Orthogonal arrays are widely used in manufacturing and high-tech industries for quality and productivity improvement experiments. For reasons of run size economy or flexibility, nearly orthogonal arrays are also used. The construction of orthogonal or nearly orthogonal arrays can be quite challenging. Most existing methods are complex and produce limited types of arrays. This article describes a simple and effective algorithm for constructing mixed-level orthogonal and nearly orthogonal arrays. It can construct a variety of small-run arrays with good statistical properties efficiently.

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## Scholarly Works (34 results)

A common problem experimenters face is the choice of fractional factorial designs. Minimum aberration designs are commonly used in practice. There are situations in which other designs meet practical needs better. A catalogue of designs would help experimenters choose the best design. Based on coding theory, new methods are proposed to efficiently classify and rank fractional factorial designs. A collection of three-level fractional factorial designs with 27, 81, 243 and 729 runs is given. This extends the work of Chen, Sun and Wu (1993), who gave a collection of fractional factorial designs with 16, 27, 32 and 64 runs.

Nonregular designs are used widely in experiments due to their run size economy and flexibility. These designs include the Plackett-Burman designs and many other symmetrical and asymmetrical orthogonal arrays. Supersaturated designs have become increasingly popular in recent years because of the potential in saving run size and its technical novelty. In this paper, a novel combinatorial criterion, called minimum moment aberration, is proposed for assessing the goodness of nonregular designs and supersaturated designs. The new criterion, which is to sequentially minimize the power moments of the number of coincidence among runs, is a good surrogate with tremendous computational advantages for many statistically justified criteria, such as minimum G2-aberrration, generalized minimum aberration and E(s2). In addition, the minimum moment aberration is conceptually simple and convenient for theoretical development. The general theory developed here not only unifies several separate results, but also provides many novel results on nonregular designs and supersaturated designs.

The Nordstrom and Robinson code is a well-known nonlinear code in coding theory. This paper explores the statistical properties of this nonlinear code. Many nonregular designs with 32, 64, 128, 256 runs and 7-16 factors are derived from it. It is shown that these nonregular designs are better than regular designs of the same size in terms of resolution, aberration and projectivity. Furthermore, many of these nonregular designs are shown to have generalized minimum aberration among all possible designs. Seven orthogonal arrays are shown to have unique wordlength pattern and four of them are shown to be unique up to isomorphism.

This paper considers the construction of minimum aberra- tion (MA) blocked factorial designs. Based on coding theory, the con- cept of minimum moment aberration due to Xu (2003) for unblocked designs is extended to blocked designs. The coding theory approach studies designs in a row-wise fashion and therefore links blocked designs with nonregular and supersaturated designs. A lower bound on blocked wordlength pattern is established. It is shown that a blocked design has MA if it originates from an unblocked MA design and achieves the lower bound. It is also shown that a regular design can be partitioned into maximal blocks if and only if it contains a row without zeros. Sufficient conditions are given for constructing MA blocked designs from unblocked MA designs. The theory is then applied to construct MA blocked designs for all 32 runs, 64 runs up to 32 factors, and all 81 runs with respect to four combined wordlength patterns.

Nowadays, the world is to develop renewable energy and algal biofuel is a promis-

ing alternative to fossil fuel. However the high cost of commercial culture median,

like Modied Bold 3N is an insuperable bottleneck in the large-scale production

of algae-based biofuels. In this paper, we use orthogonal array composite designs

(OACDs) to search optimal chemical combination settings. We initially use a 52-

run OACD with six dierent chemicals. Data analysis suggests that two chemicals

could be excluded from combination since they lack of signicance. To further

study, a follow-up 28-run OACD using four chemicals indicates the optimal com-

bination settings for these chemicals.

We have employed high resolution synchrotron radiation x-rays 3D imaging techniques, to study the electromigration (EM) failure mechanism in flip-chip solder joints. In these studies, the ex-situ imaging of the early-stage damage evolution at the interface between under bump metallurgy (UBM) layer and solder balls, revealed that the EM induced failure mode of solder joints could be described by Johnson-Mehl-Avrami (JMA) kinetics model. Thus, the JMA kinetics is proposed to serve as a new physical model for life-time prediction of Pb-free solder joints in EM tests. A corresponding Monte Carlo simulation is developed to investigate this simplified failure model and the dependence of the scale parameter and shape parameter in the statistical Weibull equation on the physical factors of the EM tests is studied.

Maximin distance Latin hypercube designs are becoming increasingly prevalent in computer experiments. As addressed by Wang,Xiao and Xu (2018), $p \times (p-1)$ optimal designs, \ or asymptotically optimal designs based on good lattice point sets have been successfully constructed using Williams transformation; in this paper, we would like to further this idea for more general, or more flexible, designs, such as $N \times \frac{(N-1)}{2}$ for N equal to primes, prime multiples and prime powers, and implement a similar construction algorithm to build optimal or asymptotically optimal designs of such dimension.

Multiple drugs is much more effective in a cancer treatment than single drugs; however, it is challenging to find a reliable model that describes the relationship between drugs and the cell activities, as well as to construct efficient designs for reducing the number of drug combinations needed to be tested. We analyze data that consists of cellular ATP-levels on lung cancer cells and normal cells with 3 inhibitor drugs, AG490, U0126 and I-3'-M, each at 8 dosage levels. We construct models with different number of predictors and methods as well as different subsets of the full data. We find that large dataset does not guarantee the accuracy of a model's prediction, but an appropriate design and a good model building method are the keys. Square root transformation on the response gives a model prediction surpassing the performance among all other models we built. A second order model constructed with a 27-run factorial design performs remarkably well for both cancer cells and normal cells. It is in general much harder to model or predict cancer cells than normal cells.

Maximin distance designs as an important class of space-filling designs are widely used in computer experiments, yet their constructions are challenging. In this thesis, we develop an efficient procedure to generate maximin Latin hypercube designs, as well as maximin multi-level fractional factorial designs, from existing orthogonal or nearly orthogonal arrays via level permutation and expansion. We show that the distance distributions of the generated designs are closely connected with the distance distributions and generalized word-length patterns of the initial designs. Examples are presented to show that our method outperforms many current prevailing methods. In addition, based on number theory and finite fields, we propose three algebraic methods to construct maximin distance Latin squares, as special Latin hypercube designs. We develop lower bounds on their minimum distances. The resulting Latin squares and related Latin hypercube designs have larger minimum distances than existing ones, and are especially appealing for high-dimensional applications. We show an application of space-filling designs in a combinatorial drug experiment on lung cancer. We compare four types of designs: a 512-run 8-level full factorial design, 80-run random sub-designs, 27-run random sub-designs and a 27-run space-filling three-level sub-design under four types of models: Kriging models, neural networks, linear models and Hill-based nonlinear models. We find that it is the best to adopt space-filling designs fitting Kriging models.