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Dewey
Massachusetts
Institute of
Technology
Sloan School of Management
Working Paper
Deciding between Sequential and Parallel
Tasks in Engineering Design
Robert P. Smith
Steven D. Eppinger
October 1995
Working Paper Number 3858
Contact Addresses:
Prof. Robert P.
Smith
Department of Industrial Engineering
University of Washington
Seattle,
A 98195
Prof. Steven D.
W
smith
Eppinger
Sloan School of Management
Massachusetts Institute of Technology
Cambridge,
@ ieng.washington.edu
MA 02142-1347
eppinger@mit.edu
MASSACHUSETTS INSTITUTE
OCT
2 4 1995
LlSnAR.'ES
Abstract
It is
generally desirable to complete design tasks in parallel in order to reduce
However, completing tasks in parallel may
sometimes increase the total amount of rework that must be done, thereby
increasing the total engineering effort, the development cost and the lead time. The
technique described in this paper helps to decide between serial and parallel
the overall development time.
scheduling of multiple tasks in a two-stage design process. Using information about
task interdependencies, this method calculates the amount of time and the amount
of effort (in engineer-weeks) required for any suggested assignment of tasks to the
two stages. The paper suggests an approach for minimizing time, or effort, or both
by adjusting the schedule of which tasks should be completed at which time. The
method is applied to data from a computer workstation design problem.
1.
Introduction
Concurrent engineering has become increasingly important in recent years.
Concurrent engineering
issues simultaneously
is
a philosophy that suggests the
need
to consider
where they may have been considered sequentially
past [Nevins and Whitney 1989,
Susman
1992].
design
in the
The division between sequential
design phases arose historically because one set of design issues (typically product
performance
issues)
were considered
design space was considered
first.
of greater importance,
The secondary
set of
manufacturing process design) were considered only
issues
had been decided ('throw
it
over the
design process has been considered
typically leads to a greater
its
after the initial set of
design
In recent years the sequential
wall'.)
type of design process
cost,
the overall profit generated
and lower
by the design.
broadest definition does not only consider
issues, but
environmental impact, service,
historically neglected in the
portion of the
design issues (typically
inefficient, since this
which lower
performance and process design
reliability,
this
development time, greater development
overall design quality, all of
Concurrent engineering in
and
many
other
testing,
product design phase.
2
life
and so
cycle issues (such as
forth)
which have been
Yet the division between design phases arose for valid reasons.
industries at
some times
there
was
The inclusion
the value be responsible for creating the design.
issues
would
in the
design process.
many people and
lead to too
It
is
difficult to
among
achieve complete consensus
who
have only those
a benefit to
many
too
of
In
some
created most of
any other design
concerns being involved early
organize complex design processes, and to
all
of the players
may
be inefficient and
expensive.
Development time
is
an important determinant of eventual success
in the
market [Blackburn 1991, Smith and Reinertsen 1991]. One of the primary reasons
that concurrent engineering has
concurrent engineering
product and bring
We believe
environment.
able to lessen the
is
that in
to
most cases
In this
method
The model
is
a
amount
paper
it is
movement
of time
it
is
we
is
takes to develop a
offer a
model which can help engineers and
way
is
appropriate in their design
based on the work transformation matrix
in a
to calculate the
amount
Our
analyze the eigenstructure of the matrix
of
earlier
in
work completed during an
paper describes
how
on the idea of the design structure matrix (DSM) which
are
is
a tool
done during every
effects of not
is
used
and coupled design tasks [Steward 1981, Eppinger and others
earlier
it is
in turn
iteration, the intent of the current
in
every
iteration.
paper
We
is
based
to identify
1994].
paper [Smith and Eppinger 1995b] assumes that
completing every task
possible to
order to determine important properties
about the design process. The work transformation matrix concept
Whereas the
(WTM)
previous paper [Smith and Eppinger 1995bl. The
engineering design iteration process.
serial
because
beneficial to separate the design process
what extent concurrent design
model, which was developed
WTM
a significant
to market.
it
into multiple phases.
managers decide
become
all
tasks
to look at the
demonstrate here
that
the
WTM can also be used
to
examine the
sequentially during a design process,
scheduling of tasks
Design
is
is
which can
assist in
deciding
when
concurrent
appropriate.
complex process of many
a
doing tasks concurrently or
effect of
interrelated tasks.
