Selection and screening procedures to determine optimal product designs

To compare several promising product designs, manufacturers must measure their performance under multiple environmental conditions. In many applications, a product design is considered to be seriously flawed if its performance is poor for any level of the environmental factor. For example, if a particular automobile battery design does not function well under temperature extremes, then a manufacturer may not want to put this design into production. Thus, this paper considers the measure of a product's quality to be its worst performance over the levels of the environmental factor. We develop statistical procedures to identify (a near) optimal product design among a given set of product designs, i.e., the manufacturing design that maximizes the worst product performance over the levels of the environmental variable. We accomplish this by intuitive procedures based on the split-plot experimental design (and the randomized complete block design as a special case); split-plot designs have the essential structure of a product array and the practical convenience of local randomization. Two classes of statistical procedures are provided. In the first, the δ-best formulation of selection problems, we determine the number of replications of the basic split-plot design that are needed to guarantee, with a given confidence level, the selection of a product design whose minimum performance is within a specified amount, δ, of the performance of the optimal product design. In particular, if the difference between the quality of the best and second best manufacturing designs is δ or more, then the procedure guarantees that the best design will be selected with specified probability. For applications where a split-plot experiment that involves several product designs has been completed without the planning required of the δ-best formulation, we provide procedures to construct a ‘confidence subset’ of the manufacturing designs; the selected subset contains the optimal product design with a prespecified confidence level. The latter is called the subset selection formulation of selection problems. Examples are provided to illustrate the procedures.     

Taguchi methods: linear graphs of high resolution

An important reason behind the success of the Taguchi methodology in qual- ity assurance has been the use of statistical methods, presented in a way that is accessible to the nonexpert user. Among the tools used to simplify the sta- tistical design of experiments has been the linear graph, apparently introduced by Taguchi. However, he did not consider the resolution of the corresponding designs (the higher the resolution, the more accurate the conclusions). For example, it will be shown that half of the linear graphs given by Taguchi for the L₁₆(2¹⁵) orthogonal array correspond to designs of resolution III, when designs of resolution IV are available (with the same lines in the linear graphs but with different assignments to the columns of the orthogonal array). A nontraditional but very straightforward method is presented for obtaining the alias chains and the linear graphs corresponding to an orthogonal array. The procedure can be easily understood and employed by nonstatisticians to find an experimental design of the highest possible resolution. The design can be used to obtain products or processes that are robust to variation.     

Combined array approach for optimal designs

Robust parameter design, originally proposed by Taguchi ( 1987 ) is an offline production technique for reducing variation and improving product's quality To achieve this objective Taguchi proposed the use of product arrays. However. the product array approach, results in an exorbitant number of runs To overcome the drawbacks of the product array Welch, Wu, Kang and Sacks ( 1990 ), Shoemaker, Tsui and Wu ( 1991 ) and Montgomery ( 1991a ) proposed the use of combined arrays, where the control factors and noise factors are combined in a single array. In this paper we study the concept of combined array for an intermediate class of designs where n = 1 (mod4), n = 2 (mod4) and n = 3 (mod4). The designs presented in this paper, though not orthogonal, offer a great reduction in the run-size.     

Catalogue of group structures for two-level fractional factorial designs

SUMMARY Taguchi introduced the concept of split-unit design to sort factors into different groups with respect to difficulties involved in changing the levels of factors. Li et al. have developed all possible group structures for eight factors in an L16 orthogonal array for resolution IV with split-plot design. Chen et al. have searched for a best design, according to the various criteria for two-level fractional factorial design and have presented a catalogue. In this paper, we have developed an algorithm for generating group structure and possible allocations for various 2n- k fractional factorial designs that correspond to the designs given by Chen et al.     

Further exploratory analysis of split-plot experiments to study certain stratified effects

Designed experiments are a key component in many companies' improvement strategies. Because completely randomized experiments are not always reasonable from a cost or physical perspective, split-plot experiments are prevalent. The recommended analysis accounts for the different sources of variation affecting whole-plot and split-plot error. However experiments on industrial processes must be run and, consequently analyzed quite differently from ones run in a controlled environment. Such experiments are typically subject to a wide array of uncontrolled, and barely understood, variation. In particular, it is important to examine the experimental results for additional, unanticipated sources of variation. In this paper, we consider how unanticipated, stratified effects may influence a split-plot experiment and discuss further exploratory analysis to indicate the presence of stratified effects. Examples of such experiments are provided, additional tests are suggested and discussed in light of their power, and recommendations given.     

Response model analysis for robust design experiments

Taguchi's robust design technique, also known as parameter design, focuses on making product and process designs insensitive (i.e., robust) to hard to control variations. In some applications, however, his approach of modeling expected loss and the resulting “product array” experimental format leads to unnecessarily expensive and less informative experiments. The response model approach to robust design proposed by Welch, Ku, Yang, and Sacks (1990), Box and Jones (1990), Lucas (1989), and Shoemaker, Tsui and Wu (1991) offers more flexibility and economy in experiment planning and more informative modeling. This paper develops a formal basis for the graphical data-analytic approach presented in Shoemaker et al. In particular, we decompose overall response variation into components representing the variability contributed by each noise factor, and show when this decomposition allows us to use individual control-by-noise interaction plots to minimize response variation. We then generalize the control-by-noise interaction plots to extend their usefulness, and develop a formal analysis strategy using these plots to minimize response variation.     

