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The screening and sequential designs of categorized mixture experiments
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Abstract
In the area of mixture experiments, specially in certain chemical industries, many times there is present a large number of potentially important components that can be considered candidates in an experiment. In this case, a study of reduction in the number of necessary components is required. Snee and Marquart (1976) develop simplex screening designs for a q-component blending system. However, the method cannot apply to the categorized mixture experiments.
The research developed screening design for the categorized mixture experiments. At first, one considered the case of mixture experiments with two categories. Consider that only the components of a certain category are required to be screened. Then generalize the concept to the case of mixture experiments with multiple categories where only the components of a certain category are required to be screened.
The research also developed a sequential design following the screening design developed. The research identified the feasibility of conducting the screening and sequential experiment based on economical consideration. Experimenters can decide which one to choose by comparing the total number of design points of the two alternative designs.
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A mixture experiment involves mixing two or more components (ingredients) together to form some end product, and measuring one or more properties of the resulting mixture or end product. In the general mixture problem, the measured response is assumed to be dependent only on the proportions of the ingredients present in the mixture and not on the amount of the mixture (Cornell 1990, Scheffe' 1958, Tian an Fang 1999). Lin and Wen (1993) applied mixture design approach to the service life and the oxygen evolving catalytic activity of Ru-Sn-Ti ternary oxide coated electrodes. A mixture experiment with components classified into categories while each category contributes a fixed proportion to the mixture is called the mixture problem with categorized components.
The designs for the mixture problem can be divided into five categories. They are:
(1) Simplex designs (Scheffe 1958 and 1962), (Gorman and Hinman 1962), (Cornell 1975), (Murty and Das 1968).
(2) Mixture experiments using ratios (Kenworthy 1963), (Fang 1996).
(3) Design using mixture-related variables (Claringbold 1955), (Draper and Lawrence 1965a and 1965b), (Fang 2000).
(4) Designs and models while process variables are also considered to be factors (Scheffe 1963), (Murty and Das 1968), (Cornell and Gorman 1984), (Cornell 1988), (Tian and Fang 1999).
(5) Computer-aided designs (Piepel and Cornell 1987), (Piepel 1994), (Fang 1999).
Lambrakis (1968), Cornell and Good(1970), and Fang(1995, 1996, 1999, 2000) develops designs for categorized mixture experiments. Snee and Marquart (1976) develop simplex screening designs for a q-component blending system. Cox (1970) defines the effect of a certain component in a mixture is the change in the value of the response resulting from a change in the proportion of component while holding constant the relative proportions of the other components. Snee and Marquart (1976) point out that the design points should be located on the component axes of the simplex for screening purpose. The method of Snee and Marquart cannot be applied to the mixture experiment with categorized components since one cannot hold the relative proportions of the other components constant while the proportion of a certain component is changed. That is, the design of the component in one category is independent of the design of the component in the other categories.
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In order to develop the screening design for categorized mixture experiment, one will do as the following steps:
select the two end points of each component axes in the simplex of the screened category.
select the vertices of the simplex that is not screened.
the final design points of the screening design are the factorial arrangement of the points selected in (1) & (2).
the model of the screening design is the multiplication of the right sides of the two first-degree models for the two categories.
In order to develop the sequential design following the screening design for categorized mixture experiment, one will do as the following steps:
Assume there are 3 important components in the first category after screening 8 components. Then the collected data in the screening experiment can be employed for further sequential design. One will observe 3/8 of the data collected in the screening design is useful for further design. These design points are part of the design points in the multiple-lattice design or multiple-centroid design.
The model for the sequential design can be second-degree or higher order polynomial.
In order to identify the feasibility of conducting the screening experiment based on economical consideration, one will do as the following steps:
(1) One fact is that the number of design points for the {q, m} simplex-lattice design is EMBED Equation.3 where q is the number of components in the mixture and m is the degree of the polynomial model. One alternative for an experimenter is to choose the second-degree model for both categories to form the multiple-lattice design for the two categories without conducting the screening experiment. The number of design points for the multiple-lattice design is EMBED Equation.3 = 216.
(2) The other alternative is to conducting the screening and sequential experiments. If there are 3 significant components after the screening experiment, the number of design points of the multiple-lattice design for the two 3-component categories is EMBED Equation.3 = 36. Deducting the number of runs conducted in the screening experiment from 36 and adding the number of runs in the screening experiment to the resulting number, the required number of runs for further experiment is 36 9 + 24 = 51.
The analysis shows that conducting the screening and sequential experiments are economical under the assumption that there are 3 significant components after the screening experiment. In some cases, it may not be economical if the number of significant components after the screening experiment is large.
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The research accomplishes:
(1) develop the screening design for the categorized mixture experiments.
develop the corresponding model of the screening design for the categorized mixture experiments.
develop a sequential design following the screening design developed.
identify the feasibility of conducting the screening and sequential experiments based on economical consideration.
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