions allow us to use a linear algebraic analytical method on the


To develop the model, we first introduce the concept of the work vector u..

This is an n-vector, where n is the number of design tasks to be completed. Each

element of the work vector contains the amount of work to be done on each task
after iteration stage t. The initial work vector Ug is a vector of ones, which

indicates that all of the work remains to be completed on every task at the
beginning of the iteration process.

During each iteration stage all work is completed on all of the design tasks.
(For a relaxation of this assumption, where a fraction of the work is completed in
every stage see Appendix 4. A.) However, work on a task will cause some rework
to be created for all other tasks which are dependent on that task for information.
We determine which tasks those are fi-om the design structure matrix. Every
iteration stage produces a change in the work vector according to:

Ut+1 = Au,
where each of the entries a-, in A implies that doing one unit of work on design
task j creates a units of rework for design task i. The matrix A is then the

strength of dependencies portion of the WTM (Figure 2a). The diagonal entries
are set to zero. The work vector u can be also be expressed by:

u, = a'uq
The sum of each of the work vectors is the total work vector U, the total
number of times that each of the tasks is attempted during the total of T iteration
stages of design process:


u = X^t



U = j^A\


which can be rewritten as:

^'(1 +

The model output U is therefore in units of the original amount of work
done on each task in the first iteration stage. (If element i in vector U is 1.6, then j
the design organization will have done 60% rework on task i in subsequent
stages.) For a time-based interpretation of the matrix A see Appendix 4.B. For
now, we scale U by the task durations to obtain units of task times. If W is a
matrix which contains the task times along its diagonal (See Figure 2b), then WU
is a vector which contains the amount of time (in engineer-hours) that each task
will require during the first T iteration stages.

B: Eligenvalue Deoompositioii

If A has linearly independent eigenvectors (the eigenvector matrix S is
invertible) then we can decompose A into:


A = SAS"""
where A is a diagonal matrix of the eigenvalues of A, and S is the corresponding
eigenvector matrix. (For S to be inveri;ible it is sufficient, but not necessary, that
none of the eigenvalues be repeated.) The powers of A can be found by:

a' = sa's"''

The total work vector U can therefore be expressed as:

U = S


vt=o y


If the magnitude of the maximum eigenvalue is less than one, then the
design process will converge (i.e. as T increases to infinity the total work vector U
remains bounded.) An eigenvalue greater than one corresponds to a design
process where doing one unit of work at some task during an iteration stage will
create more than one imit of work for itself at some future stage. Such a system is
unstable and the vector U will not converge, instead growing without bound as T
increases. (It is a sufficient, but not necessary, condition for stability that the
entries in every row sum to less than one.)

A design process which does not converge would be one where there is no
technically feasible solution to the given specifications, or one where the designers
are not willing to compromise to find the technical solution. This situation is not
likely in the design environments we are modeling, that is, routine design where
the designers are responsible for bringing out a new variation of a known,
technically successful product. The remainder of the discussion on Work
Transformation Matrices is limited to problems where a technical solution exists
and can be found in finite time (i.e. eigenvalues are less than 1.)


C: Interpreting the E^genstructure

The eigenvalues and eigenvectors of matrix A determine the rate and
nature of the convergence of the design process. Much can be learned about what
controls the iteration by looking at the eigenvalues and eigenvectors as opposed to
looking at the sequence of work vectors.

A design mode is defined as a group of design tasks which are very closely
related, and working on any one of them creates significant work, directly or
indirectly, for each of the other tasks within the mode. We use the eigenvalues
and eigenvectors of matrix A to identify the design modes.

The magnitude of each eigenvalue of A identifies the rate of convergence of
each design mode. The eigenvector corresponding to each eigenvalue
characterizes the relative contribution of each of the various tasks to the body of
work which converges, as a group, at a given rate.^

By the Perron-Frobenius Theorem (a fundamental result of matrix theory)
we know that the largest magnitude eigenvalue of a coupled non-negative matrix
will be real and positive [Marcus and Mine 1964]. Also, the eigenvector associated
with this eigenvalue will have positive elements.