Split-plot designs for multistage experimentation

Most of today’s complex systems and processes involve several stages through which input or the raw material has to go before the final product is obtained. Also in many cases factors at different stages interact. Therefore, a holistic approach for experimentation that considers all stages at the same time will be more efficient. However, there have been only a few attempts in the literature to provide an adequate and easy-to-use approach for this problem. In this paper, we present a novel methodology for constructing two-level split-plot and multistage experiments. The methodology is based on the Kronecker product representation of orthogonal designs and can be used for any number of stages, for various numbers of subplots and for different number of subplots for each stage. The procedure is demonstrated on both regular and nonregular designs and provides the maximum number of factors that can be accommodated in each stage. Furthermore, split-plot designs for multistage experiments with good projective properties are also provided.     

Catalogue of Group Structures for Three-level Fractional Factorial Designs

Taguchi (1959) introduced the concept of split-unit design to sort the factors into different groups depending upon the difficulties involved in changing the levels of factors. Li et al. (1991) renamed it as split-plot design. Chen et al. (1993) have given a catalogue of small designs for two- and three-level fractional factorial designs pertaining to a single type of factors. Aggarwal et al. (1997) have given a catalogue of group structure for two-level fractional factorial designs developed under the concept of split-plot design. In this paper, an algorithm has been developed for generating group structure and possible allocations for various 3^n-k fractional factorial designs.     

Identification of factors influencing dispersion in split-plot experiments

As split-plot designs are commonly used in robust design it is important to identify factors in these designs that influence the dispersion of the response variable. In this article, the Bergman-Hynén method, developed for identification of dispersion effects in unreplicated experiments, is modified to be used in the context of split-plot experiments. The modification of the Bergman-Hynén method enables identification of factors that influence specific variance components in unreplicated two-level fractional factorial splitplot experiments. An industrial example is used to illustrate the proposed method.     

An application of taguchi's methods reconsidered

Two aspects of Taguchi's methods for analyzing parameter design experiments that can be improved upon are considered. It is shown how using interaction graphs instead of marginal graphs, and how using the sample variance instead of a signal-to-noise ratio, can lead to product designs that are more robust to variation. The advantages of the alternative analysis will be illustrated by reanalyzing a case study considered by Barker (1986).     

Optimization of parameters for the fabrication of gelatin nanoparticles by the Taguchi robust design method

The Taguchi method is a statistical approach to overcome the limitation of the factorial and fractional factorial experiments by simplifying and standardizing the fractional factorial design. The objective of this study was to optimize the fabrication of gelatin nanoparticles by applying the Taguchi design method. Gelatin nanoparticles have been extensively studied in our previous works as an appropriate carrier for drug delivery, since they are biodegradable, non-toxic, are not usually contaminated with pyrogens and possess relatively low antigenicity. Taguchi method with L₁₆ orthogonal array robust design was implemented to optimize experimental conditions of the purpose. Four key process parameters – temperature, gelatin concentration, agitation speed and the amount of acetone – were considered for the optimization of gelatin nanoparticles. As a result of Taguchi analysis in this study, temperature and amount of acetone were the most influencing parameters of the particle size. For characterizing the nanoparticle sample, atomic force microscope and scanning electron microscope were employed. In this study, a minimum size of gelatin nanoparticles was obtained at 50 °C temperature, 45 mg/ml gelatin concentration, 80 ml acetone and 700 rpm agitation speed. The nanoparticle size at the determined condition was less than 174 nm.     

Modifying a central composite design to model the process mean and variance when there are hard-to-change factors

Summary.  An important question within industrial statistics is how to find operating conditions that achieve some goal for the mean of a characteristic of interest while simultaneously minimizing the characteristic's process variance. Often, people refer to this kind of situation as the robust parameter design problem. The robust parameter design literature is rich with ways to create separate models for the mean and variance from this type of experiment. Many times time and/or cost constraints force certain factors of interest to be much more difficult to change than others. An appropriate approach to such an experiment restricts the randomization, which leads to a split-plot structure. The paper modifies the central composite design to allow the estimation of separate models for the characteristic's mean and variances under a split-plot structure. The paper goes on to discuss an appropriate analysis of the experimental results. It illustrates the methodology with an industrial experiment involving a chemical vapour deposition process for the manufacture of silicon wafers. The methodology was used to achieve a silicon layer thickness value of 485 Å while minimizing the process variation.     

