Department of Statistics, Ewha Womans UniversityPurpose: Given the mixture components - process variables experimental data, we propose the strategy to find the proper combined model.Methods: Process variables are factors in an experiment that are not mi xture components but could affect the blending properties of the mixture ingredients. For example , the effectiveness of an etching solution which is measured as an etch rate is not only a function of the proportions of the three acids that are combined to form the mixture, but also depends on the temperature of the solution and the agitation rate. Efficient designs for the mixture components - process variables experiments depend on the mixture components - process variables model which is called a combined model. We often use the product model between the canonical polynomial model for the mixture and process variable s model as a combined model. Results: First we choose the reasonable starting models among the class of admissible product models and practical combined models suggested by Lim(2011) based on the m odel seletion criteria and then, search for candidate models which are subset models of the starting mo del by the sequential variables selection method or all possible regressions procedure. Conclusion: Good candidate models are screened by the evaluation of model selection criteria and checking the residual plots for the validity of the model assumption. The strategy to find the proper combined model is illustrated with examples in this paper.Key Words：Mixture Components - Process Variables Experiments, S trategy to Find Proper Combined Model, Model Seletion Criteria

/pdf/analysis-of-mixture-experimental-data-with-process-variables-i7mgxf2gi6.pdf

Analysis of mixture experimental data with process variables

Purpose: Given the several proper models for given mixture components-process variables experimental data, we propose a strategy to find the optimal condition in which the performance of the responses is well-behaved under those models. Methods: Given the mixture experimental data with process variables, first we choose the reasonable starting models among the class of admissible product models based on the model selection criteria and then, search for the candidate models that are the subset models of the starting model by the sequential variable selection method or all possible regressions procedure. Good candidate models are screened by the evaluation of model selection criteria and checking the residual plots for the validity of the model assumption. Results: We propose a strategy to find the optimal condition in which the performance of the responses is well-behaved under those good candidate models by adopting the optimization methods developed in multiple responses surface methodology. Conclusion: A strategy is proposed to find the optimal condition in which the performance of the responses is well-behaved under those proper combined models. This strategy to find the optimal condition is illustrated with the example in this paper.

/pdf/a-note-on-finding-optimum-conditions-using-mixture-58y3zmv9mb.pdf

A Note on Finding Optimum Conditions Using Mixture Experimental Data with Process Variables

Purpose: Ball milling is a popular process for obtaining fine powders in the part and material industry. One of important issues in the ball milling is to produce particles with a uniform size. Although many factors affect uniformity of particles, this paper focuses on the choice of ball diameter. Consider a ball milling where balls can be taken with three different diameters. The purpose of this paper is to find a ball mix which minimizes the average particle size. Methods: Ball diameters are selected as 10mm, 3mm, and 0.5mm. In order to find an optimum mixing ratio, the method of mixture experiments is employed in this paper. Taguchi``s signal-to-noise ratio (SNR) for smaller- the-better type is also used to analyze experimental data. Results: According to the experimental result, SNR is maximized when the ball mix is taken as either 7:3:0 or 6:4:0. Such mixing ratios can be technically validated in terms of porosity reduction. Conclusion: The ball mixing technique presented in this paper provides a useful way to improve the production efficiency with a low cost.

Design and Analysis of Mixture Experiments for Ball Mix Selection in the Ball Milling

In this paper, we consider the mixture components-process variables model and propose a model selection strategy using MTS. This strategy is illustrated using an example that involves three mixture components and two process variables in a bread making experiment that was studied in several literatures.

/pdf/a-note-on-model-selection-in-mixture-experiments-with-1ax08b25t4.pdf

A Note on Model Selection in Mixture Experiments with Process Variables

Process variables are factors in an experiment that are not mixture components but could affect the blending properties of the mixture ingredients. For example, the effectiveness of an etching solution which is measured as an etch rate is not only a function of the proportions of the three acids that are combined to form the mixture, but also depends on the temperature of the solution and the agitation rate. Efficient designs for the mixture components-process variables experiments depend on the mixture components-process variables model which is called a combined model. We often use the product model between the canonical polynomial model for the mixture and process variables model as a combined model. In this paper we propose three starting models for the mixture components-process variables experiments. One of the starting model we are considering is the model which includes product terms up to cubic order interactions between mixture effects and the linear & pure quadratic effect of the process variables from the product model. In this paper, we propose a method for finding robust designs and practical designs with respect to D-, G-, and I-optimality for the various starting combined models and then, we find practically efficient and robust designs for estimating the regression coefficients for those models. We find the prediction capability of those recommended designs in the case of three components and three process variables to be good by checking FDS(Fraction of Design Space) plots.

Yong-B. Lim

Papers

Practical designs for mixture component-process experiments