Supersaturated Designs Research Articles

A new class of supersaturated design: Application to a sensitivity study of a photochemical model

The use of experimental designs in the field of numerical simulation improves global sensitivity analysis compared to Monte Carlo based method by decreasing the number of simulations. More particularly, supersaturated designs can be applied to decrease the number of calculations, because the number of simulations is inferior to the number of studied factors but its use is risky and depends on “parsimony principle”. This paper shows new supersaturated designs that use the a priori knowledge and thus overcome this difficulty.

Modelling by supersaturated designs

The analysis of supersaturated designs is an interesting problem of great importance since it provides economic estimates. Moreover, this problem is challenging due to the fact that the design matrix has a complicated structure. The identification of the active factors in supersaturated designs is investigated. The singular value decomposition (SVD), principal components analysis and regression analysis are used together in an SVD principal regression method to reveal the hidden true linear model. Special cases are studied by using simulation data under idealized conditions. Simulations are used to investigate the performance of the method and also to compare the proposed method with other known methods from the literature.

A general construction of E(s 2)-optimal supersaturated designs via supplementary difference sets

Supplementary difference sets can be used to construct a variety of combinatorial designs. In this work, we construct supplementary difference sets that provide E(s 2)-optimal, two-level supersaturated designs with the equal occurrence property. Comparisons are made with previous works and it is shown that the proposed method gives promising results for the construction of supersaturated designs with good properties.

A classical dataset from Williams, and its role in the study of supersaturated designs

AbstractA Plackett‐Burman type dataset from a paper by Williams [1], with 28 observations and 24 two‐level factors, has become a standard dataset for illustrating construction (by halving) of supersaturated designs (SSDs) and for a corresponding data analysis. The aim here is to point out that for several reasons this is an unfortunate situation. The original paper by Williams contains several errors and misprints. Some are in the design matrix, which will here be reconstructed, but worse is an outlier in the response values, which can be observed when data are plotted against the dominating factor. In addition, the data should better be analysed on log‐scale than on original scale. The implications of the outlier for SSD analysis are drastic, and it will be concluded that the data should be used for this purpose only if the outlier is properly treated (omitted or modified). Copyright © 2008 John Wiley & Sons, Ltd.

Kansei engineering approach for total quality design and continuous innovation

PurposeThis paper aims at defining a structured process of continuous innovation in the product concept development phase by a statistical‐based Kansei engineering (KE) approach. It consists in the identification of quality elements satisfying both functional and emotional user needs, i.e. the total quality elements.Design/methodology/approachThe approach is developed integrating results from Kano and KE analysis. Three statistical methods considered to be suitable for KE study, are used: supersaturated design for concept configuration, ordinal logistic regression for data analysis, and EVA method for quality evaluation of the optimal concept. These methods are compared with the most used ones in KE regarding their efficacy, efficiency and easiness of use. An innovative procedure to exhibit concepts in a KE session is also presented. It uses the abstraction and association idea principles to elicit users' grade of agreement for a particular Kansei word.FindingsThe proposed approach is fully exploited through a case study on train interior design, developed in a virtual reality (VR) laboratory. The evaluation of comfort improvements obtained by means of a new handle and handrail design is carried on with expert users in VR. A consistent increase of a quality index, by using the defined approach, was obtained.Originality/valueThis work aims at contributing to the conception of new product solutions, which are appealing and saleable. The availability of virtual reality technologies and software capable to manage complex statistical analyses, will concretely aid designers and engineers in the ideation of high‐emotional‐quality products, which can be helpful for innovative enterprises to maintain and even increase their market position.

A new look at discrete discrepancy

The objective of this paper is to study the issue of discrete discrepancy [Hickernell, F.J., Liu, M.Q., 2002. Uniform designs limit aliasing. Biometrika 89, 893–904; Fang, K.T., Lin, D.K.J., Liu, M.Q., 2003. Optimal mixed-level supersaturated design. Metrika 58, 279–291; Qin, H., Fang, K.T., 2004. Discrete discrepancy in factorial designs. Metrika 60, 59–72], which has wide application to the field of fractional factorial designs. Here we present an improved lower bound to discrete discrepancy.

Model-robust supersaturated and partially supersaturated designs

Supersaturated designs are an increasingly popular tool for screening factors in the presence of effect sparsity. The advantage of this class of designs over resolution III factorial designs or Plackett–Burman designs is that n, the number of runs, can be substantially smaller than the number of factors, m. A limitation associated with most supersaturated designs produced thus far is that the capability of these designs for estimating g active effects has not been discussed. In addition to exploring this capability, we develop a new class of model-robust supersaturated designs that, for a given n and m, maximizes the number g of active effects that can be estimated simultaneously. The capabilities of model-robust supersaturated designs for model discrimination are assessed using a model-discrimination criterion, the subspace angle. Finally, we introduce the class of partially supersaturated designs, intended for use when we require a specific subset of m 1 core factors to be estimable, and the sparsity of effects principle applies to the remaining ( m - m 1 ) factors.