How
these tasks are
divided and organized has a significant effect on the eventual outcome and success
of the process [Alexander 1964,
effectively
is
not easy, but
its
von Hippel
success
is
Managing the design process
1990].
important to the long-term health of
technology companies [Whitney 1990, Clark and Fujimoto 1991].
The model
tasks
to
in this
paper addresses the issue of scheduling coupled design
by decomposition. This problem
complex design
several methods.
removed which
projects,
acknowledged
is
and has been addressed
These methods include
result in
'tearing',
to
be of practical importance
in the research literature
where
a set of feedback tasks are
an uncoupled ordering [Steward 1981, Kusiak and
1993]; the sequential iteration
by
Wang
model, where coupled tasks are placed in a sequential
ordering in order to minimize the expected amount of time taken to complete the
entire process [Smith
total length of
and Eppinger
1994a];
and
a
model intended
to
minimize the
feedback chains in coupled multi-disciplinary optimization design
problems [Altus and others
1995].
There are other recent models which address the problem of determining the
appropriate amount of parallelism in the product development process.
model explore the case where
[Hoedemaker and others
level the time taken for
a project
1994].
As
the
is
where
divided into a number of identical tasks
number
of tasks increases
communication between tasks
time for completing the project increases.
One such
An
increases,
alternative
beyond a
and the overall
model presents
tasks have a probabilistic need for repetition [AitSahlia
critical
and others
a case
1995].
Under these conditions
scheduled
in parallel in
it
is
desirable to limit the
order
to
number
of tasks
which are
minimize the cost of completing the design
project.
Both of those papers on the sequential/parallel task scheduling decision rely
on simpler mcxiels of the structure
how
asymmetrical interconnectedness between tasks, and
affects the
that interconnectedness
scheduling of tasks. Another recent paper (Krishnan and others 1995]
models the overlapping (parallelism) of two
development process
to
activities
of the
asymmetry
within a product
determine the optimal overlapping conditions based on the
nature of the information coupling the pair of tasks.
some
They do not explore the
of the design process.
of the Krishnan
Our
current paper retains
model while extending
to the multi-task
two-stage case.
The remainder
Work Transformation
of the paper
is
laid out as follows: Section 2 describes the
Matrix as given in the earlier paper [Smith and Eppinger
1994bJ. Section 3 describes
how
the
WTM can be used to describe a
two-phase design
process. Section 4 looks at the two-phase process as an optimization
suggests
some
heuristic algorithms.
Section 5 applies the
problem and
model and optimization
algorithms from sections 3 and 4 to a computer workstation design problem.
Section 6 describes
how
this
model can be extended
to
more than two-stage
design.
Section 7 contains the discussion and conclusions.
2.
Review and Extension
of the
Work Transformation Matrix Method
The design structure matrix, the precursor of the
WTM,
identifies the tasks
involved in the design process, and shows the ways that information
between tasks [Steward 1981, Eppinger and others
corresponding column
in
1994].
is
transferred
Each row and
the matrix concerns the information flows (inputs
outputs) of a design task. Each off-diagonal entry on the
row
of a given task
and
indicates the other tasks
the
column
of a given task indicates the other tasks to
information.
parallel,
from which that task needs information, and each entry on
The
DSM
method
is
used
which
that task gives
which tasks can be done
to identify
which tasks can be done purely sequentially, and which tasks have
in
cyclic
information flows and therefore require iteration to be completed.
The
WTM
information.
is
The
an extension
to the
DSM
which contains additional numeric
WTM contains two separate sets of matrix data.
There
is
a off-
diagonal matrix which contains the numbers which indicate the amount of rework
done during the
iteration process.
the times for each task. Figure
1
There
is
also a diagonal matrix
shows an example two-task
which contains
WTM containing both
rework and time data. (Because of the complementary structure of the two types of
data
it is
also possible to write both types of data in
A
(a)
Rework matrix
done completely 20% of task
Example
1.
