Theory of Constraints for Design and Manufacture of Thermoplastic ...

THEORY OF CONSTRAINTS FOR DESIGN AND MANUFACTURE OF
THERMOPLASTIC PARTS
David Kazmer, University of Massachusetts Amherst
Christoph Roser, University of Massachusetts Amherst
Stephen Shuler, GE Plastics
Abstract
A Theory of Constraints
The functionality and manufacturability of a given
From a fundamental viewpoint, the value of a product
product is largely determined by constraints imposed by
is determined by the functionality that it provides the
the end user and manufacturer. With conflicting
customer. Philosophically, this functionality is related to
constraints, the designer may be unable to satisfy the
the requirements and constraints placed on the product.
complete set and has to find the best trade off to optimize
At the beginning of the product development, no
the design. In order to find this point with the minimum
requirements have been set. While the design space is
quality loss, extensive knowledge of the design space is
totally unconstrained, vague product concepts provide no
essential but seldom readily available. The design space
value. As development continues, both the requirements
can be investigated using various Design of Experiments
and the design become more defined. Each additional
techniques, such as a Taguchi L128 experimental design
product requirement typically adds some value to the
with augmented levels [1]. The actual loss of performance
customer. Each additional product requirement, however,
or cost can be calculated using a loss function for each
provides a constraint on the design, which is another
constraint and then combined in order to find the total
possibility for the product design to become unacceptable.
quality loss and subsequently the optimal point in the
design space. Application of the developed theory will
This paper seeks to answer the question: How can the
then be demonstrated.
product and process design parameters be selected to
ensure that all product requirements are met? A theory of
Introduction
constraints has been developed elsewhere to provide a
rational basis for performance evaluation [2]. This work
Injection molding is a commonly used process for the
states that 1) feasible designs must satisfy all imposed
manufacture of commercial goods. The design of molded
constraints, and 2) the optimal design will occur at a
parts, however, poses some significant challenges to the
position such that the probability of violating any
product designer as well as the manufacturing engineer.
constraint is minimized.
In particular, there is significant coupling between the
Consider a vector composed of product and process
product geometry, material properties, and process
design variables, X. Each of the product and process
dynamics. Selection of wall thickness, for instance, is
variables can have stochastic behavior, i.e. distributed
dependent upon the required part stiffness, elastic
according to some probability density function f(X).
modulus of the polymer, and gating design with its
Defining y to be a single response g(X), the estimated
implications on pressure drop and flow length.
value of y can be defined as:
The trade-offs between the many design and process
requirements can often be made more easily by increasing
E
y
X 

g
X f
X dX



.
(1)


the part cost. In the above example, for instance, the
requirements can be more easily met by increasing the
Given this estimate of y, we must now evaluate its
wall thickness, increasing the number of gates, or using a
value relative to the product requirements. Several
more rigid material. However, each of these decisions will
different approaches have been advocated. Taguchi has
typically increase the part cost.
defined a quality loss function as the response deviates
from target [3]. A classic optimization approach might
What makes a design competitive? Largely the ability
define an objective function which represents the value of
of the development team to adeptly manage the trade-offs
the product performance [4]. Utility theory might utilize
between attributes which are critical to quality. This
customer lottery questions to explicitly evaluate the utility
paper explores these concepts in more detail.
of choosing between different performance options. [5]

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Each of these approaches, however, is deficient.
Implementation
Taguchi’s approach does not stem from a fundamental
basis and provides very low fidelity regarding the product
The described theory of constraints provides a
performance. Utility theory and classic optimization fail
rational basis for establishing the ‘feasible window’ for
on two fronts. First, these approaches are not well suited
product and process design. Given the approach’s ability
to probabilistic design and process variables. Second,
to consider stochastic behavior of product and process
determination of a single aggregate measure to determine
variables, the same theory of constraints also provides a
the product value is extremely difficult.
method for assessing product quality. The primary
drawback of the method, however, is its reliance on
As such, this approach has defined the value or
developing a global model for each performance
utility, u, of the product response as the probability that
characteristic. This implementation will be described in
the response meets the product specification. For
the next two subsections.
instance, consider a requirement that y must be less than
an upper specification limit, USL. The utility is then
Response Surface Methodology
defined by:
The calculation of design utility, eq. 2, requires the
u
x
P
y  USL
estimation of each response’s probability density
(2)
USL
function, fy. As a response, this is an extremely difficult
u
x

f y
ydy



function to obtain from experimental data, requiring
multiple samples at many points through the design
This approach is notable in that it stems from a
universe to characterize the nature of the performance
completely rational basis. As such, this measure of utility
distribution. Fortunately, the probability density function
is directly understandable as the likelihood of product
of a response can be related to the probability density
acceptance, and is operable as such. All products,
function of its independent variables [6]:
however, have multiple product and process
specifications. Injection molded parts typically have many
f x
x
dozens of specifications: product requirements such as
f