Two-level supersaturated designs for [formula omitted] runs and other cases

Two-level supersaturated designs are constructed for n = 2 k ( k ⩾ 5 ) runs and m factors where n + 3 ⩽ m ⩽ 5 ( n - 4 ) . The designs so formed are shown to have a maximum absolute correlation between factors of 1 4 and to be efficient in terms of E ( s 2 ) , particularly when the number of factors m is approximately double the number of runs n or greater. Thus, supersaturated designs with favourable properties are found for much higher numbers of runs than would be possible solely using algorithms.

A hybrid SAGA algorithm for the construction of [formula omitted]-optimal cyclic supersaturated designs

In this paper, we propose a hybrid simulated annealing genetic algorithm (SAGA) for generating cyclic structured supersaturated designs. The hybrid SAGA combines features such as the power of the GA and the speed of a local optimizer such as SA, merging the previous metaheuristics into a powerful hybrid optimization algorithm. This class of hybrid metaheuristics enabled us to build supersaturated designs for q = 2 , 3 , … , 14 generators. Comparisons are made with previous works and it is shown that the hybrid SAGA is a powerful tool for the construction of E ( s 2 ) -optimal supersaturated designs.

An algorithmic approach to constructing mixed-level orthogonal and near-orthogonal arrays

Due to run size constraints, near-orthogonal arrays (near-OAs) and supersaturated designs, a special case of near-OA, are considered good alternatives to OAs. This paper shows (i) a combinatorial relationship between a mixed-level array and a non-resolvable incomplete block design (IBD) with varying replications (and its dual, a resolvable IBD with varying block sizes); (ii) the relationship between the criterion E ( d 2 ) proposed by Lu and Sun [Lu, X., Sun, Y., 2001. Supersaturated designs with more than two levels. Chinese Ann. Math. B 22, 183–194] or E ( f N O D ) proposed by Fang et al. [Fang, K.T., Lin, D.K.J., Liu, M.Q., 2003b. Optimal mixed-level supersaturated design. Metrika 58, 279–291] used in the (near-) OA construction and the ( M , S ) -optimality criterion used in the IBD construction; (iii) the derivation of a tighter bound for E ( d 2 ) ; (iv) how to modify the IBD algorithm of Nguyen [Nguyen, N.-K., 1994. Construction of optimal incomplete block designs by computer. Technometrics 36, 300–307] to obtain efficient (near-) OA algorithms. Some new (near-) OAs are presented and some near-OAs are compared with arrays constructed by other authors. Examples showing the use of the constructed arrays are given.

On E( s2)-optimal supersaturated designs

A popular measure to assess 2-level supersaturated designs is the E ( s 2 ) criterion. In this paper, improved lower bounds on E ( s 2 ) are obtained. The same improvement has recently been established by Ryan and Bulutoglu [2007. E ( s 2 ) -optimal supersaturated designs with good minimax properties. J. Statist. Plann. Inference 137, 2250–2262]. However, our analysis provides more details on precisely when an improvement is possible, which is lacking in Ryan and Bulutoglu [2007. E ( s 2 ) -optimal supersaturated designs with good minimax properties. J. Statist. Plann. Inference 137, 2250–2262]. The equivalence of the bounds obtained by Butler et al. [2001. A general method of constructing E ( s 2 ) -optimal supersaturated designs. J. Roy. Statist. Soc. B 63, 621–632] (in the cases where their result applies) and those obtained by Bulutoglu and Cheng [2004. Construction of E ( s 2 ) -optimal supersaturated designs. Ann. Statist. 32, 1662–1678] is established. We also give two simple methods of constructing E ( s 2 ) -optimal designs.

Optimal mixed-level k-circulant supersaturated designs

Supersaturated designs (SSDs) offer a potentially useful way to investigate many factors with only few experiments in the preliminary stages of experimentation. This paper explores how to construct E ( f NOD ) -optimal mixed-level SSDs using k-cyclic generators. The necessary and sufficient conditions for the existence of mixed-level k-circulant SSDs with the equal occurrence property are provided. Properties of the mixed-level k-circulant SSDs are investigated, in particular, the sufficient condition under which the generator vector produces an E ( f NOD ) -optimal SSD is obtained. Moreover, many new E ( f NOD ) -optimal mixed-level SSDs are constructed and listed. The method here generalizes the one proposed by Liu and Dean [2004. k -circulant supersaturated designs. Technometrics 46, 32–43] for two-level SSDs and the one due to Georgiou and Koukouvinos [2006. Multi-level k-circulant supersaturated designs. Metrika 64, 209–220] for the multi-level case.