40%
of
A
needs
to
takes 4 units of time to complete,
are linear factors, so doing
40%
doing 20% of rework on task
A
The assumptions made
All tasks are
(b)
done
its
WTM
Every time task
be redone. From Figure
A
generates
A
work redone. Likewise, when
and task B takes 7
of task
B
Time matrix
in Figure 1(a) are interpreted as follows:
completely, task B needs to have
•
A
B
Figure
The data
one matrix.)
1(b)
we
is
task B
is
see task
A
units of time to complete. These
requires 1.6 units of time (=0.4x4),
8%
(=0.4x0.2) of
in constructing the
in every stage.
6
done
rework
and
for task B.
WTM model are as
follows:
Rework performed
•
is
a linear function of the
work done
in the
previous iteration
stage.
The work transformation parameters do not vary with
•
time.
Discussion of the vahdity of these assumptions can be found
in the earlier
paper
[Smith and Eppinger 1995b]. The model described in Section 3 below explicitly
relaxes the
first
assumption above.
Each iteration
much
of each task
vector of
Is,
is
is
characterized by a
worked on
which indicates
first iteration.
that
work
vector
in the t-th iteration.
all
work needs
During each iteration work
to
u,
.
The
how
This vector indicates
work
initial
be done on
all
vector u^
is
a
tasks during the
created for the next iteration according
is
to the linear rule
m,„=Am,
where the matrix
The
the
all
total
work
A
is
the rework matrix from Figure 1(a) above.
amount
of
work done during
total
work
vector
U
is
is
is
U, the
sum
the vector of
amount
= t^i.
of
W
is
(2)
W in order to get the total amount of time
multiplied by
spent on each design task, where
R
the iteration process
vectors.
U
The
(1)
a diagonal matrix of task times (Figure 1(b)).
of time spent
on each task during the
iteration process.
R = WU = WY,u,
(3)
(=0
which can also be written as
R = wf^A'u„
(4)
(=0
or,
if
the
maximum
matrices, see Smith
can be simplified to
eigenvalue of
A
is
less
and Eppinger [1995b]
than
1
(these are
known
as stable
for a discussion of this issue), equation (4)
R = W{I-A)-'u,
The
total
(5)
time spent to complete the design process
The time spent on each
for each iteration stage.
T
is
the
sum
of the times
iteration stage is the longest time
taken for any task in that stage.
T = £max[WM,f
where
[]*'
the f-th element of the vector within the brackets.
is
Under worst-case conditions T may be
is
(6)
no closed-form expression equivalent
The convergence
calculated explicitly.
magnitude of the
eigenvalue of
A
maximum
is
a difficult quantity to calculate. There
summation must be
to (5) for T; the
rate of the infinite series
(positive) eigenvalue of A.
sufficiently less than unity
it is
If
the
is
controlled
by the
maximum
possible to approximate
T
to
reasonable accuracy using only a few terms of the infinite series.
Effort
amount
the total
is
the design process.
It is
the
sum
of engineering time (in engineer-weeks) spent
of
all
the time spent
on the individual design
£ = IR'"
on
tasks.
(7)
1=1
The quantities time
the design process.
while effort
is
Time
(T)
is
and
effort (£) are
important quantities in managing
an important determining
an indicator of the development
cost.
factor of time-to-market,
Formulas
in calculating these quantities for the fully parallel case.
the concept of a serial structure of the process
to calculate time
3.
Using the
and
effort
(6)
and
(7)
are useful
The next section introduces
and develops the formulas necessary
under those circumstances.
WTM to Describe Two-Phase Design Processes
We now
work on some
assume
that the coupled design process
is
to
be structured such that
of the coupled tasks can be delayed until later in the process.
WTM can be extended to consider multiple-phase design processes.
8
The
The simplest
version (and the one which
the design process,
and
more than two design
on
It is
two
that there are
(See Section 6 for including
sets of tasks.
