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Cost optimisation by using DoE
The work of a paint formulator is strongly dependent of
personal experience, since most properties cannot be
predicted from mathematical equations. The use of a Design
of Experiments methodology and the selection of the right
constrains aiming to optimise paint properties could be
helpful. This methodology can be used in the context of
product's development final optimisation or for tune existing
ones. Result could be cost reduction of a product, within
keeping or improving the original characteristics and
properties.
Hugo Machado, Vasco Coelho, Inês Feyo, Filomena Braga,
Fernanda Oliveira, José Nogueira, Adélio Mendes.
Paints are complex multicomponent systems. The paint
formulation process is complex and will determine the
characteristics of the product. A systematic approach is
desirable allowing to investigate the effect of different
variables and their relationship in the product's final
attributes.
Usually a product born from the formulator's experience and
inspiration and can depend of his/her state of mind and
humour. This is hard to co-ordinate and a lot of important
combinations and interactions can be lost with high costs.
Formulators should use a methodical approach that allows
getting the greatest desirability at the lowest cost.
Optimise design of experiments
Design of Experiments (DoE) is used to identify or screen
the important factors affecting a process or product and to
develop empirical models. These techniques enable to get a
maximum amount of information from a minimum number of
runs. There are different kinds of designs, such as: Latin
Square, Factorial Design, Taguchi Methods or Mixture
Design, among others. Previously, other works were
published applying a DoE methodology to paint formulations
optimisation [1, 2]. However, little emphasis was given on
how to choose variable limits, restrictions and how to
perform the tests. This paper aims to treat these issues
more deeply and to provide another example of the use of
DoE methodology for paint formulation optimisation.
Mixture design and analysis
In many designs the range of variable such as temperature
is limited only by physical constrains within the system.
When dealing with mixtures there also is a mathematical
constrain for the mixture variables - the component
proportions must sum to unity. Thus, for example, with a
three-variable mix it is impossible to vary all the three
variables independently of each other and the mathematical
constrain must be taken into consideration in both design
and analysis.
A one at a time variable study is not an efficient approach
for mixtures because can require a lot of human effort and is
inefficient capturing subtleties. In mixture experiments, the
measured response is assumed to depend only on the
relative proportions of the ingredients or components in the
mixture and not on the amount of the mixture. Factorial
Design and Taguchi Methods are not appropriate for
mixtures since do not take into account the dependence of
response on proportionality of variables [3].
When the mixture components are only subject to the
constrain that they must some one, standard mixture
designs, such as "Simplex Lattice" or "Simplex Centroid"
can be used. When mixture components are subject to
additional constrains, such as maximum and/or minimum
value for each component, designs other than the standard
mixture designs, such as D-Optimal or Extreme Vertices
designs are appropriate [4].
In mixture problems, the purpose of the experiments is to
model the design space with some form of mathematical
equation or using other appropriate method.
D-optimal point selection
This optimality criterion results in minimising the generalised
variance of the parameters estimates for a pre-specified
model. As a result, the "optimality" of a given D-optimal
design is model dependent. That is, the experimenter must
specify a model for the design before generate the specific
treatment combinations for the design. Given the total
number of treatment runs for an experiment and a specified
model, the algorithm chooses the optimal set of design runs
from a candidate set of possible treatment runs. In other
words, the candidate set is a collection of treatment
combinations from which the D-optimal algorithm chooses
the treatment combinations to include in the design.
This candidate set of treatment points usually consists of all
possible combinations of various factor levels that one
wishes to use in the experiment.
D-optimal point selection maximises the determinant of the
Fisher information matrix [5]. The optimisation is performed
to minimise the general variance of the coefficients in the
model.
Scheffe's method a classical approach
When the number of parameters of interest is extremely
large, such as in the case of quantifying the uncertainty in
the estimate of a regression curve or a response surface,
the standard multiple comparisons methods for a finite
number of parameters often lead to an infinite confidence
bound or a test too conservative to be useful. In these
cases, the methods designed for a continuos domain must
be used. The Scheffe's methods is a classical approach for
such purpose [6]. It provides a simultaneous confidence
bound for a regression function when errors are Gaussian,
independents, homoscedastic and the predictor space is
unconstrained, i.e., the domain of interest is the whole
dimensional Euclidean space.
