Abstract: Orthogonal arrays are used widely in manufacturing and high-technology 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 that can construct a variety of small-run designs with good statistical properties efficiently.

Abstract: A covering array of sizeN, strengtht, degree k, and order υ is a k × N array on υ symbols in which every t × N subarray contains every possible t × 1 column at least once. We present explicit constructions, constructive upper bounds on the size of various covering arrays, and compare our results with those of a commercial product. Applications of covering arrays include software testing, drug screening, and data compression. © 2002 Wiley Periodicals, Inc. J Combin Designs 10: 217–238, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/jcd.10002

Abstract: Orthogonal arrays are widely used in industrial experiments for factor screening. Suppose only a few of the factors are important. An orthogonal array can be used not only for screening factors but also for detecting interactions among a subset of active factors. In this article, a set of optimality criteria is proposed to assess the performance of designs for factor screening, pro jection, and interaction detection, and a three-step approach is proposed to search for opti- mal designs. Combinatorial and algorithmic construction methods are proposed for generating new designs. Level permutation methods are used for improving the eligibility and estimation efficiency of the pro jected designs. The techniques are then applied to search for best three-level designs with 18 and 27 runs. Many new, efficient and practically useful nonregular designs are found and their properties discussed.

Abstract: We define a $k$-plex to be a partial latin square of order $n$ containing $kn$ entries such that exactly $k$ entries lie in each row and column and each of $n$ symbols occurs exactly $k$ times. A transversal of a latin square corresponds to the case $k=1$. For $k>n/4$ we prove that not all $k$-plexes are completable to latin squares. Certain latin squares, including the Cayley tables of many groups, are shown to contain no $(2c+1)$-plex for any integer $c$. However, Cayley tables of soluble groups have a $2c$-plex for each possible $c$. We conjecture that this is true for all latin squares and confirm this for orders $n\leq8$. Finally, we demonstrate the existence of indivisible $k$-plexes, meaning that they contain no $c$-plex for $1\leq c

Abstract: The issue of uniformity is crucial in quasi-Monte Carlo methods and in the design of computer experiments. In this paper we study the role of uniformity in fractional factorial designs. For fractions of two- or three-level factorials, we derive results connecting orthogonality, aberration and uniformity and show that these criteria agree quite well. This provides further justification for the criteria of orthogonality or minimum aberration in terms of uniformity. Our results refer to several natural measures of uniformity and we consider both regular and nonregular fractions. The theory developed here has the potential of significantly reducing the complexity of computation for searching for minimum aberration designs.

Abstract: Generalization properties of support vector machines, orthogonal least squares and zero-order regularized orthogonal least squares algorithms are studied using simulation. For high signal-to-noise ratios (40 dB), mixed results are obtained, but for a low signal-to-noise ratio, the prediction performance of support vector machines is better than the orthogonal least squares algorithm in the examples considered. However, the latter can usually give a parsimonious model with very fast training and testing time. Two new algorithms are therefore proposed that combine the orthogonal least squares algorithm with support vector machines to give a parsimonious model with good prediction accuracy in the low signal-to-noise ratio case.

Abstract: The current literature on fractional factorial plans in block designs centres around orthogonal blocking which may not, however, always be attainable because of practical restrictions on the block size. For general factorials, including asymmetric ones, sufficient conditions are indicated in this paper for a main effect plan to be universally optimal under possibly non-orthogonal blocking. A construction procedure is given using generalised Youden designs in conjunction with orthogonal arrays. We also illustrate how the procedure can be applied to obtain optimal main effect plans in the practically important situation where each factor has two or three levels and the block size is small.

Abstract: (t,m,s)-nets are point sets in Euclidean s-space satisfying certain uniformity conditions, for use in numerical integration. They can be equivalently described in terms of ordered orthogonal arrays, a class of finite geometrical structures generalizing orthogonal arrays. This establishes a link between quasi-Monte Carlo methods and coding theory. The ambient space is a metric space generalizing the Hamming space of coding theory. We denote it by NRT space (named after Niederreiter, Rosenbloom and Tsfasman). Our main results are generalizations of coding-theoretic constructions from Hamming space to NRT space. These comprise a version of the Gilbert-Varshamov bound, the (u,u+υ)-construction and concatenation. We present a table of the best known parameters of q-ary (t,m,s)-nets for qe{2,3,4,5} and dimension m≤50. © 2002 Wiley Periodicals, Inc. J Combin Designs 10: 403–418, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/jcd.10015

Abstract: Fractional factorial design is arguably the most widely used design in experimental investigation, and uniformity has gained popularity in experimental designs in recent years. In this present paper, a suitable measure of uniformity, i.e. a discrete discrepancy, is defined by the reproducing kernel Hilbert space, and is used to evaluate the uniformity of fractional factorial designs. Some relations between orthogonality and uniformity in fractional factorial designs are obtained. The results show that orthogonality and uniformity are strongly related to each other and the discrepancy plays an important role in evaluating such experimental designs.

