Foldover Plans Research Articles

Uniformity and projection uniformity of combined designs

The purpose of this paper is to study the issue of employing the uniformity criterion measured by the mixture discrepancy to assess the optimal foldover plans for q-level factorials. The average mixture discrepancy and the average projection mixture discrepancy based on the level permutation method are respectively defined for combined designs, and the optimal foldover plan in terms of the overall uniformity and the uniformity of each dimension are also explored. The tight lower bounds of the average mixture discrepancy and the uniformity pattern under the general foldover plan are obtained respectively, which can be used as a benchmark for searching optimal foldover plans. Some illustrative examples are provided to show the theoretical results.

부분요인배치법과 2개의 반사배열을 이용한 코코아 용해도 실험연구

It is essential to employ design of experiments (DOE) to secure key properties of materials and major functions of important components. To enhance the level of understanding in DOE of classroom students or company engineers, it is very helpful for the students or engineers to do real projects which make use of some of the DOE methods. In this paper, a project of implementing a fractional factorial design and two foldover plans are presented. Two foldover plans are compared with each other and an integrated optimal conditions are presented and confirmation experiments are also conducted. From this research it is expected that 1) students recognize the importance of sequential experiments based on foldover designs, 2) they can feel confidence in applying DOE when they have an opportunity to do projects of developing materials or components, 3) R&D personnel of various organizations implementing DOE can benefit from the experimental process described in this paper.

A new foldover strategy and optimal foldover plans for three-level design

The foldover is a useful technique in construction of factorial designs. It is also a standard follow-up strategy discussed in many textbooks by adding a second fraction called a foldover design. In this paper uniformity criterion measured by the wrap-around $$L_2$$ -discrepancy is used to further distinguish the optimal foldover plan for three-level designs. For three-level fractional factorials as the original designs, a new foldover strategy is provided based on level permutation of each factor, which vastly enlarge the full foldover space. Some theoretical properties of the defined foldover plans are obtained, a tighter lower bound of the wrap-around $$L_2$$ -discrepancy of combined designs is also provided, which can be used as a benchmark for searching optimal foldover plans. For illustration of our theoretical results and comparison with the existing results, a catalog of optimal foldover plans of the new strategy for uniform initial designs with s three-level factors is tabulated, where $$2\le s \le 11$$ .

A catalog of optimal foldover plans for constructing U-uniform minimum aberration four-level combined designs

ABSTRACTThe foldover plan is a transformation map for adding a foldover design to the initial design, thus resulting in a combined design which can be used for breaking the links between aliased effects. This paper discusses the optimality of foldover plans for four-level designs via the most common criteria: the generalized word-length pattern (GWLP), Lee discrepancy (LD), wrap-around discrepancy (WD), centered discrepancy (CD) and mixture discrepancy (MD). We prove that the LD, WD and GWLP are equivalent for: any initial design and the corresponding foldover designs; any foldover design and its complementary foldover design; and any combined design and its complementary combined design. However, these interesting properties do not necessarily take place in the cases of the CD and MD. New analytical expressions and lower bounds of these discrepancies are given for initial and combined designs, which can be used as benchmarks for constructing uniform designs. For illustration of the usage of our theoretical results, a catalog of optimal foldover plans for constructing U-uniform minimum aberration four-level combined designs that involve factors with run is tabulated, which can be used for investigating either qualitative or quantitative factors.

Optimal foldover plans of asymmetric factorials with minimum wrap-around $$L_2$$ L 2 -discrepancy

Literatures reveal that foldover is a useful technique in construction of factorial designs. The objective of this paper is to study the issue of employing the uniformity criterion measured by the wrap-around \(L_2\)-discrepancy to assess the optimal foldover plans for asymmetric fractional factorials. A general foldover strategy and combined design under a foldover plan are developed for asymmetric fractional factorials, some theoretical properties on the equivalence between the defined foldover plan and its complementary foldover plan are discussed. A new lower bound for the wrap-around \(L_2\)-discrepancy of combined designs is obtained, which can be used as a benchmark for searching optimal foldover plans. Moreover, it also provides a theoretical justification for optimal foldover plans in terms of uniformity criterion.

A powerful and efficient algorithm for breaking the links between aliased effects in asymmetric designs

SummaryFractional factorial (FF) designs are no doubt the most widely used designs in experimental investigations due to their efficient use of experimental runs. One price we pay for using FF designs is, clearly, our inability to obtain estimates of some important effects (main effects or second order interactions) that are separate from estimates of other effects (usually higher order interactions). When the estimate of an effect also includes the influence of one or more other effects the effects are said to be aliased. Folding over an FF design is a method for breaking the links between aliased effects in a design. The question is, how do we define the foldover structure for asymmetric FF designs, whether regular or nonregular? How do we choose the optimal foldover plan? How do we use optimal foldover plans to construct combined designs which have better capability of estimating lower order effects? The main objective of the present paper is to provide answers to these questions. Using the new results in this paper as benchmarks, we can implement a powerful and efficient algorithm for finding optimal foldover plans which can be used to break links between aliased effects.

Some new lower bounds to various discrepancies on combined designs

ABSTRACTThe foldover is a useful technique in construction of two-level factorial designs. A foldover design is the follow-up experiment generated by reversing the sign(s) of one or more factors in the initial design. The full design obtained by joining the runs in the foldover design to those of the initial design is called the combined design. In this article, some new lower bounds of various discrepancies of combined designs, such as the centered, symmetric, and wrap-around L2-discrepancies, are obtained, which can be used as a better benchmark for searching optimal foldover plans. Our results provide a theoretical justification for optimal foldover plans in terms of uniformity criterion.