.
(5)
y
y 


g
x
weight, size, stiffness, strength, aesthetics coupled with
process requirements such as cycle time, moldability, and
warpage.
Calculation of the derivative of the response, g’(x), is
not trivial, however. One approach to estimation of g(x)
Each of these specifications will have a utility
is the use of response surface modeling [7] through
associated with it. Remembering that the utility represents
Design of Experiments (DOE). The current
the likelihood of acceptance, a single measure of global
implementation is able to use two different Design of
utility, R, can be estimated by multiplying all the the
Experiments to model the response surfaces, the Box
utilities together:
Behnken Method and the Central Composite Design
(CCD). Both designs incorporate three levels and are
 
u .
(3)
i
therefore able to model response surfaces including
i
n
:
1

constant, first order, linear interactions, and quadratic
By this definition, R is the joint probability that the
effects. Throughout this paper, these three levels are
product will meet all of the specifications. A higher R
denoted as –1, 0, +1 for the lower, middle and upper level
represents a more robust design where the limit of 100%
of each design and process variable.
indicates that the product will be acceptable given the
defined product and process noise. If a constraint is
violated, then the likelihood of the specification being met
approaches 0 and the global measure of utility will
likewise approach 0.
This global norm is founded on a rational basis and is
also directly operable. For instance,
can be related to
the process capability index, Cp, through the inverse
normal cumulative probability density function, F-1:
Figure 1: Box Behnken Design
The Box Behnken Design investigates every possible
  
1
1
F 

 2 
combination of two factors out of all factors, with those
Cp 
(4)
two factors set to the extreme limits while the other
3
factors are set to zero. After investigating every
combination, a center point run is added. This