Abstract: In this paper we use incidence matrices of block designs and row–column designs to obtain combinatorial inequalities. We introduce the concept of nearly orthogonal Latin squares by modifying the usual definition of orthogonal Latin squares. This concept opens up interesting combinatorial problems and is expected to be useful in planning experiments by statisticians. © 2002 John Wiley & Sons, Inc. J Combin Designs 10: 17–26, 2002

Abstract: Taguchi's idle column method for constructing arrays is investigated and generalized. A general construction method is proposed via a necessary and sufficient condition on two-level orthogonal arrays to which the method is applicable. A method is developed for identifying the idle column and pairs of columns in the arrays. Mixed two- and three-level idle column arrays with 64 or fewer runs are tabulated. Some guidelines are offered on when the idle column arrays should be used and when they should be replaced by orthogonal main-effect plans or D-optimal designs.

Abstract: It is demonstrated, through a case study, how statistically designed experiments can help in improving online process control schemes remarkably. The study was conducted in the packing plant of a company manufacturing urea. The packing machine had an in-built computerized control mechanism that was supposed to control the variation in packed bag weights by automatic measurement and adjustment. However, despite following the instruction manual rigorously, the machine was unable to deliver a satisfactory performance from the point of view of variation. An orthogonal array experiment conducted with some of the machine parameters could bring about a dramatic improvement in the effectiveness of the control scheme.

Abstract: A novel experimental scheme is proposed to integrate orthogonal function approximation with Taguchi's method (orthogonal array) for designing the optimal manipulated trajectory of a batch process. The orthogonal function approximation finds a set of orthonormal functions as the basis to represent the batch trajectory. The optimal trajectory can be obtained if the location of the coefficients of the orthonormal functions is properly adjusted in the function space. The Taguchi approach is used to design and analyze the effect of each coefficient on reaching the optimal objective (quality) function. Because the coefficients are implicitly related to the objective function, they simply vary over two levels in a systematic way, and they would be moved into the optimal design condition. A search procedure for the optimal design coefficients is also proposed. As opposed to model-based design, the proposed method utilizes the simplicity of Taguchi methods to determine the potentially available knowledge of dynami...

Abstract: The orthogonality is an important property in experimental designs. We use the saturated or supersaturated designs when we cannot execute a full factorial experiment under practical situations in which cost constraint does not allow a full factorial experiment. Unfortunately, there does not exist orthogonal designs in many cases. Therefore, we need evaluate the degree of the non-orthogonality of given experimental designs. Several measures for evaluating non-orthogonality of the experimental designs are suggested.

Abstract: If the number of runs in a (mixed-level) orthogonal array of strength 2 is specified, what numbers of levels and factors are possible? The collection of possible sets of parameters for orthogonal arrays with N runs has a natural lattice structure, induced by the ``expansive replacement'' construction method. In particular the dual atoms in this lattice are the most important parameter sets, since any other parameter set for an N-run orthogonal array can be constructed from them. To get a sense for the number of dual atoms, and to begin to understand the lattice as a function of N, we investigate the height and the size of the lattice. It is shown that the height is at most [c(N-1)], where c= 1.4039... and that there is an infinite sequence of values of N for which this bound is attained. On the other hand, the number of nodes in the lattice is bounded above by a superpolynomial function of N (and superpolynomial growth does occur for certain sequences of values of N). Using a new construction based on ``mixed spreads'', all parameter sets with 64 runs are determined. Four of these 64-run orthogonal arrays appear to be new.

Abstract: The goal of this paper was to design the optimal time-varying operating pH profile in the asymmetric reduction of ethyl 4-chloro-3-oxobutyrate by baker's yeast. Ethyl (S)-4-chloro-3-hydroxybutyrate was produced to reach two important quality indices: reaction yield and product optical purity. The method integrated an orthogonal function approximation and an orthogonal array. The technique used a set of orthonormal functions as the basis for representing the possible profile. The optimal profile could be obtained if the orthogonal coefficients were properly adjusted. The orthogonal array was used to design and analyze the effect of each orthogonal coefficient in order to reach the optimal objective (quality) function. The performance based on the proposed strategy was significantly improved by over 10% compared with the traditional fixed pH or uncontrolled pH values during the reaction. The proposed method can be applied to the required dynamic profile in the bioreactor process to effectively improve the product quality, given good design directions and the advantage of the traditional statistical approach.

Abstract: This study describes an application of the Taguchi method in optimizing the fused process parameters that have been made in the development of fused bi-conic taper couplers, to improve the performance and reliability of the 1% (1/99) single-window broadband tap coupler. A Taguchi 18(21×37) orthogonal array is used to study the effects of six control factors on the six interested responses. Implementation results reveal that the Taguchi method outperforms the traditional engineering judgment method. This study has now been extensively and successfully used to develop the optimal fuse parameters for other coupling ratio tap couplers such as 2/98, 3/97, 4/96, …, 50/50.