During the
phases.)
a limited set of tasks.
tasks, including
explored here) assumes that there are two phases to
is
first
phase,
all
During the second phase, work
rework of the
and
first set
all
of the
of the
is
work
is
completed
completed on both sets of
work on
the remaining tasks.
possible to interpret a two-phase design process as corresponding to a
product design phase and a process design phase, but the model need not be limited
to that case.
where
The example given
the phases are product
Using the
WTM
in Section 5 discusses a
two-phase design process
and process design.
method,
it
is
possible to calculate the design time and the
design effort for a two-phase design process. The calculations are similar in
the calculation of time
Section
effort required
•
compare
The
on the
in the
design process described in
WTM do
amount
of time
and
assumptions:
not change as tasks switch from
first-
to second-
(or vice versa).
quality of final product
assignment of tasks
produced by the design process
assumption
constraints
is
reasonable
if
may
independent of
will not
not always represent design practice. The
the data are indicative of fundamental technical
and relationships between
dependency
is
to phases.
These assumptions are strong and
first
fully parallel
different serial divisions for the
we must make some
The parameters
phase
effort
2.
In order to
•
and
spirit to
tasks, in
which case the strength
of
change a great deal whether or not the sequence changes.
nature of the tasks changes
parameters remain constant
when an
is
ordering changes, then assuming that the
not realistic.
If
the
Both assumptions require that the information needs between the tasks be
order-independent, in the sense that the types of inputs
The work required
by a
to
task should not
and outputs are comparable.
complete a task and the type of output information produced
change depending on the order of the
For example, suppose that an engineer
a fan necessary to cool electronic
is
components
tasks.
attempting to determine the size of
in a
A
computer.
separate engineer
is
determining the size of the power supply. Both of these are routine design choices,
however these
tasks are coupled.
will be chosen after,
process used to
If
the
power supply
make
Whether
a selection of size of either
a parallel
either design task
specified
and then the power supply may have
coming
design structure
the
first,
work
is
that
to
then the fan
first,
be changed. The
component involves
the total load (thermal or power) of other elements
device.
is
calculating
and choosing an appropriate
chosen, or a sequential structure with
must be completed during any one
undertaking of a task would not change, the amount of rework created for the other
task
would not change, and the
independent
would not change. These
is
exemplified by two tasks such as choosing a material for
computer case and choosing the manufacturing process used
These are coupled
first.
are order-
tasks.
The converse case
a
final result
tasks, yet
we have
the opportunity to
make
to
make
the case.
either design choice
This choice will strongly constrain the range of options available to the other
design task, and therefore affects the amount of time taken and the amount of
rework created on subsequent
iterations.
Given these assumptions,
process.
During the
first
we
These are order-dependent
can build a model of the two-phase design
phase, only the indicated subset of tasks
These tasks create work for each other as described
need
to find a
companion equation
tasks.
to (5)
above
10
is
worked
in Section 2 above.
We
on.
therefore
for the total time vector of the first
We
phase.
need
to consider only a restricted portion of the matrix A.
we mtroduce
calculate the total time vector
specify the portion of
vector of the
first
A which
phase
is
K. is
''
1 if
K
/
=
/
known
is
which were not done
on any
task,
whether
k^^
and the
total
(8)
such that
Jth task is in the first
phase
a
it is
work
is initial
in the first phase.
first
to
Iterative
be completed only on the
rework may need
to
be done
or a second phase task.
R„=W{I-Ar\l-K)u,
The time
each of which
is
for the
two-phase process T
calculated as
shown
is
a
sum
(10)
of the times for the
for the fully parallel
WTM
two phases,
(see section
2).
T = J^max[WKA'Ku,,f^ + J^max[WA' {I - K)u^f
(=0
The
total effort
time
as a division or state of the system.
During the second phase there
tasks
The
phase.
to
otherwise
[0
of matrix
first
used
is
=W(/-K^K)"'Kmo
defined with elements
f
The value
considered during the
which
is:
R,
the matrix
the division matrix K,
order to
In
(=0
'
(U)
'
E of the two-phase process
is
a
sum
of the efforts in each of the
separate phases.
E=
'^(R]-^ + R<;')
r
Using equations
split
between
(11)
and
(12)
we
and second-phase
first-
can find the time and effort for any suggested
tasks.
For any set of coupled tasks, any subset of them
constitute the
first
calculate the time
(12)
=l
may be
considered to
phase of the design process. Given that choice,
and
effort for that division
it is
using equations (11) and
possible to
(12).
It is
desirable to simultaneously minimize both the time and effort for any given
process.
However,
it
is
not necessarily possible to lower both for
11
all
design projects.