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experimental design is useful to estimate linear effects,
The first column is the constant coefficient, followed
quadratic effects and all linear 2-way interactions.
by the main effects, all interactions and finally the
However, because no corner points are used it is not
quadratic effects. This provides a system of equations,
possible to incorporate factors with only two possible
with:
settings, like e.g. two types of material. In addition, for
more than four factors the number of runs required is
X*B=Y,
(6)
greater than the number of runs for a CCD.
where X is the design matrix and B is a column vector
with the model parameters for each column of X. Y is a
matrix with the experimental results for each investigated
quality specification in the columns and the related results
in the corresponding rows. Transforming the matrices in
the following way solves this system of equations:
B= (XT*X)-1* XT *Y
(7)
Figure 2: Central Composite Design
Thus B provides the coefficients for the prediction
equation. The result Y is obtained by multiplying a row
The CCD uses a different approach to model the
vector X, incorporating the settings, the interaction
response surface. This experimental design consists of
values, and their quadratic values with the vector B.
three parts. First, the CCD utilizes a normal full factorial
design. Then, these runs are augmented with a center
Performance Data
point run. Finally, the quadratic effects are modeled by
making two runs for each factor, where the factor is set to
While the theory surrounding response surface
+ and - while all other factors are zero. The value for 
modeling is fairly involved, its implementation can be
is recommended to be n ¼
embedded in software and completely removed from the
f , where nf is the number of runs
in the factorial part of the design.
design or process engineer. The described response
surface modeling provides a method for efficiently
Unfortunately, this valuation of  could require
sampling the design space and characterizing the product
experimentation at extreme settings for a large number of
response.
factors. For example, ten factors would require 1024
The product performance data can originate from
factorial runs and a value of  equal to 5.65. For a melt
either simulation runs or molding trials depending on the
temperature with a normal range from 200 to 240 ºC (380
current state of product development. The authors
to 460 F) as ±1 would have to be run at 127 and 331 ºC
envision a ‘Design for Six Sigma’ strategy utilizing these
(260 and 595 F) for =± 5.65. These melt temperatures
same concepts throughout the product development cycle.
are way outside any useful range and the experimentation
For instance, application engineers could use this
would not prove feasible reducing the characterization of
methodology with finite element analysis to ensure robust
the response models. This problem can be overcome by
geometric designs. Then mold-filling analyses could be
setting =±1. This greatly lowers the prediction of the
utilized to ensure moldability across a wide range of
variance of the results, but the variance is of no concern
process conditions. Finally, the same tool can be used by
for the response surface modeling since the variances will
the processing engineer to set-up and optimize the
be estimated through (5). A representative CCD design
injection molding process.
for three factors is shown in Table 1.
Once performance data is supplied, the engineer can
Table 1: Central Composite Design
evaluate the global product performance. The effect of
changing product specifications or input variable
b0
A
B
C
AB
AC
BC
ABC
A²
B²
C²
1
-
-
-
+
+
+
-
1
1
1
distributions can be immediately witnessed. This
1
-
-
+
+
-
-
+
1
1
1
information is critical is assessing the engineering
1
-
+
-
-
+
-
+
1
1
1
1
-
+
+
-
-
+
-
1
1
1
feasibility and economic viability during product
1
+
-
-
-
-
+
+
1
1
1
1
+
-
+
-
+
-
-
1
1
1
development.
1
+
+
-
+
-
-
-
1
1
1
1
+
+
+
+
+
+
+
1
1
1
1
0
0
0
0
0
0
0
0
0
0
Software
1
+
0
0
0
0
0
0
+
0
0
1
-
0
0
0
0
0
0
+
0
0
1
0
+
0
0
0
0
0
0
+
0
The described theory has been implemented in a
1
0
-
0
0
0
0
0
0
+
0
1
0
0
+
0
0
0
0
0
0
+
software program named Looking Glass to provide the
1
0
0
-
0
0
0
0
0
0
+
industry practitioner a dynamic view of the injection
molding (IM) process. The user will be able to determine
critical process parameter and find a process window for

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their application and molding process. Graphics visualize



PL
h, E 
3
(9)
the process characteristics throughout the program to

3
4Ewh
improve the understanding of the IM process for the user.
The question we want to address is: “what are the
Using this software, the user can select the critical to
design and process parameters that will result in minimal
quality specifications (CTQs). For each product
cooling time and deflection?” This question is made
requirement, the user must specify lower and/or upper
significantly more difficult given that there may be issues
specification limits. Depending on the product
surrounding the exact mold temperature, melt
requirements, the user chooses up to eleven parameters
temperature, elastic modulus, and load that the plaque will
for investigation and gives a range, e.g. melt temperatures
experience. Given an input distribution, the utility of
from 160 to 260 C and fiber contents from 5 to 20 %. It is
cooling time as a function of wall thickness can be plotted
interesting to note that the selection of independent
(Figure 4).
variables will vary with the background of the user and
stage of product development. For instance, product
1
designers may be more interested in wall thickness,
0.9
elastic modulus, and number of ribs while a process
0.8
engineering may be more interested in melt temperature,
injection speed, and pack pressures. However, all design
0.7
and process variables are available for investigation.
0.6
0.5
Next the program creates a DOE including quadratic
0.4
effects as previously discussed. Finally, the performance
0.3
data is provided through simulation or experimentation.
according to the settings provided by the DOE. The
0.2
software automatically solves the prediction equations
0.1
and provides many different types of results to the user.
01
1.5
2
2.5
3
Results and Discussion
Figure 4: Utility of Cooling Time vs. Thickness
The previously described theory will now be
demonstrated through a simple molding application.
The curve indicates that wall thicknesses less than 2
Consider a flat plaque to be molded of ABS with
mm will result in an adequately small cooling time of less
approximate dimensions of 100 mm by 20 mm by 2 mm.
than 10 sec. At higher wall thicknesses, machine settings,
A typical commercial application may be concerned
material properties, and process dynamics will reduce the
about the trade-off between cooling time, material
probability that the cooling time will be acceptable. It is
utilization, and stiffness.
important to realize that this is one plot of utility for a
constant melt temperature and Young's Modulus.
Twall
Changes in the input distributions will change the utility
curves, which can fortunately be modeled through the use
Tmelt
of equations 1 and 2.
Twall
The one utility curve for cooling time as a function
of wall thickness does not provide sufficient information
P
to make an informed design decision. In fact, the
relationships between cooling time and deflection are
quite involved – yet some trade-off has to be made
between cooling time and deflection. The global metric
of utility,
, can be calculated as shown in equation 3 as a
Figure 3: Geometries for cooling and deflection analysis
function of all the performance specifications. The
resulting plot of global utility is shown in Figure 5. In the
Given the variable declarations as defined in the
figure, utility is indicated on the vertical axis as a function
Nomenclature, the solution to these problems can be
of melt temperature and wall thickness. The results
roughly estimated as:
indicate the feasible region in which cooling time and
deflection are acceptable. If the design was not feasible,
T
z,t
T
(8)
then the utility across the entire design and process space
wall 
Tmelt Twall 