Abstract: A critical set in a Latin square of order n is a set of entries in an n×n array which can be embedded in precisely one Latin square of order n, with the property that if any entry of the critical set is deleted, the remaining set can be embedded in more than one Latin square of order n. The number of critical sets grows super-exponentially as the order of the Latin square increases. It is difficult to find patterns in Latin squares of small order (order 5 or less) which can be generalised in the process of creating new theorems. Thus, I have written many algorithms to find critical sets with various properties in Latin squares of order greater than 5, and to deal with other related structures. Some algorithms used in the body of the thesis are presented in Chapter 3; results which arise from the computational studies and observations of the patterns and subsequent results are presented in Chapters 4, 5, 6, 7 and 8. The cardinality of the largest critical set in any Latin square of order n is denoted by lcs(n). In 1978 Curran and van Rees proved that lcs(n) =2) and m×2^α+1 (m odd, α>=2 and α≠3), and a new lower bound on lcs(4m). It also discusses critical sets in intercalate-rich Latin squares of orders 11 and 14. In Chapter 6 a construction is given which verifies the existence of a critical set of size n²÷ 4 + 1 when n is even and n>=6. The construction is based on the discovery of a critical set of size 17 for a Latin square of order 8. In Chapter 7 the representation of Steiner trades of volume less than or equal to nine is examined. Computational results are used to identify those trades for which the associated partial Latin square can be decomposed into six disjoint Latin interchanges. Chapter 8 focusses on critical sets in Latin squares of order at most six and extensive computational routines are used to identify all the critical sets of different sizes in these Latin squares.

Abstract: Orthogonal functions play an important role in factorial experiments and time series models. In the latter half of the twentieth century orthogonal functions became prominent in industrial experimentation methodologies that employ complete and fractional factorial experiment designs, such as Taguchi orthogonal arrays. Exact estimates of the parameters of linear model representations can be computed effectively and efficiently using “fast algorithms.” The origin of “fast algorithms” can be traced to Yates in 1937. In 1958 Good created the ingenious fast Fourier transform, using Yates’s concept as a basis. This paper is intended to illustrate the fundamental role of orthogonal functions in modeling, and the close relationship between two of the most significant of the fast algorithms. This in turn yields insights into the fundamental aspects of experiment design.

Abstract: When an orthogonal array is projected on a small number of factors, as is done in screening experiments, the question of interest is the structure of the projected design, by which we mean its decomposition in terms of smaller arrays of the same strength. In this paper we investigate the decomposition of arrays of strength t having t + 1 factors. The decomposition problem is well-understood for symmetric arrays on s = 2 symbols. In this paper we derive some general results on decomposition, with particular attention to arrays on s = 3 symbols. We give a new proof of the regularity of arrays of index 1 when s = 2 or 3, and show by counterexample that the result doesn’t extend to larger s. For s = 3 we also construct an indecomposable array of index 2. Finally, we determine the structure of completely decomposable arrays on 3 symbols having strength 2 and index 2, 3 or 4.

Abstract: The topic of this paper is experiment planning, particularly fractional factorial designs or orthogonal arrays, for computer experiments to assess important inputs. The work presented in the paper is motivated by considering a non-stochastic computer simulation which has many inputs and which can, in a reasonable period of time, be run thousands of times. With many inputs, information that allows focus on a subset of important inputs is valuable. The characterization of 'importance' is expected to follow suggestions in McKay (1995) or McKay, et al. (1992). This analysis approach leads to considering factorial experiment designs. Inputs are associated with a finite number of discrete values, referred to as levels, so if each input has K levels and there are p inputs then there are K{sup P} possible distinct runs which constitute the K{sup P} factorial design space. The suggested size of p has been 35 to 50 so that even with K=2 the complete 2{sup P} factorial design space would not be run. Further, it is expected that the complexity of the simulation code and discrete levels possibly associated with equi-probable intervals from the input distribution make it desirable to consider more than 2 level inputs. Inputs levels of 5more » and 7 have been investigated. In this paper, orthogonal array experiment designs, which are subsets of factorial designs also referred to as fractional factorial designs, are suggested as candidate experiments which provide meaningful basis for calculating and comparing R{sup 2} across subsets of inputs.« less

Abstract: In this study, the geometric representation of an Orthogonal Array is obtained using flnite analytic projective geometry of the Galois fleld GF(s) of t-dimensions, which can be denoted by PG(t; s), where s is a prime or a power of a prime number. We give relations between the parameters of Orthogonal Arrays and properties of the projective geometry and of related geometries. We ofier some geometrical examples.