A new model and class of designs for mixture experiments with process variables

Experiments that involve the blending of several components are known as mixture experiments. In some mixture experiments, the response depends not only on the proportion of the mixture components, but also on the processing conditions, A new combined model is proposed which is based on Taylor series approximation and is intended to be a compromise between standard mixture models and standard response surface models. Cost and/or time constraints often limit the size of industrial experiments. With this in mind, we present a new class of designs that will accommodate the fitting of the new combined model.     

Saturated main-effect designs for factorial experiments

The results of a computer search for saturated designs for 2ⁿ factorial experiments with n runs is reported, (where n = 2 mod 4). A complete search of the design space is avoided by focussing on designs constructed from cyclic generators. A method of searching quickly for the best generators is given. The resulting designs are as good as, and sometimes better than, designs obtained via search algorithms reported in the literature. The addition of a further factor having three levels is also considered. Here, too, a complete search is avoided by restricting attention to the most efficient part of the design space under p-efficiency.     

Experimental designs for mixture systems with multicomponent constraints

In an earlier paper it was recommended that an experimental design for the study of a mixture system in which the components had lower and upper limits should consist of a subset of the vertices and centroids of the region defined by the limitson the components. This paper extends this methodology to the situation where linear combinations of two or more components (e.g., liquid content=x₃+x₄+≦0.35) are subject to lower and upper constraints. The CONSIM algorithm, developed by R. E. Wheeler, is recommended for computing the vertices of the resulting experimental region. Procedures for developing linear and quadratic mixture model designs are discussed. A five-component example which has two multiple-component constraints is included to illustrate the proposed methods of mixture experimentation.     

Efficient designs for constrained mixture experiments

Many practical experiments on mixtures (where the components sum to one) include additional lower or upper bounds on components, or on linear combinations of them. Usually theory cannot be used to obtain a good design, and algorithmic methods are necessary. Some of the available methods are discussed. Their performance is evaluated on some examples, and the form of the optimal design is investigated.     

A simple method for constructing confounded designs for mixed factorial experiments

This paper presents the sinplesr procedure that uses wodular aryithmetic for constructing confounded designs for mixed factorial experiments. The present procedure and the classical one for confounding in symmetrical factorial experiments are both at the same mathema.tical level. The present procedure is written for

practitioners and is lllustrared with several examples.     

Optimal designs for experiments with mixtures: a survey

This is a survey article on known results about analytic solutions and numerical solutions of optimal designs for various regression models for experiments with mixtures. The regression models include polynomial models, models containing homogeneous functions, models containing inverse terms and ratios, log contrast models, models with quantitative variables, and mod els containing the amount of mixture, Optimality criteria considered include D-, A-, E-,φ_p- and I_λ-Optimalities. Uniform design and uniform optimal design for mixture components, and efficiencies of the {q,2} simplex-controid design are briefly discussed.     

Uniform coverage designs for mixture experiments

We investigate an optimization problem for mixture experiments. We consider the case when a large number of ingredients are available but mixtures can contain only a few number of ingredients. These conditions are held in experiments for self assembling molecular systems. First, we introduce a concept of uniform coverage design specialized for the situation. Next, we propose to use the stepwise technique for estimating coefficients of third-order Scheffe model which describes a response surface. After that, we propose a method of adding new mixtures for a movement to an extremum region. By this method, additional mixtures of experiments are extremum points of current estimated model and points which lead to more accurate estimation of current model prediction. This methodology is studied numerically for a model constructed from real data.     

A-optimal designs for heteroscedastic multifactor regression models

Abstract

This paper searches for A-optimal designs for Kronecker product and additive regression models when the errors are heteroscedastic. Sufficient conditions are given so that A-optimal designs for the multifactor models can be built from A-optimal designs for their sub-models with a single factor. The results of an efficiency study carried out to check the adequacy of the products of optimal designs for uni-factor marginal models when these are used to estimate different multi-factor models are also reported.     

Optimal designs for asymmetrical factorial paired comparison experiments

Three forms of a general null hypothesis Ho on the factorial parameters of a general asymmetrical factorial paired comparison experiment are considered. A class of partially balanced designscorresponding to each form of H₀ is constructed and the A,D and ioptimal design, minimizing the trace, determinant and largest eigenvalue of a defined covariance matrix of related maximumlikelihoodestimators, in that class is determined. Moreover, the optimal design in each class maximizes the noncentrality parameter λ² of the asymptotic noncentral chi-square distribution of the likelihood ratiostatistic -2 log λ for testing H_o under defined local alternatives. These results apply directly to symmetrical factorial paired comparison experiments as special casesExamples are given forillustrating applications of the developed results     

Construction of uniform designs for mixture experiments with complex constraints

Abstract

Mixture experiments have attracted increasingly attention due to their great practical value in production and living, while uniform designs over irregular experimental regions have become a hot topic in the area of experimental designs in the past two decades. Noting that the experimental region of a mixture experiment with q components under some constraints is in fact a (q ? 1)-dimensional geometry, this article proposes a new method for searching nearly uniform designs for mixture experiments with any complex constraints. Two examples with some tables and figures are given to illustrate this method.     

