A graph-aided method for planning two-level experiments when certain interactions are important
TL;DR: In this paper, a graph-aided method is proposed to solve the problem of fractional factorial factorial experiment planning, where prior knowledge may suggest that some interactions are potentially important and should therefore be estimated free of the main effects.
Abstract: In planning a fractional factorial experiment prior knowledge may suggest that some interactions are potentially important and should therefore be estimated free of the main effects. In this article, we propose a graph-aided method to solve this problem for two-level experiments. First, we choose the defining relations for a 2 n–k design according to a goodness criterion such as the minimum aberration criterion. Then we construct all of the nonisomorphic graphs that represent the solutions to the problem of simultaneous estimation of main effects and two-factor interactions for the given defining relations. In each graph a vertex represents a factor and an edge represents the interaction between the two factors. For the experiment planner, the job is simple: Draw a graph representing the specified interactions and compare it with the list of graphs obtained previously. Our approach is a substantial improvement over Taguchi's linear graphs.
Citations
More filters
01 Jan 2005
TL;DR: Fractional factorial plans for asymmetric factorial experiments were shown to be universally optimal within the class of all plans involving the same number of runs under a model that includes the mean, all main eects and a specied set of two-factor interactions as discussed by the authors.
Abstract: Fractional factorial plans for asymmetric factorial experiments are ob- tained. These are shown to be universally optimal within the class of all plans involving the same number of runs under a model that includes the mean, all main eects and a specied set of two-factor interactions. Finite projective geometry is used to obtain such plans for experiments wherein the number of levels of each of the factors and the number of runs is a power of m, a prime or a prime power. Methods of construction of optimal plans under the same model are also discussed for the case where the number of levels as well as the number of runs are not necessarily powers of a prime number.
3 citations
Cites background from "A graph-aided method for planning t..."
...The issue of estimability and optimality in situations of this kind has been addressed by Hedayat and Pesotan (1992, 1997), Wu and Chen (1992) and Chiu and John (1998) in the context of two-level factorials, and by Dey and Mukerjee (1999b) and Chatterjee, Das and Dey (2002) for arbitrary factorials…...
[...]
TL;DR: This paper considers the construction of D-optimal two-level orthogonal arrays that allow for the joint estimation of all main effects and a specified set of two-factor interactions and a sharper upper bound on the determinant of the related matrix is derived.
3 citations
TL;DR: In this paper, a maximum estimability (maxest) criterion is proposed for design classification and selection, which is an extension and refinement of Webb's resolution criterion for general factorial designs.
3 citations
Cites background from "A graph-aided method for planning t..."
...Wu and Chen (1992) proposed the concept of clear effects for regular designs, which states that a main effect or 2fi is clear if it is not aliased with any other main effects or 2fi’s (hence it is estimable in the second-order model) and is strongly clear if it is not aliased with any 3fi’s as well…...
[...]
...Wu and Chen (1992) proposed the concept of clear effects for regular designs, which states that a main effect or 2fi is clear if it is not aliased with any other main effects or 2fi’s (hence it is estimable in the second-order model) and is strongly clear if it is not aliased with any 3fi’s as well (therefore, it is estimable in the third-order model)....
[...]
TL;DR: This note builds on results from Wu, Mee, and Tang's article (henceforth WMT) on admissible fractional factorial two-level designs by concentrating on the “dominating designs” that were introduced but not further pursued in WMT.
Abstract: This note builds on results from Wu, Mee, and Tang's article (henceforth WMT) on admissible fractional factorial two-level designs, specifically concentrating on the “dominating designs” that were introduced but not further pursued in WMT. WMT's work has been used for increasing the efficiency of the author's graph-based algorithm for creation of minimum aberration designs that keep a requirement set of two-factor interactions clear: That algorithm originally searched aberration-sorted complete catalogs of nonisomorphic resolution IV designs; according to WMT's proposal, it suffices to search the dominating designs. For implementing this reduction of search space, aberration-sorted catalogs of dominating resolution IV regular fractional factorial two-level designs with up to 128 runs were created; this was achieved with the help of a recent subgraph isomorphism algorithm by Solnon that proved to be particularly efficient for this search task. The online supplementary materials provide the code used for id...
3 citations
TL;DR: In this article, the existence of clear two-factor interaction components in mixed-level factorial designs with resolution III or IV was investigated and the results revealed the structures of these designs. But the results were limited to the case of two high-level factors.
