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, ie, 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
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 606) 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 40 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, WR 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://wwwitlnistgov/div898/handbook/pri/section5/pri54ht m [5] http://wwwitlnistgov/div898/handbook/pri/section5/pri521ht 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 76% 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
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 SA 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
Figure 1: Multi - layer perceptron Figure 2: Selected formulations composition
Figure 3: Real values and predicted values for Contrast Ratio Figure 4: Errors obtained for contrast ratio prediction