Using the chinese remainder theorem in constructing confounded designs for mixed factorial experiments

A procedure for constructing confounded designs for mixed factorial experiments derived from the Chinese Remainder Theorem is presented. The present procedure as well as others, all through use of modular arithmetic, are compared.     

Orthogonal three-level parallel flats designs for user-specified resolution

Orthogonal factorial and fractional factorial designs are very popular in many experimental studies, particularly the two-level and three-level designs used in screening experiments. When an experimenter is able to specify the set of possibly nonnegligible factorial effects, it is sometimes possible to obtain an orthogonal design belonging to the class of parallel flats designs, that has a smaller run-size than a suitable design from the class of classical fractional factorial designs belonging to the class of single flat designs. Sri-vastava and Li (1996) proved a fundamental theorem of orthogonal s-level, s being a prime, designs of parallel flats type for the user-specified resolution. They also tabulated a series of orthogonal designs for the two-level case. No orthogonal designs for three-level case are available in their paper. In this paper, we present a simple proof for the theorem given in Srivastava and Li (1996) for the three-level case. We also give a dual form of the theorem, which is more useful for developing an algorithm for construction of orthogonal designs. Some classes of three-level orthogonal designs with practical run-size are given in the paper.     

Projection designs for mixture experiments in orthogonal blocks

Mixture experiments are often carried out in the presence of process variables, such as days of the week or different machines in a manufacturing process, or different ovens in bread and cake making. In such experiments it is particularly useful to be able to arrange the design in orthogonal blocks, so that the model in tue mixture vanauies may ue iitteu inucpenuentiy or tne UIOCK enects mtrouuceu to take account of the changes in the process variables. It is possible in some situations that some of the ingredients in the mixture, such as additives or flavourings, are present in soian quantities, pernaps as iuw a.s 5% ur even !%, resulting in the design space being restricted to only part of the mixture simplex. Hau and Box (1990) discussed the construction of experimental designs for situations where constraints are placed on the design variables. They considered projecting standard response surface designs, including factorial designs and central composite designs, into the restricted design space, and showed that the desirable property of block orthogonality is preserved by the projections considered. Here we present a number of examples of projection designs and illustrate their use when some of the ingredients are restricted to small values, such that the design space is restricted to a sub-region within the usual simplex in the mixture variables.     

Optimal designs for second-degree Kronecker model mixture experiments

The paper investigates optimal designs in the second-degree Kronecker model for mixture experiments. Three groups of novel results are presented: (i) characterization of feasible weighted centroid designs for a maximum parameter system, (ii) derivations of D-, A-, and E-optimal weighted centroid designs, and (iii) numerically φ_p-optimal weighted centroid designs. Results on a quadratic subspace of invariant symmetric matrices containing the information matrices involved in the design problem serve as a main tool throughout the analysis.     

Restricted minimax robust designs for misspecified regression models

The authors propose and explore new regression designs. Within a particular parametric class, these designs are minimax robust against bias caused by model misspecification while attaining reasonable levels of efficiency as well. The introduction of this restricted class of designs is motivated by a desire to avoid the mathematical and numerical intractability found in the unrestricted minimax theory. Robustness is provided against a family of model departures sufficiently broad that the minimax design measures are necessarily absolutely continuous. Examples of implementation involve approximate polynomial and second order multiple regression.     

Robust designs for experiments with blocks

ABSTRACT

For experiments running in field plots or over time, the observations are often correlated due to spatial or serial correlation, which leads to correlated errors in a linear model analyzing the treatment means. Without knowing the exact correlation matrix of the errors, it is not possible to compute the generalized least-squares estimator for the treatment means and use it to construct optimal designs for the experiments. In this paper, we propose to use neighborhoods to model the covariance matrix of the errors, and apply a modified generalized least-squares estimator to construct robust designs for experiments with blocks. A minimax design criterion is investigated, and a simulated annealing algorithm is developed to find robust designs. We have derived several theoretical results, and representative examples are presented.     

Some Restricted randomization rules in sequential designs

This paper presents a new class of designs (Big Stick Designs) for sequentially assigning experimental units to treatments, when only the time covariate is considered. By prescribing the degree of imbalance which the experimenters can tolerate, complete randomization is used as long as the imbalance of the treatment allocation does not exceed the prescribed value. Once it reaches the value, a deterministic assignment is made to lower the imbalance. Such designs can be easily implemented with no programming and little personnel support. They compare favorably with the Biased Coin Designs, the Permuted Black Designs, and the Urn Designs, as far as the accidental bias and selection bias are concerned. Generalizations of these designs are considered to achieve various purposes, e.g., avoidance of deterministic assignments, early balance, etc.     

Algebraic statistics of level codings for fractional factorial designs

We discuss the applications of algebraic statistics to fractional factorial design with special emphasis on the choice of level coding. In particular, we deal with the theory of Bayley's level codings in that framework.