Uniform Designs For Mixture Experiments

Fang and Wang(1990)devised the 'uniform design for mixture experiments' to overcome some of limitations of the optimal design that become apparent in certain situations. Their method transformed the uniform design of factorial experiments in to the mixture design.This approach has many benefits and is often sufficient;however,the nonlinear transformation cannot maintain the uniformity.The mixture design achieved by their method may not be the best one.In this thesis,I find a criterion to construct the uniform design for experiments with mixture.Specifically,I propose the discrepancy DM₂ and CDM₂ as a new method to measure the uniformity of mixture design.The computational formula of the discrepancy, by the functional method,was also determined.The superior design in any two mixture designs was demonstrated conveniently by comparing the values of DM₂ and CDM₂.This thesis advances two approaches to construct the uniform design for unconstrained mixture experiments.One is the "Transform of U-type Matrix",and the second is the "Search in interior of Simplex-lattice".Both achieved a good design when the number of ingredients and design points were small.For a specific type of constrained mixture experiment, i.e.isotonic restriction mixture experiment,I demonstrate that the variables still follow a uniform distribution after being ordered.Hence,a uniform design is achieved for the isotonic restriction mixture experiment if a uniform design for unconstrained mixture experiment exists.Last,this thesis applies the 'Threshold Accepting' and 'NTLBG algorithm' to construct the uniform design for experimental design with mixture.The results showed that the algorithm achieved a good design after the complicate of computation was reduced. Further,the weighted NTLBG algorithm,which was advanced based on the NTLBG algorithm, achieved a good design under the DM₂ criterion but not under the MSE criterion, i.e.the NTLBG algorithm.In conclusion,this new method develops the theory of the uniform experimental design with mixture,and can help to advance its utility in academia and industry.