The slowest design mode (largest eigenvalue) will therefore have an
eigenvector which is strictly positive. This design mode gives us little problem
with interpretation. Other design modes are, however, less obvious. Also by the
Perron-Frobenius Theorem, there is only one eigenvector which is strictly
positive. We must be able to interpret negative and complex nimibers in the
eigenvectors as well as negative and complex eigenvalues.

^ The interpretation of the eigenvalues and eigenvectors for design problems is similar to

the eigen structure analysis used to examine the dynamic motion of a physical system. In the
discrete time description of linear dynamic systems, each eigenvalue corresponds to a rate of
convergence of one of the modes of the system (a natural frequency.) The eigenvectors identify the
mode shapes of natural motion, quantifying the participation of each of the state variables in each
mode [Ogata 1967].


Recalling that the total work vector U is calculated by:

U = S X^' S'''uo

vt=o y
we will look at each of the elements in the above formula for U to see how the
eigenstructure of matrix A can be used to interpret the design modes. If we take
the limit as T approaches infinity we can use the formula:


lim Xa' = (I-A)'^

If the maximum eigenvalue is not close to one, then the limit will be approached
within relatively few iterations. For the remainder of this discussion the limit
will be used, although the analysis can also be completed for finitely many

This limit is also a diagonal matrix, where each entry along the diagonal
corresponds to one eigenvalue and has the form:


1 -X

where X is an eigenvalue.

In the next two subsections, both real and complex eigenvalues are
discussed. In subsection the interpretation of eigenvectors is considered.

Real Eigenvalues

The function:

1 -X

is strictly increasing over (-1,1). The graph of this function is shown in Figure 3.


-0.8 -0.6








Figure 3. Graph of Magnitude vs. X for Real Eligenvalues

We see that all positive eigenvalues have the greater contribution to the series
sum than do negative eigenvalues. Therefore, as we consider which are the more
important design modes, we restrict our attention among real eigenvalues to the
positive eigenvalues.

Complex Eigenvalues

For complex eigenvalues we also wish to find the magnitude of the limit of
the sum of the infinite series. For a complex eigenvalue a + pi the magnitude of
the limit is:


Or, alternatively:

1 - (a -H pi)



V(1-a)^ + P^

1 -(a-^po


Which, using the fact that:


allows us to find an upper bound on the limit:


1 -(a + pi)

1 -a

Also, we can find a lower bound using the fact that:

a^ + p^ < 1

to show:


1 - (a + pi)


The graph of the upper and lower bounds is shown in Figure 4.











Figure 4 Graph of Bounds on Magnitude vs. a for Complex Eigenvalues

We see that the real part of complex eigenvalues gives bounds on the magnitude of
the siun of the infinite series corresponding to that eigenvalue. We also see that
complex eigenvalues with negative real part are not going to contribute
significantly to the sum, and can therefore be ignored.


By the previous argviment we need only consider those real eigenvalues
which are positive. We therefore need to consider only those eigenvalues which
have a positive real component, whether they are real or complex.

The Eligenvectors

This section discusses how the relative importance of each task within an
eigenvectors is interpreted, given that we know the eigenvalue corresponding to
that eigenvector. We want to be able to interpret the eigenvectors so that we can
distinguish which of the tasks are importsmt contributors to each design mode.

Again, consider the formula:

f T \


u = s


Vt=0 J

We see that the final two terms in this formula:


give a weight for each eigenvector which is both a magnitude and a direction.

The eigenvector corresponding to real eigenvalues is real. Each weight for
a real eigenvector is also real. Therefore, the direction is either positive or
negative. The important quantities in a real eigenvector are therefore the large
positive values if the weight is positive, and large negative values if the weight is

Complex eigenvalues have complex eigenvectors and complex weights.
Determining how the direction of the weight and the direction of the eigenvector
interact is difficult. The best way to look at the interaction is to calculate the
contribution of the mode to the total work vector U and see which the tasks give
large contribution to the total work.

Positive eigenvalues correspond to non-oscillatory design modes. Negative
and complex eigenvalues describe damped oscillations. Oscillatory design modes

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Online LibraryRobert P SmithIdentifying controlling features of engineering design iteration → online text (page 1 of 3)