Therefore,
we must sometimes
We
the effort, or vice versa.
divisions for
must therefore attempt
to identify a set of
good
any one design matrix, where goodness indicates superior performance
on one or both
4,
attempt to lower the time at the expense of raising
Finding these good divisions
criteria.
the focus of Section
is
4.
Considering the Two-Phase Division as an Optimization Problem
It is
desirable to lower both the effort
described in Section
we have
3,
the tasks to complete in the
As
phase.
As
such, this
is
and the development
control over these quantities
first
time.
As
by choosing which
of
phase, and which of them to delay until the second
a two-criteria combinatorial optimization problem.
a first attempt at looking for optimal points,
minimum-time and minimum-effort
solutions.
we have examined
The remainder
of this sections
discusses finding solutions which are locally or globally optimal on one of the
criteria.
4.1.
Minimizing Time and
Finding
minimum
known
Effort
optimal solutions
effort) is difficult.
We
which finds such solutions short
discuss
some
(in
the sense of
have not been able
to
of exhaustive search.
heuristic algorithms
which
minimum
time or
determine an algorithm
In the next subsection
we
find locally optimal solutions, although
they are not guaranteed to be globally optimal.
4.2.
Heuristic for Finding Locally Optimal
One simple
Time and
(but effective) heuristic solution
current best
known
phase. This
is
solution
does quite well
when
far
attempt to improve the
to
by switching one task of the division
not a sophisticated algorithm, and
optima which are quite
is
Effort Solutions
may be
from the global optimum. In
the optimization criterion
12
is effort,
to the other
trapped finding local
practice, this algorithm
but not as well when the
optimization criterion
is
For discussion on the effectiveness of this heuristic
time.
on an industrial example see Section
5.
Heuristic Algorithm (1-opt)
Generate a random starting division.
Switch one task from the phase in which it is to the other phase.
3.
Evaluate the current division to see if it is an improvement (on either
time or effort, whichever we are considering), if not then switch that
task back to its previous phase.
Step 4. Go back to step 2 until there are no more improvements available.
Step
Step
Step
A
1.
2.
(slightly)
counterpart. That
more complex version
is its
2-opt
replace step 2 with:
is,
Improved Heuristic
of the previous algorithm
(2-opt)
Step 2a. Switch two tasks from the phase which they are to the other phase.
The 2-opt algorithm
global optimal value, but
the global optimum.^
solutions
is
still
more
likely to find solutions
with values close to the
can be fooled into finding local optima which are not
Again, evaluating the time criterion
is
more
likely to find
which are not the global optimal values than when looking
for effort
solutions for our example problem.
Changing the
of time
and
effort
criterion of interest in Step 3 to incorporate linear
would allow
other changes to the algorithm
combinations
the finding of other Pareto-optimal solutions.
would be
No
required.
^As an example of a matrix where the 2-opt heuristic does
consider the case where
is the identity matrix and
r\ot
always terminate on the global optimum
W
0.7
A
0.1
0.2'
0.6
0.6
=
0.1
0.5
0.3
0.2
When
using the effort criterion there are two matrices
depending on the
initial starting state.
13
K
at
which the 2-opt
heuristic
may
terminate,
5.
Application
We
to
Workstation Design
have previously modeled workstation design
which are
different tasks
related in a
WTM
be composed of 45
to
[Smith 1992]. The dependencies and task
times have been suggested by engineers from the project. The
list
of tasks
and
their
corresponding times are given in Appendix A. The dependencies are shown in
Figure A.l. In the text of the paper
Using the equations
strategy,
where
all
tasks will be referred to
all
and
effort for the
The workstation design process
design phases: product design
appropriate
better
ways
weeks and
way
numbers.
tasks are attempted from the beginning of the design process,
parallel design strategy has time 51.597
(all
their
in Section 2 for describing the fully parallel design
possible to calculate the time
process design
by
(all
to divide the
to divide
it.
workstation design process. The fully
weeks and
effort 319.98 engineer-weeks.
as described
tasks with task
tasks beginning with A2).
it is
is
naturally divided into
number begirming with
We wish
to
two
Al), and
analyze whether this
is
an
design process into two phases or whether there are
The product/process two-phase solution has time 55.590
effort 234.45 engineer-weeks.
