7 Analysis and Optimization of Constrained Mixture-Design Formulations Stephen E. Krampe
Downloaded by COLUMBIA UNIV on February 27, 2013 | http://pubs.acs.org Publication Date: June 27, 1986 | doi: 10.1021/bk-1986-0313.ch007
Medical-Surgical Division, 3M, St. Paul, M N 55144
Techniques are presented for the analysis of mixture design responses and for the optimization of formulations comprised of components constrained by lower and upper limits. The analysis of component effects by the use of tricoordinate contour plots is enhanced by an algorithm which determines the feasible experimental region defined by the component limits. The direct optimization of a single response formulation modelled by either a normal or pseudocomponent equation is accomplished by the incorporation of the component constraints in the Complex algorithm. Multiresponse optimization to achieve a "balanced" set of property values is possible by the combination of response desirability factors and the Complex algorithm. Examples from the literature are analyzed to demonstrate the utility of these techniques. A considerable amount of s t a t i s t i c a l e f f o r t has been devoted to the design of mixture experiments, the proper selection of regression models to analyze the designs, and the determination of the i n d i v i d u a l component e f f e c t s on the response. Such research i s warranted due to the unique dependencies inherent i n mixture designs, the r e s t r i c t i o n s on the component l e v e l s which the experimenter t y p i c a l l y imposes on the system, and the considerable number of applications u t i l i z i n g formulations. This paper takes the end r e s u l t of the mixture design, the equations describing the responses i n terms of the component l e v e l s , and presents three techniques which enhance the information available from the design. These techniques assume that good s t a t i s t i c a l practice has been used f o r the establishment of the mixture design and the development of the response equation. 0097^6156/ 86/0313-0058$06.00/0 © 1986 American Chemical Society
In Computer Applications in the Polymer Laboratory; Provder, T.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.
7.
KRAMPE
Constrained Mixture-Design Formulations
59
Boundary Region Determination on Tricoordinate Contour P l o t s Unlike conventional experimental designs which have independent v a r i a b l e s , mixture designs possess v a r i a b l e s which are interdependent i n that the summation of the q component proportions must be u n i t y . T y p i c a l l y , the i n d i v i d u a l component l e v e l s are r e s t r i c t e d by lower (a^) and upper (b^) constraints imposed on the system by p h y s i c a l or chemical l i m i t a t i o n s of the formulation or by the s e l e c t i o n of the l e v e l values by the formulator. These constraints are represented as: q E x. = 1 i=l Downloaded by COLUMBIA UNIV on February 27, 2013 | http://pubs.acs.org Publication Date: June 27, 1986 | doi: 10.1021/bk-1986-0313.ch007
0 < a
< x^^ < b
i
i
< 1
Any component l e v e l change must be compensated by changes of the remaining component l e v e l s to maintain the u n i t y constraint. To describe the r e l a t i o n s h i p s between the response and the component l e v e l s , polynomial models of s p e c i a l forms are f i t to the data. The Scheffe model (1), expressed i n quadratic canonical form as q E(Y) » Z g X i=l 1
1
+
q I I