Optimal mixed-level supersaturated design with general number of runs

Supersaturated design (SSD) has received much interest because of its potential in factor screening experiments. Most studies focused mainly on balanced designs. This paper considers SSDs with general run sizes. The lower bounds of the χ 2 proposed by [Yamada, S., Lin, D.K.J., 1999. Three-level supersaturated designs. Statist. Probab. Lett. 45, 31–39] and E ( f NOD ) proposed by [Fang, K.T., Lin, D.K.J., Liu, M.Q., 2003b. Optimal mixed-level supersaturated design. Metrika 58, 279–291] are obtained along with the sufficient and necessary conditions for achieving them, respectively. Some construction methods for optimal mixed-level SSDs with general run sizes are proposed. Many new designs are also tabulated for practical uses.

Supersaturated Designs for Asymmetrical Factorial Experiments

This article describes a general method of construction of supersaturated designs for asymmetric factorials obtained by exploiting the concept of resolvable orthogonal arrays and Hadamard matrices. The supersaturated design constructed here has a restricted form of t:q z n, where z factors are at q-levels each and one factor is at t-levels and the number of runs is n. The designs obtained have the factor at t-levels always orthogonal to the z factors with q levels each in the symmetric design q z n. The method of construction is illustrated with the help of examples. A catalogue of designs obtained is prepared and fNOD-efficiency and c 2 -efficiency of the designs are given. Many designs are optimal while other designs have high efficiencies. The efficiency of the resulting design is better than that of the symmetric design q z n. AMS Subject Classification: 62K15, 62K10

New E(S2)-Optimal Supersaturated Designs Constructed From Incomplete Block Designs

We present a method for constructing two-level supersaturated designs (SSDs) from incomplete block designs. A lower bound of E(s2) that also covers the case of odd run sizes is given. This bound is attained by SSDs constructed from balanced incomplete block designs. We study SSDs that can be constructed from regular graph designs when balanced incomplete block designs do not exist. A computer search is conducted to find SSDs with 5 ≤ n ≤ 50 and n ≤ m ≤ 2n that can be constructed from regular graph designs, where m is the number of factors and n is the run size. Many SSDs derived from regular graph designs are optimal. The best E(s2)-optimal SSDs with respect to additional optimality criteria are tabulated. Some notes on the construction of saturated designs also are given.

A Method for Analyzing Supersaturated Designs with a Block Orthogonal Structure

Supersaturated designs is a large class of factorial designs which can be used for screening out the important factors from a large set of potentially active variables. The huge advantage of these designs is that they reduce the experimental cost drastically, but their critical disadvantage is the confounding involved in the statistical analysis. In this article, we propose a method for analyzing data using a specific type of supersaturated designs. This method heavily uses the special block orthogonal structure of the supersaturated designs given by Tang and Wu (1997). Also, we compare our method with several known statistical analysis methods by using some of the existing supersaturated designs. The comparison is performed by some simulating experiments and the Type I and Type II error rates are calculated. The results are presented in tables and the discussion to follow.

Applying Box-Meyer Method for Analyzing Supersaturated Designs

Supersaturated designs can be used for screening purposes to investigate a number of factors which is at least as large as the number of runs. Several methods have been proposed for the challenging data analysis of such designs. In this paper Box-Meyer method for finding active factors in fractionated designs is adapted for the analysis of supersaturated designs. By restricting search of important factors to a main-effect analysis, as is typically the case for supersaturated designs, Box-Meyer method is compared to other existing analysis methods. A numerical simulation and a well-known dataset are used to illustrate the ideas.

Understanding a ZnO nanorods fabrication process: a supersaturated design approach

Nanomanufacturing promises to increase quality, productivity and efficiency of existing technologies and has the potential to accelerate commercialisation of products and benefit various industries. As this technology is still in its discovery stage, there is a tremendous amount of experimentation occurring every day. Often in a nanomanufacturing setup, a large number of factors can be listed as possible sources of effects and among those, only a few are actually significant. A problem frequently encountered in nanomanufacturing is how to reduce the total number of experiments while estimating a large number of effects. Realising this challenge and the growing application of statistical techniques to discover relationships at the nanoscale, we advocate the use of supersaturated designs to nanomanufacturing settings. Here, we develop a supersaturated design, which is effective in identifying significant effects with minimal experimental runs for a process for the fabrication of ZnO nanorods, and discuss the analysis techniques.

Optimal Factor Assignment for Supersaturated Designs

Optimal Factor Assignment for Supersaturated Designs

Supersaturated designs: set-ups, data interpretation, and analytical applications

For screening purposes, two-level screening designs, such as fractional factorial (FF) and Plackett-Burman (PB) designs, are usually applied. These designs enable examination of, at most, N-1 factors in N experiments. However, when many factors need to be examined, the number of experiments still becomes unfeasibly large. Occasionally, in order to reduce time and costs, a given number of factors can be examined in fewer experiments than with the above screening designs, by using supersaturated designs. These designs examine more than N (SS)-1 factors in N (SS) experiments. In this review, different set-ups to construct supersaturated designs are explained and discussed, followed by several possible data interpretations of supersaturated design results. Finally, some analytical applications of supersaturated designs are given.