Minimum Aberration 2 k–p Designs
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.
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18 Feb 2020
TL;DR: In this paper, the incremental construction of nested designs having good spreading properties over the d-dimensional hypercube is considered, for values of d such that the 2 d vertices of the hypercube are too numerous to be all inspected.
Abstract: The incremental construction of nested designs having good spreading properties over the d-dimensional hypercube is considered, for values of d such that the 2 d vertices of the hypercube are too numerous to be all inspected. A greedy algorithm is used, with guaranteed efficiency bounds in terms of packing and covering radii, using a 2 d−m fractional-factorial design as candidate set for the sequential selection of design points. The packing and covering properties of fractional-factorial designs are investigated and a review of the related literature is provided. An algorithm for the construction of fractional-factorial designs with maximum packing radius is proposed. The spreading properties of the obtained incremental designs, and of their lower dimensional projections, are investigated. An example with d = 50 is used to illustrate that their projection in a space of dimension close to d has a much higher packing radius than projections of more classical designs based on Latin hypercubes or low discrepancy sequences.
3 citations
TL;DR: Minimum resolution IV and Plackett and Burman designs are popular screening designs because of their run size efficiency and good projection properties and it came out that for most of these designs, it was possible to find blocking schemes satisfying that for any three factors, either all main effects and their interactions or all main effect and their two-factor interactions could be estimated with a reasonable high efficiency when blocked.
Abstract: Minimum resolution IV (MinResIV) and Plackett and Burman (PB) designs are popular screening designs because of their run size efficiency and good projection properties. The purpose of this investigation is to find out which projection properties that can be maintained when these designs are blocked and also the efficiency by which the effects of interest can be estimated. MinResIV designs with 10–20 runs and PB designs with 12 and 20 are investigated. For the PB designs, design factor columns were used as blocking columns, while for the MinResIV designs, the main rule consisted of finding good arrangements for allocating mirror-image pair runs to the same blocks, a method inspired from Jacroux. As a criterion for a good blocking arrangements, we used maximum Ds - efficiency, which we found to be more generally applicable and provide better projection properties than previous suggested methods. It came out that for most of these designs, it was possible to find blocking schemes satisfying that for any three factors, either all main effects and their interactions or all main effects and their two-factor interactions could be estimated with a reasonable high efficiency when blocked. Copyright © 2016 John Wiley & Sons, Ltd.
3 citations
01 Jan 2009
TL;DR: In this article, a brief introduction on how to design fractional factorial split-plots for multi-step situations using real industrial examples, and shows how JMP® users can take advantage of these new capabilities using a JMP ® application.
Abstract: In the past 10 years, there has been an increase in research and software development in the design of experiments for split-plot situations. As supply chains grow across the globe and complexity increases, it is necessary to design experiments for processes involving three, four, or more steps. New features in PROC FACTEX in SAS® 9.2 make it possible to design fractional factorial split-plots for multi-step situations. This paper gives a brief introduction on how to design fractional factorial split-plots for multi-step situations using real industrial examples, and shows how JMP® users can take advantage of these new capabilities using a JMP® application.
3 citations
Additional excerpts
...Huang et al (1998), and Bingham and Sitter (1999, 2001) have applied the concept of minimum aberration (Fries and Hunter (1980)) to split-plot designs, giving comprehensive tables for small to moderately sized minimum aberration splitplot designs, while Kulahci et al (2006) have discussed…...
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25 Oct 2011
TL;DR: In this paper, the concept of minimum aberration has been extended to choose blocked fractional factorial designs (FFDs) and a method is then proposed for constructing minimum-aberration blocked FFDs without using defining contrast subgroups and alias sets.
Abstract: The concept of minimum aberration has been extended to choose blocked fractional factorial designs (FFDs). The minimum aberration criterion ranks blocked FFDs according to their treatment and block wordlength patterns, which are often obtained by counting words in the treatment defining contrast subgroups and alias sets. When the number of factors is large, there are a huge number of words to be counted, causing some difficulties in computation. Based on coding theory, the concept of minimum moment aberration, proposed by Xu (2003) for unblocked FFDs, is extended to blocked FFDs. A method is then proposed for constructing minimum aberration blocked FFDs without using defining contrast subgroups and alias sets. Minimum aberration blocked FFDs for all 32 runs, 64 runs up to 32 factors, and all 81 runs are given with respect to three combined wordlength patterns.
