Experimental designs when there are one or more factor constraints

In response surface methodology, designs of orders one or two are often needed such that some or all the factor levels satisfy one or more linear constraints. A method is discussed for obtaining such designs by projection of a standard design onto the constraint hyperplane. It is shown that a projected design obtained from a rotatable design is also rotatable, and for a rotatable design that is also orthogonal (in particular any orthogonal first-order design) a least squares analysis carried out on the generating design supplies a least squares solution for the constrained design subject to the constraints. Some useful properties of the generating design, such as orthogonal blocking and fractionation are retained in the projected design. Some second-order mixture designs generated by two-level factorials are discussed.

[1]  Virgil L. Anderson,et al.  Extreme Vertices Design of Mixture Experiments , 1966 .

[2]  Tormod Næs,et al.  An application of projection design in product development , 2000 .

[3]  Angela M. Dean,et al.  Mixture designs for four components in orthogonal blocks , 1993 .

[4]  Mixture Designs for Q Components in Orthogonal Blocks , 1994 .

[5]  M. Agha,et al.  Experiments with Mixtures , 1992 .

[6]  H. Scheffé.  The Simplex‐Centroid Design for Experiments with Mixtures , 1963 .

[7]  Ronald D. Snee.  Techniques for the Analysis of Mixture Data , 1973 .

[8]  G. Box,et al.  On the Experimental Attainment of Optimum Conditions , 1951 .

[9]  H. Scheffé.  Experiments with Mixtures , 1958 .

[10]  Ian Hau.  Constrained Experimental Designs Part II: Analysis of Projection Designs , 1990 .

[11]  J. S. Hunter,et al.  MULTIFACTOR EXPERIMENTAL DESIGN FOR EXPLORING THE RESPONSE SURFACES , 1957 .

[12]  CONSTRAINED DESIGNS. PART I. FIRST ORDER DESIGNS. , 1966 .

[13]  J. S. Hunter,et al.  Multi-Factor Experimental Designs for Exploring Response Surfaces , 1957 .

[14]  John A. Cornell,et al.  EXPERIMENTS WITH MIXTURES: A REVIEW , 1973 .

[15]  Ronald D. Snee,et al.  Developing Blending Models for Gasoline and Other Mixtures , 1981 .

[16]  George E. P. Box,et al.  Empirical Model‐Building and Response Surfaces , 1988 .

[17]  Angela M. Dean,et al.  Mixture designs for five components in orthogonal blocks , 1993 .

[18]  Ian Hau,et al.  Constrained Experimental Designs Part I: Construction of Projection Designs , 1990 .