Mixture designs with orthogonal blocks

Constructions of blocked mixture designs are considered in situations where BLUEs of the block effect contrasts are orthogonal to the BLUEs of the regression coefficients. Orthogonal arrays (OA), Balanced Arrays (BAs), incidence matrices of balanced incomplete block designs (BIBDs), and partially balanced incomplete block designs (PBIBDs) are used. Designs with equal and unequal block sizes are considered. Also both cases where the constants involved in the orthogonality conditions depend and do not depend on the factors have been taken into account. Some standard (already available) designs can be obtained as particular cases of the designs proposed here.     

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.     

Pairwise orthogonal F-rectangle designs

The concept of pairwise orthogonal Latin square design is applied to r row by c column experiment designs which are called pairwise orthogonal F-rectangle designs. These designs are useful in designing successive and/or simulataneous experiments on the same set of rc experimental units, in constructing codes, and in constructing orthogonal arrays. A pair of orthogonal F-rectangle designs exists for any set of v treatment (symbols), whereas no pair of orthogonal Latin square designs of order two and six exists; one of the two construction methods presented does not rely on any previous knowledge about the existence of a pair of orthogonal Latin square designs, whereas the second one does. It is shown how to extend the methods to r=pv row by c=qv column designs and how to obtain t pairwise orthogonal F-rectangle design. When the maximum possible number of pairwise orthogonal F-rectangle designs is attained the set is said to be complete. Complete sets are obtained for all v for which v is a prime power. The construction method makes use of the existence of a complete set of pairwise orthogonal Latin square designs and of an orthogonal array with vⁿ columns, (vⁿ−1)/(v−1) rows, v symbols, and of strength two.     

Mixture and mixture–process variable experiments for pharmaceutical applications

Many experiments in research and development in the pharmaceutical industry involve mixture components. These are experiments in which the experimental factors are the ingredients of a mixture and the response variable is a function of the relative proportion of each ingredient, not its absolute amount. Thus the mixture ingredients cannot be varied independently. A common variation of the mixture experiment occurs when there are also one or more process factors that can be varied independently of each other and of the mixture components, leading to a mixture–process variable experiment. We discuss the design and analysis of these types of experiments, using tablet formulation as an example. Our objective is to encourage greater utilization of these techniques in pharmaceutical research and development. Copyright © 2004 John Wiley & Sons Ltd.     

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.     

Optimization of a product performance using mixture experiments including process variables

This article presents a case study of a chemical compound used in the delay mechanism to start a rocket engine. The compound consists in a three-component mixture. Besides the components proportions, two process variables are considered. The aim of the study is to investigate the mix components proportions and the levels of process variables that set the expected delay time as close as possible to the target value and, at the same time, minimize the width of prediction interval for the response. A linear regression model with normal responses was fitted. Through the model developed, the optimal components proportions and the levels of the process variables were determined. For the model selection, the use of the backward method with an information criterion proved to be efficient in the case under study.     

Mixture designs for product improvement studies

Many products are mixtures of several components (ingredient). Characteristics of the products such as the strength of steel, the efficacy of a chemical pesticide, or the viscosity of a liquid detergent, depend only on the relative proportions of the components in the mixture. Studying changes in a product' properties caused by varying the ingredient proportions is the objective of performing mixture experiments. The inherent restriction that the sum of the component proportions equal unity creates different design strategies than are usually employed with independent factors where factorial arrangements are quite common. Experimental designs for exploring the entire mixture simplex region as well as for exploring only a subregion of the simplex are presented. In those cases where four or more components are considered and a subregion is to be investigated, computer-aided designs are the rule rather than the exception. Design criterion based on the properties (variance and bias) of the prediction equation are mentioned briefly and some suggestions are made for future research in mixture experiments.     

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.     

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.     

Small orthogonal main effect plans with four factors

In this paper we study orthogonal main effect plans with four factors, A table of such designs, where each factor has at most 10 levels, and there are at most 40 runs, is generated. We determine the spectrum of the degrees of freedom of pure error for these designs.     