This division has greatly reduced the effort
while slightly increasing the time.
The product /process two-phase solution
improvement, but there
may be
still
is
therefore potentially an
better solutions.
We
have attempted
such solutions using the heuristics described in subsection
4.2.
the structure of the best found two-phase solutions for time
discuss
how
effective the
two
and
effort
first
discuss
and secondly
heuristics are at finding such solutions.
The best minimum-effort solution found
is
given in Appendix
vector which corresponds to the minimum-effort solution
from the matrix K. All entries with
indicate second-phase tasks.)
We will
to find
It
is
the
a 1 indicate first-phase tasks,
B.
(The
main diagonal
and
entries with
has an effort of 134.59 engineer-weeks and a time of
14
17.060 weeks.
This
a significant
is
as well as the product /process
second phase (11 of the
a
low
the product design
effort
it
Both the
anomalous
if it is
makes sense
it
minimum
results,
is
given
is
in
Appendix
B.
It
section
The
3.
tasks
in the
minimum
effort solution).
important to minimize the amount of time spent
begin more tasks earlier in the project.
to
time and the
minimum
some
effort divisions give
such as delaying choosing the architecture (task Allll) until the
do not
Even though there are feedbacks
it
performed
A11333
after task
To handle such
in
one phase or the
constraint
is
(logic
it
other,
and timing
which cannot
difficult to
feasibly
tasks,
then
it
tasks.
If
there
would be
to explicitly require certain tasks to
and then reapply the
have
is
heuristic.
Adding
this
type of
implement. The search space considered
would simply be
their schedule
Another alternative would be
between the
example) the task of choosing the
validation).
would be possible
2 (or step 2a) of the heuristic algorithm
tasks
to (for
not meaningful to suggest that this task could be
cases,
would not be
in the
the definition of order-independent tasks as described in
fit
product architecture,
be
until
has a time of 14.746
second phase. The anomalies arise because of errors in specifying the tasks
matrix.
to
engineer-weeks. Only nine of the design tasks are
effort of 151.59
design process,
expect to see in
substantially complete.
is
This matches intuition that
a
we would
in the
complete the process design
to
delayed until the second phase (as opposed to 16
on
design tasks
criterion for the design process
makes sense not
The minimum time solution
weeks and an
of tht
design tasks, which
an important
If
either the fully parallel solution
two-phase solution. Mo'^t
16) are process
total-effort process.
minimize engineering
improvement on
to
in step
restricted to exclude those
changed.
change the weights of the interdependencies
a strict ordering relationship
between two
(or
more)
possible to increase the magnitudes of the feed-forward
15
dependencies, and decrease the magnitudes of any feed-back dependencies. The
iteration
model and scheduling algorithm
will then
be compelled
to
schedule the
tasks to reflect the effective ordering structure.
The
effort
minimization problem
very well behaved for the two heuristic
(Appendix B contains the values of
algorithms tested.
the heuristics
is
came
to rest, as well as the
number
local
it
was
run, starting from
optima have
effort
within
found the best known solution
generated
initial
randomly generated
1%
all
27 times
was
On
on which
the given
solution 14 out of the 20
initial conditions.
solution.
The other
six
The 2-opt solution
run, again starting from
randomly
conditions.
The time minimization problem
opt solution found the best
(starting
it
known
known
of the best
of the solutions
of times observed.)
45-task example, the 1-opt algorithm found the best
times
all
known
from randomly generated
is
also algorithmically well behaved.
The
1-
time solution eight out of the 23 times run
initial conditions).
The other
fifteen local
optima have time within 9.5% of the best known solution. The 2-opt solution
found the best known time solution
all
24 times executed.
Finding one 2-opt solution using
hours of
CPU
time on a
10 minutes of
6.
CPU
Extension to
There
is
contemplated.
VAX
8800.
MATLAB
requires approximately three
Finding a 1-opt solution requires approximately
time on the same machine.
More than Two-Phase Design
no particular reason
It is
why
only two-phase design should be
possible to extend the ideas expressed in this paper to design
processes which have
more than two
phases.
process would involve:
Three-Phase Process:
Divide tasks into groups
I, II,
and
HI.