Abstract: Mixed-level designs are widely used in factorial experiments. Clear effects criterion is one of the important rules for selecting optimal fractional factorial designs. It is highly desirable to know when mixed-level designs with resolution III or IV can have clear two-factor interaction components. This paper considers mixed-level designs with one or two high-level factors and some two-level factors, denoted as \((2^{r})\times 2^n\) and \((2^{r_1})\times (2^{r_2})\times 2^n\), respectively, and gives a complete classification of the existence of clear two-factor interaction components in such designs with resolution III or IV. The results reveal the structures of these designs.
3 citations
References
More filters
Book•
01 Jan 2015TL;DR: This book offers a complete blueprint for structuring projects to achieve rapid completion with high engineering productivity during the research and development phase to ensure that high quality products can be made quickly and at the lowest possible cost.
Abstract: From the Publisher:
Phadke was trained in robust design techniques by Genichi Taguchi, the mastermind behind Japanese quality manufacturing technologies and the father of Japanese quality control. Taguchi's approach is currently under consideration to be adopted as a student protocol with the US govrnment. The foreword is written by Taguchi. This book offers a complete blueprint for structuring projects to achieve rapid completion with high engineering productivity during the research and development phase to ensure that high quality products can be made quickly and at the lowest possible cost. Some topics covered are: orthogonol arrays, how to construct orthogonal arrays, computer-aided robutst design techniques, dynamic systems design methods, and more.
3,928 citations
"A graph-aided method for planning t..." refers methods in this paper
...For a review on efficient algorithms for testing graph isomorphism, see Read and Corneil (1977) and Hoffman (1982)....
[...]
...For simplicity we do not include in this article the column numbers for the lines, which can be easily read from the interaction tables given by Phadke (1989) and Taguchi (1987)....
[...]
TL;DR: The present state of the art of isomorphism testing is surveyed, its relationship to NP-completeness is discussed, and some of the difficulties inherent in this particularly elusive and challenging problem are indicated.
Abstract: The graph isomorphism problem—to devise a good algorithm for determining if two graphs are isomorphic—is of considerable practical importance, and is also of theoretical interest due to its relationship to the concept of NP-completeness. No efficient (i.e., polynomial-bound) algorithm for graph isomorphism is known, and it has been conjectured that no such algorithm can exist. Many papers on the subject have appeared, but progress has been slight; in fact, the intractable nature of the problem and the way that many graph theorists have been led to devote much time to it, recall those aspects of the four-color conjecture which prompted Harary to rechristen it the “four-color disease.” This paper surveys the present state of the art of isomorphism testing, discusses its relationship to NP-completeness, and indicates some of the difficulties inherent in this particularly elusive and challenging problem. A comprehensive bibliography of papers relating to the graph isomorphism problem is given.
519 citations
TL;DR: In this article, the concept of aberration is proposed as a way of selecting the best designs from those with maximum resolution, and algorithms are presented for constructing these minimum aberration designs.
Abstract: For studying k variables in N runs, all 2 k–p designs of maximum resolution are not equally good. In this paper the concept of aberration is proposed as a way of selecting the best designs from those with maximum resolution. Algorithms are presented for constructing these minimum aberration designs.
420 citations
01 Jan 1980
TL;DR: The concept of resolution was introduced by Box and Hunter as discussed by the authors, who defined the resolution of a two-level fractional factorial design as the length of the shortest word in the defining relation.
Abstract: Fractional factorial designs-especially the twolevel designs-are useful in a variety of experimental situations, for example, (i) screening studies in which only a subset of the variables is expected to be important, (ii) research investigations in which certain interactions are expected to be negligible and (iii) experimental programs in which groups of runs are to be performed sequentially, ambiguities being resolved as the investigation evolves (see Box, Hunter and Hunter, 1978). The literature on fractional factorial designs is extensive. For references before 1969, see the comprehensive bibliography of Herzberg and Cox (1969). For more recent references, see Daniel (1976) and Joiner (1975-79). A useful concept associated with 2k-P fractional factorial designs is that of resolution (Box and Hunter, 1961). A design is of resolution R if no cfactor effect is confounded with any other effect containing less than R c factors. For example, a design of resolution III does not confound main effects with one another but does confound main effects with two-factor interactions, and a design of resolution IV does not confound main effects with two-factor interactions but does confound two-factor interactions with one another. The resolution of a two-level fractional factorial design is the length of the shortest word in the defining relation. Usually an experimenter will prefer to use a design which has the highest
354 citations