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CAFWIN/NNOpt Computer Aided Formulation Formulated products are those produced according to a formula (e.g. plastics, rubber, alloys, composites, coatings, chemicals, food, adhesives, paper, pharmaceuticals etc.). In each case, a number of ingredients are combined in accordance with a specific formula. They are mixed or otherwise processed to provide a product with properties tailored to meet functional requirements. Typically, the ingredients interact and consequently, product properties are sensitive to and governed by exact ratios of ingredients. Further, the processing of the ingredients also affects product properties. For these reasons, processing techniques and parameters are often considered an integral part of products' formulation. The task of formulation is complex because there is often no model or detailed understanding of how changes in formulation affects product properties. Traditional formulation approaches include statistical methods and experimentation. Both methods are expensive. Experimentation is expensive in terms of laboratory and staff time and in terms of opportunities missed through slow response to new customer requirements. The implementation of statistical methods requires a highly trained staff. In addition, in most cases the conventional, linear statistical tools do not work at all. A significant opportunity exists to improve operations and resulting profitability by streamlining the formulation design task. Artificial Neural Network (ANN) approximation addresses this opportunity which is most useful in an environment where theoretical descriptions are difficult to obtain, but partial knowledge about the process is known and input-output data are available. Quantitative relationships between the composition (and process variables) of formulation and the properties of the product are modeled by an Adaptively Learning Artificial Neural Network (ALANN) based on M. Smith: Neural Networks for Statistical Modeling, Van Nostrand Reinhold, New York , 1993. The trained ALANN is then used as an interpolating function to estimate product performance when given specific formulations and processing requirements (direct modeling). The trained ALANN is also used as the object function of a Nelder-Mead simplex to optimize formulation and processing to accomplish desired product characteristics (inverse modeling).
CAFWin 2.0 is the ideal tool for performance estimation and optimization of formulations. To have the possibility to see how CAFWin 2.0 works let consider 300 'measured' data points (sets) obtained using a known function: z = cos(x) + sin(y) + y / 2
It is easily observed that this is a transcendent function that is the equation of a surface represented in the picture. It is known that modeling of such functions using polynomials is rather difficult.The 'measured' (x, y, z) sets can be obtained as values of the function at uniform distributed random (x, y) points. Having the trained neural network it can be used in estimating output values. The next picture is the image of the z = f(x, y) surface as it is 'seen' by the neural network. This plot was obtained using estimated z values for x Î [0, 6.5] and y Î [0, 3.5], varying them by 0.5. Comparing the two figures we can observe a good matching, excepting the boundaries of the represented domain (they were not covered by the 300 sample points).
The obtained neural network also can be used in inverse modeling. The aim of the optimization (or of the inverse modeling) is the following: we want to find the values of the input properties those give the desired output properties, respecting some conditions. These conditions can be: - the value of the output property has to be inside a prescribed interval (enter the lower and upper bound of this definition interval); - the value of the output property ought to be as close to the middle of the definition interval as possible (choose 'tent' optimization); - the value of the output property ought to be as close to the upper bound of the definition interval as possible (choose 'up-hill' optimization); - the value of the output property ought to be as close to the lower bound of the definition interval as possible (choose 'down-hill' optimization); - because sometimes all conditions cannot be satisfied in meantime, an order of the priorities can be established (enter higher values for higher priorities); - the value of the input property has to be inside a prescribed interval enter the lower and upper bound of this constrained interval); - some input properties (e.g. components) must have a prescribed ratio (replace the asterisks with these values); - the sum of some input properties (components) must be inside a constrained sum (replace the asterisks with these values and place a minus sign in the 'ratio' column at those inputs those are not wanted in this constrained sum). Let's consider a view of our surface:
The highest zone is white, the lowest black. The dark blue and the dark green lines have the meaning of lower and the upper limit of the constrained sum: (x + y) must be between 1.0 and 2.0. The light blue line represents another constrain: the ratio between the input properties must be x : y = 2 : 1.
Let's suppose that we have studied the mechanical properties of some welding rod steels and the obtained mean values are summarized in the next table:
We can find the relation between the composition and the mechanical properties of these steels using CAFWin 2.0 . Let suppose that we have to determine the composition of a ductile steel unpurified with P and S with the possible highest mechanical properties. The ratio of the components must be Mn : Si = 2.5 : 1. The sum of the alloying elements must be under 1.2%. To resolve this problem we have to make an optimization (inverse modeling) in the next conditions: - constrain the S & P property on the [1.0, 1.0] interval (it is forced to 1.0); - set the ratio value 2.5 for Mn and 1.0 for Si; - set the ratio for S & P to '-' and the interval of the sum to [0.78, 1.2] (the ratio of 'C' must be '*'); - set the kind of optimization to 'up-hill' for l1 and l2 (possible highest mechanical properties); - set the kind of optimization to 'up-hill' for d (ductile steel). (S & P = 1 means that the steel is unpurified with S and P over 0.04 % - it corresponds to 'yes' in our original data sheet). The obtained properties will be: steel with components C = 0.08%, Mn = 0.86 %, Si = 0.34 % unpurified with S and P with d (Breaking strain) = 26.8 %, l1 (Tensile strength) =529 N/mm 2 , l2 (Ductility limit) = 452 N/mm 2
The theoretical basis of CAFWin 2.0 is in its theoretical book and the two presented examples are described in the "Resolved examples" electronic book". Reading them and using the provided on-line help CAFWin 2.0 will be very easy to use. After the new revision: - the introduction of the data to be analyzed is very user-friendly (possibility of exporting from Excel...) - the user can decide about the parameters of the training process - in manual training: the evolution of the errors is graphically represented - the separate training for each output is resolved - the optimization can be constrained in a detailed manner (intervals, ratios, sums, kind of optimum to be searched) - the latest version(s) of CAFWin/NNOpt are programs running under most Windows systems - it has no special requirements regarding hardware New features are the on-line help, the graphical representation of the dependencies and the "best matching" of properties.
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