3 citations
Cites background from "Minimum Aberration 2 k–p Designs"
...Because resolution alone cannot determine the best design, Fries and Hunter (1980) further proposed the MA criterion as its refinement....
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...2 Review of optimality criteria Because resolution alone cannot determine the best design, Fries and Hunter (1980) further proposed the MA criterion as its refinement....
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...Experimenters would often face the problem of choosing optimally blocked FFDs. FFDs are typically chosen according to the maximum resolution criterion (Box and Hunter, 1961) and its refinement, the minimum aberration (MA) criterion (Fries and Hunter, 1980)....
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...FFDs are typically chosen according to the maximum resolution criterion (Box and Hunter, 1961) and its refinement, the minimum aberration (MA) criterion (Fries and Hunter, 1980)....
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Posted Content•
TL;DR: In this paper, the authors investigated properties of Ehrhart polynomials of matroid polytopes and showed that for fixed rank their polynomial time computations are computable.
Abstract: This dissertation presents new results on three different themes all related to matroid polytopes. First we investigate properties of Ehrhart polynomials of matroid polytopes, independence matroid polytopes, and polymatroids. We prove that for fixed rank their Ehrhart polynomials are computable in polynomial time. The proof relies on the geometry of these polytopes as well as a new refined analysis of the evaluation of Todd polynomials.
Second, we discuss theoretical results regarding the algebraic combinatorics of matroid polytopes. We discuss two conjectures about the h *-vector and coefficients of Ehrhart polynomials of matroid polytopes and provide theoretical and computational evidence for their validity. We also explore a variant of White’s conjecture which states that every matroid polytope has a regular unimodular triangulation. We provide extensive computational evidence supporting this new conjecture and propose a combinatorial condition on simplices sufficient for unimodularity. Lastly we discuss properties of two dimensional faces of matroid polytopes.
Finally, motivated by recent work on algorithmic theory for non-linear and multicriteria matroid optimization, we have developed algorithms and heuristics aimed at practical solutions of large instances of these difficult problems. Our methods primarily use the local adjacency structure inherent in matroid polytopes to pivot to feasible solutions which may or may not be optimal. We also present a modified breadth-first-search heuristic that uses adjacency to enumerate a subset of feasible solutions. We present other heuristics, and provide computational evidence supporting these new techniques. We implemented all of our algorithms in the software package MOCHA (Matroids Optimization Combinatorial Heuristics and Algorithms).
3 citations
References
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471 citations
Book•
23 Jun 1976
TL;DR: In conclusion, the size of Industrial Experiments, Fractional Replication--Elementary, and Incomplete Factorials are found to be about the same as that of conventional comparison experiments.
Abstract: Introduction. Simple Comparison Experiments. Two Factors, Each at Two Levels. Two Factors, Each at Three Levels. Unreplicated Three--Factor, Two--Level Experiments. Unreplicated Four--Factor, Two--Level Experiments. Three Five--Factor, Two--Level Unreplicated Experiments. Larger Two--Way Layouts. The Size of Industrial Experiments. Blocking Factorial Experiments, Fractional Replication--Elementary. Fractional Replication--Intermediate. Incomplete Factorials. Sequences of Fractional Replicates. Trend--Robust Plans. Nested Designs. Conclusions and Apologies.
311 citations
09 Jun 1976
271 citations
TL;DR: Incomplete Factorials, Fractional Replication, Intermediate Factorial, and Nested Designs as discussed by the authors are some of the examples of incomplete Factorial Experiments and incomplete fractional replicates.
Abstract: Introduction. Simple Comparison Experiments. Two Factors, Each at Two Levels. Two Factors, Each at Three Levels. Unreplicated Three--Factor, Two--Level Experiments. Unreplicated Four--Factor, Two--Level Experiments. Three Five--Factor, Two--Level Unreplicated Experiments. Larger Two--Way Layouts. The Size of Industrial Experiments. Blocking Factorial Experiments, Fractional Replication--Elementary. Fractional Replication--Intermediate. Incomplete Factorials. Sequences of Fractional Replicates. Trend--Robust Plans. Nested Designs. Conclusions and Apologies.
252 citations