Note on a mixture experiment with process variables

The experimental design to model the response of a mixture experiment in three blending components in the presence of process variables is considered. Czitrom (1988) gave an experimental design in two orthogonal blocks of blends that was "possibly" D-Optimal in the case of arbitrary restrictions on the blending component proportions. It will be shown that the design is indeed D-Optimal. The pair of orthogonal D-Optimal blocks of blends can be used with an arbitrary number of process variables and require a reduced number of observations     

Construction of orthogonal factorial designs controlling interaction efficiencies

This paper employs some variants of the usual Kronecker product to construct orthogonal factorial designs controlling the interaction efficiencies. The methods suggested have a fairly wide coverage and the resulting designs involve a small number of replicates.     

Orthogonal blocking of response surface split-plot designs

When all experimental runs cannot be performed under homogeneous conditions, blocking can be used to increase the power for testing the treatment effects. Orthogonal blocking provides the same estimator of the polynomial effects as the one that would be obtained by ignoring the blocks. In many real-life design scenarios, there is at least one factor that is hard to change, leading to a split-plot structure. This paper shows that for a balanced ordinary least square–generalized least square equivalent split-plot design, orthogonal blocking can be achieved. Orthogonally blocked split-plot central composite designs are constructed and a catalog is provided.     

On the construction of orthogonal factorial designs of resolution 11

A new method of construction of orthogonal resolution IV designs for symmetrical and asymmetrical factorials has been presented. Many new series of orthogonal factorial designs of resolution IV can be obtained by the above general method.     

Experimental design for four mixture components with process variables

The experimental design to model the response of a mixture in four components in the presence of process variables is considered. Two different blocks of blends that are orthogonal for linear or quadratic blending are D-optimized. The two orthogonal blocks of blends are generalized and D-optimized in some cases (and possibly Doptimized in others) to deal with restrictions on the blending component proportions. The pair of orthogonal D-optimal blocks of blends can be used with an arbitrary number of process variables, and requires a reduced number of observations.     

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.     

Some robust parameter designs from orthogonal arrays

Robust parameter design, originally proposed by Taguchi [System of Experimental Design, Vols. I and II, UNIPUB, New York, 1987], is an offline production technique for reducing variation and improving a product's quality by using product arrays. However, the use of the product arrays results in an exorbitant number of runs. To overcome this drawback, several scientists proposed the use of combined arrays, where the control and noise factors are combined in a single array. In this paper, we use non-isomorphic orthogonal arrays as combined arrays, in order to identify a model that contains all the main effects (control and noise), their control-by-noise interactions and their control-by-control interactions with high efficiency. Some cases where the control-by-control-noise are of interest are also considered.     

Optimal orthogonal block designs for four-mixture components in two blocks based on F-squares for Becker's models and K-model

Orthogonal block designs in mixture experiments have been extensively studied by various authors. Aggarwal et al. [M.L. Aggarwal, P. Singh, V. Sarin, and B. Husain, Mixture designs in orthogonal blocks using F-squares, METRON – Int. J. Statist. LXVII(2) (2009), pp. 105–128] considered the case of components assuming the same volume fractions and obtained mixture designs in orthogonal blocks using F-squares. In this paper, we have used the class of designs presented by Aggarwal et al. and have obtained D-, A- and E-optimal orthogonal block designs for four components in two blocks for Becker's mixture models and K-model, respectively. Orthogonality conditions for the considered models are also given.     

Efficient maximin distance designs for experiments in mixtures

In this paper, different dissimilarity measures are investigated to construct maximin designs for compositional data. Specifically, the effect of different dissimilarity measures on the maximin design criterion for two case studies is presented. Design evaluation criteria are proposed to distinguish between the maximin designs generated. An optimization algorithm is also presented. Divergence is found to be the best dissimilarity measure to use in combination with the maximin design criterion for creating space-filling designs for mixture variables.     

Optimum mixture designs in some constrained experimental regions

In a mixture experiment, the response depends on the mixing proportions of the components present in the mixture. Optimum designs are available for the estimation of parameters of the models proposed in such situations. However, these designs are found to include the vertex points of the simplex Ξ defining the experimental region, which are not mixtures in the true sense. Recently, Mandal et al. (2015 Mandal, N.K., Pal, M., Sinha, B.K., and Das, P. (2015). Optimum mixture designs in a restricted region. Stat. Pap. 56(1):105–119.[Crossref], [Web of Science ®] , [Google Scholar]) derived optimum designs when the experiment is confined to an ellipsoidal region within Ξ, which does not include the vertices of Ξ. In this paper, an attempt has been made to find optimum designs when the experimental region is a simplex or is cuboidal inside Ξ and does not contain the extreme points.