16
For example, a three-phase design
work on tasks in group
Second phase: work on tasks in groups
First phase:
I.
have
1
initial
bird phase:
We
I
and
but only the tasks in group
II,
II
work.
work on
but only tasks in group
all tasks,
must define two matrices
K,
with elements
III
have
initial
work.
and K„ with elements
k'
k'!
such that
[1 if
,
;
=
/
and the
zth task is in the first
phase
^/,=L otherwise
_...:....
(13)
*''
1
f
„
''
The
1 if
= / and the
/th task is in the first or
second phase
[O otherwise
work time
total
/
vectors for the three phases are calculated by:
Ri=W{I-K,AK,y'K,Ua
(15)
R,=W{I-K„AK,r\K„-K,)u,
(16)
R„=W{I-Ar\l-K,)u,
(17)
Calculating the total time and /or total effort for the three-phase process
to
is
analogous
using equations (11) and (12) for the two-phase process. Minimizing the
time or the
the
total effort of the
design project as a function of whether each task
second, or third phase leads to an optimization problem which
first,
complex than the one which
As
the
number
is
discussed in Section
of phases
on the
total
time
the effects through analysis of the
best possible two-phase ordering.
would have
which minimizes the
number
the
The
effect of
problem under study.
We
minimum
total
parallel.
it is
We
in
more
work
an increase
in the
necessary to determine
observe
in the
example
has a longer development time that the
would not expect
time.
is
4.
difficult to predict;
is
in section 5 that the fully parallel strategy
strategy
is
of phases increases, the total effort will decrease, since
becomes increasingly sequential rather than
number
total
that a fully sequential
Therefore finding the
number
of phases
time requires restructuring the problem to allow the
of phases be an independent variable.
17
7.
Discussion and Conclusion
We
when
to
have given an
do design
explicit representation to the difficult
Our model
sequentially rather than in parallel.
work transformation matrix (WTM) model
able to suggest sequential
and
parallel orderings of tasks
we
set,
is
based on the
The model
of product development.
product development time and lower development
For our example data
problem of deciding
which lead
to
is
lower overall
effort.
believe that the divisions suggested by this
system do indicate ways in which the time or
suggested by the algorithm seem in
many
effort
can be reduced. The divisions
respects reasonable.
therefore be useful in identifying advantages
The method may
and disadvantages of
serial or parallel
tasks in the design process.
In our experience, the data required to build this
The engineers involved with design
collect.
the information needs of the tasks with
model are not
have
projects typically
a
which they work as well as
difficult to
good idea
of
task times.
Polling each of the people involved with a design project creates a complete set of
WTM data, which can then be used
The assumptions required
strong.
Further investigation
less restrictive
is
in the
manner
of the
model
in order to build the
underway
to build
in the paper.
in this
paper are
more general models based on
assumptions.
The simple
heuristic algorithms
used
in this
paper seem to do a good job of
finding locally optimal solutions for the example data
solution techniques
The
model
may
More
exploration of
be required for poorly behaved or larger problems.
class of design projects to
the class of projects with
set.
which
this
type of analysis can best be applied
which an organization already has
from similar products. This
is
a large class, as
many
significant experience
design organizations have
experience working with the technologies being used in their
18
new
products.
is
Computer workstation design
project
it is
falls into this class
of design projects.
For this type of
reasonable to assume that the data from previous projects can be used
with useful predictive value.
If
the designers had
no experience working with the
technology involved in the project, then any estimates of task times and
dependencies are
likely to
be inaccurate and the model would lose
its
predictive
utility.
The problem
design
is
interesting
of determining parallel or series division of tasks in engineering
and important. Shortening the lead time
is
a major source of
competitive advantage in design, and concurrency has a strong effect on this score.
We know
that other
models of design-task sequencing and scheduling are
and we suggest our model as
a starting point in building other
models
feasible
of this
important problem.
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20
in
Engineering
Appendix A. Data from Workstation Design Example
Tciblf A.l.
Aim
Appendix
B.
Table
Best Oh>er\t'd Solutions for Effort
Task
15.1.
Results from Heuristic Algorithms
and Time
Table
T
B.2.
Locally Optimal Effort Solutions
2218
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