Software with human intelligence
Intelligent software gets this name because it's based on
artificial intelligence technology, a prosper field which aims
at having the computers mimic the human intelligence.
Specifically, "Artificial Neural Networks" (ANN) has been
used successfully to discover hidden cause and effect
relationships within data.
ANN can be considered "universal approximation
machines". The application in the resolution of complex
regression or classification problems is nowadays a reality in
many areas of knowledge and in the technological
development of intelligent systems [7].
The scheme of multi-layer perceptron, one of the most used
ANN architectures, is shown in Figure 1. It's a very
appropriate architecture for regression problems, like in
design of experiments.
It was decided to add two more constrains, based on the
same benchmarking results: PVC and Volume of solids
(VS).
Variability of test methods has to be screen out
The next step was to select the response properties or
characteristics. After a brainstorming session the contrast
ratio, viscosity and cost were chosen.
Special care should be take when measuring the contrast
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ratio. It's a test method with low repeatability and with a lot
of uncertainty. Trying to reduce the measurement errors, 5
cards have been applied for each of the 30 design
formulations. For each card, 5 readings were performed, in
a total of 25 readings for each design formulation (5 cards x
5 readings). This represents a lot of extra effort, but there is
need to be sure that the differences in the contrast ratio
result only from the differences in the formulation, instead of
differences due to the test method variability.
Using all this information, the "Design Expert" (version 6.06)
software and choosing the D-optimal mixture design, a
design matrix with 5 replicate runs was obtained see Table
3. These replicate runs were made to estimate the method
repeatability (it considers variability in product making and
the analysis method).
Find a formulation with lower cost
The next step is the truly optimisation one. A formulation
that meets the standard formulation properties with lower
cost must be found. The following parameters have been
established see Table 4.
Setting these targets, the following possibilities were
obtained from a second degree interpolator polynomial
(Table 5).
Since some of the formulations obtained were similar, four
were selected to be experimentally formulated and
evaluated to be compared with the standard formulation.
The formulations chosen were No 1, 4, 5 and 6. A
comparison of composition of the four formulations and the
standard can be found in Figure 2.
The experimental values obtained for formulations No. 1, 4,
5 and 6 can be observed in Table 6.
The formulation chosen was No. 4. An exhaustive
comparison against the standard was performed and can be
observed in Table 7.
Using the design matrix, neural networks have been trained
to predict the contrast ratio. The architecture used was
(4-8-1): 4 inputs (raw materials), 4 hidden nodes
(logistic = 1./. 1 + e-x)
and 1 output (linear), the contrast ratio. The software
package used was "Statistica Neural Networks" (Release
4.0 F), from Statsoft.
The suggestions obtained after the optimisation step have
been used to test the neural networks model see Table 5.
The predictions obtained for contrast ratio by the second
order polynomial have been compared with the ones
obtained from the neural network model. As can be seen
from Figure 3, neural networks fits better the experimental
data. The same conclusion can be obtained from Figure 4.
Acknowledgements
The authors wish to thank to José Alves and all the analysts
from CIN research lab.
