An analysis of certain composite designs and their response surfaces
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Use of factorial experiments and factorial designs in experimental design work to obtain optimum results is the gist of the thesis. A treatment, both descriptive and mathematical, of experimental designs, response surfaces, and optimum conditions, based on the greater portion of the bibliography, is given. The problem dealt with is the exploration and exploitation of response surfaces for the attainment of optimum conditions. A descriptive account and representation of a response surface by a response function is given. Methods of finding an optimum response on the response surface employing certain experimental strategy in the experimental region, using the method of steepest ascent to approach a near-stationary point, and local exploration of the surface in a near-stationary region are treated. Types of surfaces encountered in such a region are indicated. Experimental and composite designs are described and several examples given. The matrix theory applicable to experimental designs for the estimation of coefficients of the response surface equation is presented. It applies to both orthogonal and non-orthogonal designs. The second degree equation representing a response surface is discussed. Contours generated in both two and three dimensions, with surfaces and hypersurfaces represented by the respective contours, are discussed. The canonical analysis of the second degree equation, consisting of the determination of stationary points, of the canonical form of the equation, and of the new axes in terms of the original axes, is carried out. Next, several examples of surfaces met in actual experimental research are presented. The 22 and 23 composite, pentagonal, and cubic-octahedral designs are used. All their corresponding matrices necessary for the estimation of the coefficients of the response surfaces, are tabulated. Hypothetical experiments are planned to indicate the possible variety of response surfaces. Each experimental design presents a different type of response surface. There are surfaces giving ellipses for contours, surfaces containing a rising ridge, surfaces giving ellipsoids for contours, and surfaces containing a stationary plane ridge in three variables. Calculations and computations, based on the theory in Chapter II, are carried out and the results are stated. The cubic-octahedral design, incidentially, is a more compact design than the 23 composite, and its required matrices are more conveniently determined. A statement of how the study of this thesis topic may be extended is given in Chapter V.