z
erf




t


2

would approach zero.

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variables. The goal of this paper was to develop a
1
measure of design optimality and relate that measure to
the constraints placed on a product. The resulting
0.8
methods can be used to satisfy multiple objectives in
0.6
product and process design, even subject to stochastic
0.4
variation and parameter uncertainty.
0.2
Nomenclature
0
260


thermal diffusivity
3
250
B
matrix of coefficients relating X to Y
2.5
2
240
Cp
process capability index
1.5
230
E
elastic modulus
Me lt Te mpe ra ture (C )
1
Wa ll Thickne s s (mm)
f
probability density function
F-1
inverse normal cumulative density function
Figure 5: Utility vs. Thickness and Temperature
g
performance function relating X to Y
h
part thickness
The figure indicates that a wall thickness close to 2
mm is optimal with a melt temperature at 230 C. The
L
part length
parabolic shape is due to the conflicting objectives placed
P
load
on the design. At thicker wall sections, the cooling time

global measure of design utility
is unacceptable and the utility drops off. At thinner wall
t
time
sections, the deflection becomes significant and the
Tmelt
melt temperature
design is unacceptable. The surface does show an
Twall
mold wall temperature
interaction: at lower melt temperatures a thicker wall
u
single measure of design utility
section does have higher utility since the cooling time is
USL
upper specification limit
reduced. Similar graphs of utility can be drawn for other
w
part width
design variables such as elastic modulus, mold
X
product and process design variables
temperature, etc. In this way, the global measure of
utility can be used for many product and process design
Y
product requirements
decisions.
z
thickness coordinate
The plaque example in this paper was provided to
References
demonstrate the theory of constraints and associated
[1] Schmidt, S. R., Launsby, R. G., Understanding
utility functions. The resulting design solution could
industrial Designed Experiments, 4th ed., Air
likely be improved by changing the design configuration.
Academy Press 1994.
As such, this design approach is being applied to the
development of more complex commercial applications
[2] Kazmer, D., Roser, C., “Rational Bases for
including both parametric changes (melt temperature,
Evaluation of Design Utility,” ASME Design
mold temperature, thickness) and alternative design
Engineering Technical Conference, 1998.
concepts (topology, number of gates, number of ribs).
[3] Belavendram, N., Quality by Design, Prentice Hall,
1995.
Finally, it should be noted that the estimate of utility,
[4] Foulds, L. R., Optimization Techniques: an
R, stems from a rational basis and can be interpreted
Introduction, Springer-Verlag, 1981.
directly as the likelihood of success. As such, the metric
can be related to other common measures of quality such
[5] Poulton, E. C., Behavioral Decision Theory : A New
as the process capability, Cp, and Six Sigma principles
Approach, Cambridge University Press, 1994.
through use of the metric described in equation 4.
[6] Papoulis, A., Probability, Random Variables, and
Looking Glass then become a powerful tool for not only
Stochastic Processes, 3rd Edition, McGraw-Hill, New
predicting product performance, but also providing an
York, p. 93, 1991.
estimate of the quality levels related to that product
[7] Myers, R. H., Response Surface Methodology:
performance.
Process and Product Optimization, Wiley, 1995.
Conclusions
Design of injection molded parts can be a difficult
Key Words
task due to the number of specifications and their
robust design; molded part design; quality issues;
complex relationships to product and process design
design of experiments

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