References
[1] M. J. Anderson, P. J. Whitcomb, Optimization of Paint
Formulations Made Easy with Computer-Aided Design of
Experiments for Mixtures, Journal of Coatings Technology,
Vol. 68, No. 858, July (1996)
[2] K. W. Chau, W.R. Kelley, Formulation of Printable
Coatings via D-Optimality, Journal of Coatings Technology,
Vol. 65, No. 821, June (1993)
[3] M. J. Anderson, P. .J. Whitcomb, Find the Optimal
Formulation for Mixtures, April (1998)
[4]
http://www.itl.nist.gov/div898/handbook/pri/section5/pri54.ht
m
[5]
http://www.itl.nist.gov/div898/handbook/pri/section5/pri521.ht
m
[6] D. C. Montgomery, Design and Analysis of Experiments,
John Wiley & Sons, New York (2001)
[7] J. Dayhoff, Neural Network Architectures: An
Introduction, Van Nostrand Reinhold, New York (1990)
Experimental
The major contributor for a paint formulation cost is usually
titanium dioxide. For the formulation studied it represents
approximately 47% of the formulation cost and contributes
only with 14% for the mass. To reduce the amount of
titanium dioxide, keeping the original characteristics of a
product, the extenders and the opacifying agents amounts
must be changed. This is the first important step of a design
of experiments, choosing the important variables. To vary
the amount of titanium dioxide, calcium carbonate slurry,
emulsion and organic opacifier was chosen by keeping the
other raw materials amounts unchangeable.
The next thing to do is to set limits for each variable. This
means that the design can be made within the range
specified for each raw material. This is probably the most
important step in the optimisation process, since a bad
choice of the variables limits can ruin the project.
Nevertheless this can be made without any theoretical or
empirical knowledge, using benchmarking results for similar
products. For this project all high pigment volume
concentration (PVC) with medium titanium dioxide content
products, produced at CIN - Portugal, were compared. From
these results the following limits were set (see Table 1 and
2).
Results at a glance
A reduction of 7.6% in product's cost (by volume) has been
achieved keeping or improving most of the original
waterborne paint properties.
The design of experiments methodology is very easy to use
and should be applied in product's fine tuning or in the
reformulation of products, namely to replace raw materials in
formulation.
The optimisation can be performed by a non-expert
technician, using benchmarking results.
Extra care should be taken when performing the analysis,
specially for contrast ratio, due to uncertainty linked with the
measurement method.
The optimisation using neural networks instead of the
interpolator 2nd order polynomial showed to fit better the
experimental data and will be object of further investigation.
LIFELINE
-> Hugo Machado is a Quality Engineer at CIN. He
graduated in Chemical Engineering from the Faculty of
Engineering at the University of Porto in 2000 and works for
CIN since then. He is currently managing Six Sigma
projects. His areas of interest are continuos improvement,
products and processes optimisation and paint properties
prediction.
-> Vasco Coelho is a first-degree engineering at Megadur
(CIN's Powder Coatings). He graduated in Chemical
Engineering from the Faculty of Engineering at the
University of Porto in 2002. His main interests include
Process Control Technologies.
-> Inês Feyo de Azevedo is graduated in Chemical
Engineering from the Faculty of Engineering at the
University of Porto in 2002. She is currently attending to a
training course on Promoting of Company Innovation and
Internationalisation.
-> FILOMENA BRAGA works in Research and Development
of Decorative products at CIN. She is also Decorative
products direction assistant. She graduated in Chemical
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Engineering from the Faculty of Engineering at the
University of Porto in 1980 and works in the paint industry
since 1979.
-> FERNANDA OLIVEIRA studied Chemical Engineering at
the University of Porto, Portugal. She has joined CIN S.A. in
1990, where she is working on the application of new
technologies to the coatings industry. In 2000, she has
received a master degree in Paint Technology from the
University of Barcelona.
-> JOSÉ NOGUEIRA is the CIN's Technical Director. He
graduated in Chemical Engineering from the Faculty of
Engineering at the University of Porto in 1969 and he works
for CIN since then.
-> ADÉLIO M. MENDES is Associate Professor of Chemical
Engineering at the Faculty of Engineering at the University
of Porto, Portugal. He graduated in Chemical Engineering
(1987) and earned his PhD (1993) from the same school.
He teaches Chemical Engineering Laboratories, Separation
Processes, Numerical Methods and Statistics. His main
research interests include membrane and sorption gas
separations, catalytic membrane reactors and fuel cells.
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Figure 1: Multi - layer perceptron.
Figure 2: Selected formulations composition.
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Figure 3: Real values and predicted values for Contrast Ratio.
Figure 4: Errors obtained for contrast ratio prediction.
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