Covariance Analysis for Split-Plot and Split-Block Designs

Commentaries are informative essays dealing with viewpoints of statistical practice, statistical education, and other topics considered to be of general interest to the broad readership of The American Statistician. Commentaries are similar in spirit to Letters to the Editor, but they involve longer discussions of background, issues, and perspectives. All commentaries will be refereed for their merit and compatibility with these criteria.

Proper methodology for the analysis of covariance for experiments designed in a split-plot or split-block design is not found in the statistical literature. Analyses for these designs are often performed incompletely or even incorrectly. This is especially true when popular statistical computer software packages are used for the analysis of these designs. This article provides several appropriate models, ANOVA tables, and standard errors for comparisons from experiments arranged in a standard split-plot, split–split-plot, or split-block design where a covariate has been measured on the smallest size experimental unit.     

Projection array based designs for computer experiments

In this paper we define a new class of designs for computer experiments. A projection array based design defines sets of simulation runs with properties that extend the conceptual properties of orthogonal array based Latin hypercube sampling, particularly to underlying design structures other than orthogonal arrays. Additionally, we illustrate how these designs can be sequentially augmented to improve the overall projection properties of the initial design or focus on interesting regions of the design space that need further exploration to improve the overall fit of the underlying response surface. We also illustrate how an initial Latin hypercube sample can be expressed as a projection array based design and show how one can augment these designs to improve higher dimensional space filling properties.     

Response surface design evaluation and comparison

Designing an experiment to fit a response surface model typically involves selecting among several candidate designs. There are often many competing criteria that could be considered in selecting the design, and practitioners are typically forced to make trade-offs between these objectives when choosing the final design. Traditional alphabetic optimality criteria are often used in evaluating and comparing competing designs. These optimality criteria are single-number summaries for quality properties of the design such as the precision with which the model parameters are estimated or the uncertainty associated with prediction. Other important considerations include the robustness of the design to model misspecification and potential problems arising from spurious or missing data. Several qualitative and quantitative properties of good response surface designs are discussed, and some of their important trade-offs are considered. Graphical methods for evaluating design performance for several important response surface problems are discussed and we show how these techniques can be used to compare competing designs. These graphical methods are generally superior to the simplistic summaries of alphabetic optimality criteria. Several special cases are considered, including robust parameter designs, split-plot designs, mixture experiment designs, and designs for generalized linear models.     

Generalized minimum aberration two-level split-plot designs

Split-plot experiments may arise when it is impractical to completely randomize the treatment combinations of a designed experiment. To provide more flexible design choices in the nonregular split-plot setting, we describe an approach for constructing minimum aberration orthogonal two-level split-plot designs having 12, 16, 20 and 24 runs. We consider five design scenarios that may be of importance to practitioners, and then propose an approach for assigning word lengths under these five scenarios. We then use the extended word length patterns to rank both regular and nonregular orthogonal split-plot designs. While most existing papers concerning orthogonal split-plot designs focus on regular orthogonal designs, we find that many minimum aberration split-plot designs are nonregular orthogonal designs.     

A simple method to obtain minimum aberration split-plot designs

ABSTRACT

Split-plot designs have been utilized in factorial experiments with some factors applied to larger units and others to smaller units. Such designs with low aberration are preferred when the experimental size and the number of factors considered in both whole plot and subplot are determined. The minimum aberration split-plot designs can be obtained using either computer algorithms or the exhausted search. In this article, we propose a simple, easy-to-operate approach by using two ordered sequences of columns from two orthogonal arrays in obtaining minimum aberration split-plot designs for experiments of sizes 16 and 32.     

Robust parameter design: Optimization of combined array approach with orthogonal arrays

Robust parameter design, originally proposed by Taguchi (1987. System of Experimental Design, vols. I and II. UNIPUB, New York), is an off-line production technique for reducing variation and improving a product's quality by using product arrays. However, the use of product arrays results in an exorbitant number of runs. To overcome the drawbacks of the product array 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 certain orthogonal arrays that are embedded into Hadamard matrices as combined arrays, in order to identify a model that contains all the main effects (control and noise) and their control-by-noise interactions with high efficiency. Aliasing of effects in each case is also discussed.     

A comparison of inference procedures in unbalanced split-plot designs

Several procedures for constructing confidence intervals and testing hypotheses about fixed effects in unbalanced split-plot experiments are described in this paper. These procedures can also be used for unbalanced repeated measures experiments when the repeated measures satisfy the Huyhn-Feldt (1970) conditions. A number of these procedures require that the whole plot error mean square has a distribution proportional to a chi-square distribution and that it be independent of estimators of the parameter functions. Often, neither of these conditions are met in unbalanced split-plot experiments. Simulation studies of a small design of eight observations and larger designs with 34 to 48 observations are used to investigate the performance of the different procedures.     

Correlation coefficient of response variances in compound noise and product array experiments

In robust parameter design, a compound noise experiment has been frequently used for reducing the number of experimental runs compared to a product array experiment. However, the results obtained by the compound noise experiment and the product array experiment are often much different. This paper derives an expression of the correlation coefficient of response variances in the compound noise and the product array experiments, which gives an explanation of that difference.