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.
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Dissertation•
01 Dec 2011
TL;DR: In this article, a simple method for generating its pattern is proposed, which can be linked to the three main criteria for measuring the degree of aliasing in a manner devoid of mathematical complications.
Abstract: Design of Experiments (DOE) is a powerful technique for understanding, characterising and modelling products and processes and improving their performance. Whilst the bulk of its literature revolves around how it should be applied, little attention, if any, is devoted to the manner in which it is being implemented in practice particularly in manufacturing. One objective of this study was to bridge this gap by reviewing practical applications in three manufacturing journals. This revealed not only limited use but also multiple deficiencies. Many of these concerned a lack of familiarity with the concept of aliasing; the use of fractional factorial designs and pooling methods to analyse unreplicated trials; and a misunderstanding of the concepts underpinning the use and interpretation of p-values and factorial effects’ importance measures. With respect to aliasing, a novel simple method for generating its pattern is proposed. Besides its ease of application, it can be linked to the three main criteria for measuring the degree of aliasing (maximum resolution, minimum aberration and generalised minimum aberration) in a manner devoid of mathematical complications. Regarding the use of fractional factorial designs and pooling methods, simulation experiments were used to assess the performance of certain experimentation strategies to arrive at the same conclusions had a full factorial trial been performed. In the context of two-level designs, the L\(_{16}\) together with the Pooling Up method or the Half Normal Probability plot yielded a satisfactory performance. Similarly, the strategy of using the Best Subset selection procedure in conjunction with the L\(_{18}\) design was the best among the examined three-level ones. To attain a robust performance, it was found that the use of small designs such as the L\(_8\) and the L\(_9\) should, as far as possible, be avoided. The concepts concerning the use of the p-values and the effect’s importance measures are clarified and to facilitate communication between Engineers, Managers and Statisticians, an importance measure that can be related to three quality engineering techniques is suggested.
10 citations
TL;DR: In this article, the construction of optimal two-level block designs with respect to the B1-GMC criterion is considered. And some optimal block designs are obtained by utilizing doubling theory and MaxC2 design.
Abstract: In practice, to reduce systematic variation and increase precision of effect estimation, a practical design strategy is then to partition the experimental units into homogeneous groups, known as blocks. It is an important issue to study the optimal way on blocking the experimental units. Blocked general minimum lower order confounding (B1-GMC) is a new criterion for selecting optimal block designs. The paper considers the construction of optimal two-level block designs with respect to the B1-GMC criterion. By utilizing doubling theory and MaxC2 design, some optimal block designs with respect to the B1-GMC criterion are obtained.
10 citations
Cites background from "A graph-aided method for planning t..."
...A main effect or two-factor interaction (2fi) of a 2n−m design is called clear if it is not aliased with any other main effect or 2fi (Wu and Chen, 1992)....
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...The criteria include maximum resolution (Box and Hunter, 1961), minimum aberration (Fries and Hunter, 1980), clear effects (Wu and Chen, 1992) and maximum estimation capacity (Sun, 1993)....
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TL;DR: In this article, the authors considered the clear effects criterion for selecting good asymmetric or mixed-level factorial designs and gave the complete classification of (s2)sn designs according to the clear 2fi components, main effects and 2fis they have.
Abstract: The asymmetrical or mixed-level factorial design is a kind of important design in practice. There is a natural problem on how to choose an optimal (s2)sn design for the practical need, where s is any prime or prime power. This paper considers the clear effects criterion for selecting good designs. It answers the questions of when an (s2)sn design with fixed number of runs contains clear two-factor interaction (in brief 2fi) components and when it contains clear main effects or clear 2fis. It further gives the complete classification of (s2)sn designs according to the clear 2fi components, main effects and 2fis they have.
9 citations
TL;DR: An algorithm is proposed for constructing 2"I"V^m^-^p:2^l designs with the maximum number of clear 2fi's, and some theoretical results on the existence of clear2fi's in these designs are obtained.
9 citations
Cites background from "A graph-aided method for planning t..."
...The ‘‘if’’ part was shown by Chen et al. [3]....
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...Fortunately, researchers have tabulated all 16-, 32-, 64-, and 128-run 2m−p designs with resolution at least IV and at least one clear 2fi; see Chen et al. [4] and Xu [11]....
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...Zhang and Park [13] and Cheng and Wu [5] independently proposed the same blockedminimumaberration criterion, which selects the optimal design to be thatwhich sequentially minimizes W = (A3,0, A2,1, A4,0, . . . , A2j−1,0, Aj,1, A2j,0, . . .)....
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...Note that Part (2) of Theorem 5 of Chen et al. [3] is the special case of Theorem 2 for l = 2....
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...For l = 1, 2, Chen et al. [3] provided sufficient conditions for a 2m−pIV : 2 l design to contain clear 2fi’s....
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References
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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)....
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...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)....
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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