In robust designs, the control-by-noise interactions are usually considered to be more important than other two-factor interactions. We extend the work on model-robust factorial designs of Li and Nachtsheim (2000) to robust designs, where not all factors are treated equally. A new criterion is proposed to maximize a design's ability to estimate models with at least one control-by-noise interaction. Optimal designs are chosen from the following three groups: regular orthogonal arrays (OAs), non-regular OAs, and balanced designs. Designs with economic run-sizes are constructed algorithmically and are tabulated for practical use.Key Words: Model Ordering, Model-Robust Design, Optimal Design.

*By* **DEREK R. BINGHAM, University of Michigan, Ann
Arbor, MI 48109
WILLIAM LI, University of Minnesota, Minneapolis, MN 55455**

Introduction

ROBUST parameter design, or simply parameter design, is an approach to planned experimentation that aims to reduce the variability of a process or product. An experiment is performed by varying the level settings of controllable process factors (control factors) and difficult-to-control noise factors to determine which of the control factors have dispersion effects. Variance reduction is achieved by identifying the settings of the control factors that make the system robust to variation in the noise factors.

Taguchi (1987) recommended the product array approach, which crosses separate designs for the control and noise factors. The experiment is analyzed using location-dispersion modeling, where location and dispersion measures are computed separately to see how each depends on the level settings of the control factors (Box (1988)). This approach does not explicitly include the noise factors in the analysis of the experiment.

Product arrays often result in designs that require a large number of trials (see Wu and Hamada (2000, p. 461) and references therein). An alternative approach that frequently results in smaller designs than product arrays is the use of a single array to set the levels of both the control and noise factors (Welch, Yu, Kang, and Sacks (1990) and Shoemaker, Tsui, and Wu (1991)). A location model, considering both control and noise factors explicitly, is tted, and significant control-by-noise interactions are taken as evidence of dispersion effects. A dispersion model is constructed from the location model by viewing the noise factors as random effects (Myers, Khuri, and Vining (1992)).

The primary aim of a robust parameter design is variance reduction. Identi cation of the correct dispersion model is essential, so that the control factors can be adjusted to dampen the impact of variation in the levels of the noise factors. Therefore, some effects, and by extension, models, are more important than others. For instance, models containing interactions between control and noise factors are very important because they provide the potential for variance reduction and the attainment of experimental goals. Parameter designs are performed when it is likely that a dispersion e ect is present. Thus, models where the only interactions are those between noise factors are of less interest.

The choice of the best experimental plan for a robust parameter design is different from standard situations, where e ects of the same order are all treated in the same manner. Independently, Bingham and Sitter (2001) and Wu and Hamada (2000, p. 462) have developed criteria for selecting optimal fractional factorial robust parameter designs with factors at 2-levels. Wu and Hamada (2000, p. 462) selected optimal designs based on an e ect ordering principle. The optimal design sequentially maximizes the number of clear 2-factor interactions (2 s) for the more important e ects (a 2fi is called clear if it is not aliased with any main effect or other 2fi's.). Bingham and Sitter (2001) took a different approach and ranked the words of the fractional factorial design s defining contrast sub-group to account for the ranking of effects and the likely significance of each effect. They modified the definition of word-length by considering the number of control and noise factors in each word, and they then ranked designs using the minimum aberration (MA) criterion (Fries and Hunter (1980)). Their criterion is used to find optimal fractional factorial split-plot designs for robust parameter design.

One of the motivations for this work
"is the model uncertainty" of the parameter design,
i.e., some main effects and 2fi's are assumed to be important,
but it is unknown in advance which of them will be actually
present. In a recent paper, Li and Nachtsheim (2000) proposed
a class of designs, model-robust factorial designs (MRFDs),
that are optimal over a group of models consisting of all
main effects and a certain number of 2fi's. Estimation capacity
(*EC*) is the ratio of the number of estimable models
to the number of possible models (following Cheng, Steinberg,
and Sun (1999) and Li and Nachtsheim (2000)), and the MRFD
maximizes the EC criterion.

The model-robust design approach
can also be applied to parameter designs. Note that the approaches
of both Wu and Hamada (2000) and Bingham and Sitter (2001)
take into account the model uncertainty, but they do not maximize
the *EC* directly. Consider, for example, a 16-run robust
parameter design with 4 control factors and 2 noise factors.
The optimal fractional factorial robust parameter design given
by Wu and Hamada (2000, p. 486) is D_{1} : *I =
ABCD = Apq = BCDpq*; and the optimal MA robust parameter
design using the generalized word length definition in Bingham
and Sitter (2001) is D_{2} : *I = ABDp = ABCq =
CDpq*. Suppose that the number of actual important 2fi's
is *g* = 3, then *EC*(*D*_{1}) = .36
and *EC*(*D*_{2} = .78. From Li and Nachtsheim
(2000), the optimal MRFD yields *EC* = 1.00.

The MRFD usually has high estimation capacities. However, it does not take into account the special structure of the robust parameter design, in which some 2fi's (specifically, control by noise interactions) are more important than other 2fi's. In this article, we extend Li and Nachtsheim s (2000) approach to the optimal model-robust parameter design (MRPD). We introduce a new optimal design criterion for ranking two-level robust parameter designs. Because the analysis of non-regular designs attempts to identify the best model (Hamada and Wu (1992) and Chipman, Hamada, and Wu (1997)), we present a model ordering principle that emphasizes the estimation of the models of interest. A combined estimation capacity and information capacity criterion is proposed to rank order designs (Li and Nachtsheim (2000)). The criterion is general and is applied to the construction of optimal MRPDs from the following three groups:

- Regular orthogonal designs,
- Non-regular orthogonal designs,
- Balanced designs in which there are the same numbers of (+)s and (-)s in each column.

The article is outlined as follows. After briefly introducing robust parameter design and the response model analysis, we introduce the model ordering principle and use it to develop a new criterion to rank designs. Then, the criterion is applied to some regular and non-regular 2-level designs. Finally, using the criterion, the construction of optimal balanced MRPDs is considered, and the methodology is demonstrated for some applications.

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