A note on optimal foldover four-level factorials

The foldover is a quick and useful technique in construction of fractional factorial designs, which typically releases aliased factors or interactions. The issue of employing the uniformity criterion measured by the centered L2-discrepancy to assess the optimal foldover plans was studied for four-level design. A new analytical expression and a new lower bound of the centered L2-discrepancy for fourlevel combined design under a general foldover plan are respectively obtained. A necessary condition for the existence of an optimal foldover plan meeting this lower bound was described. An algorithm for searching the optimal four-level foldover plans is also developed. Illustrative examples are provided, where numerical studies lend further support to our theoretical results. These results may help to provide some powerful and efficient algorithms for searching the optimal four-level foldover plans.

An efficient methodology for constructing optimal foldover designs in terms of mixture discrepancy

Mixture discrepancy criterion (Zhou et al., 2013) is more reasonable than other discrepancies criteria for measuring the uniformity of experimental designs. In this paper, we take the mixture discrepancy criterion as the optimality measure to assess optimal foldover plans, which serve as benchmarks for constructing optimal foldover and combined designs (see Definition 2). New analytical expressions as well as new lower bounds of the mixture discrepancy criterion for both symmetric two-level and three-level combined designs under general foldover plans are obtained. We also describe necessary conditions for the existence of optimal combined designs meeting these lower bounds. An algorithm for searching the optimal foldover plans is also developed. Illustrative examples are provided, where numerical studies lend further support to our theoretical results. These results may help to provide some powerful and efficient algorithms for searching the optimal foldover and combined designs.

Foldover Plans for Resolution IV Designs with 11 to 16 Factors

When there is a concern about the best way to apply the foldover technique to a Resolution IV fraction, it is often helpful to consult the literature available. Unfortunately, foldover plans for Resolution IV designs with 11 ≤ k factors are not provided. This article shows a methodology for the selection and classification of foldover plans for two‐level fractional factorial designs with 11 to 16 factors based on an exhaustive search and computer programming. The recommended foldovers are presented. This research extends the current foldover plans available for Resolution IV designs. Copyright © 2015 John Wiley & Sons, Ltd.

A new strategy for optimal foldover two-level designs

This paper studies the issue of optimal foldover plans for two-level designs in terms of the uniformity criteria measured by Lee, symmetric, wrap-around and centered discrepancy. An algorithm for searching the optimal foldover plans is also developed. New analytical expressions and new lower bounds for the above uniformity criteria of two-level combined designs are obtained.

Optimal foldover plans of three-level designs with minimum wrap-around L 2-discrepancy

The objective of this paper is to study the issue of employing the uniformity criterion measured by the wrap-around L 2-discrepancy to assess the optimal foldover plans for three-level designs. For three-level fractional factorials as the original designs, the general foldover plan and combined design under a foldover plan are defined, some theoretical properties of the defined foldover plans are obtained, a tight lower bound of the wrap-around L 2-discrepancy of combined designs under a general foldover plan is also obtained, which can be used as a benchmark for searching optimal foldover plans. For illustration of the usage of our theoretical results, a catalog of optimal foldover plans for uniform initial designs with s three-level factors is tabulated, where 2 ⩽ s ⩽ 11.

Lee discrepancy on symmetric three-level combined designs

This article develops a new mechanism to foldover designs involving factors with multi-level. By exhaustive search we identify the optimal foldover plans. This paper aims to study the issue of the optimal foldover plans for three-level designs, in view of the uniformity criterion measured by the Lee discrepancy.

A Lower Bound for the Wrap-around L2-discrepancy on Combined Designs of Mixed Two- and Three-level Factorials

The objective of this article is to study the issue of employing the uniformity criterion measured by wrap-around L2-discrepancy to assess the optimal foldover plans. For mixed two- and three-level fractional factorials as the original designs, general foldover plan and combined design under a foldover plan are defined, and the equivalence between any foldover plan and its complementary foldover plan is investigated. A lower bound of wrap-around L2-discrepancy of combined designs under a general foldover plan is obtained, which can be used as a benchmark for searching optimal foldover plans. Moreover, it also provides a theoretical justification for optimal foldover plans in terms of uniformity criterion.

A lower bound for the centred L 2-discrepancy on combined designs under the asymmetric factorials

The foldover is a useful technique in the construction of two-level factorial designs for follow-up experiments. To search an optimal foldover plans is an important issue. In this paper, for a set of asymmetric fractional factorials such as the original designs, a lower bound for centred L 2-discrepancy of combined designs under a general foldover plan is obtained, which can be used as a benchmark for searching optimal foldover plans. All of our results are the extended ones of Ou et al. [Lower bounds of various discrepancies on combined designs, Metrika 74 (2011), pp. 109–119] for symmetric designs to asymmetric designs. Moreover, it also provides a theoretical justification for optimal foldover plans in terms of uniformity criterion.

Alternative Foldover Plans for Two-Level Nonregular Designs

Foldover is a classic technique used to select follow-up experimental runs when an initial experiment yields ambiguities. While foldover has been soundly investigated for regular designs, less research has been devoted to this technique for nonregular designs. Previous work focuses on the use of the generalized minimum aberration criterion to obtain optimal foldover plans. In contrast, this article utilizes the concept of minimal dependent sets (MDSs) and associated criteria to rank foldovers of nonregular designs. We propose an integer programming-based solution to aid in the location and enumeration of MDSs. MDS-optimal foldovers for selected nonregular designs are presented and discussed.

Alternative optimal foldover plans for regular fractional factorial split-plot designs

McLeod and Brewster (Technometrics 46:135–146, 2008) considered the construction of optimal foldover plans for regular fractional factorial split plot designs. The construction process was based on the minimum aberration criterion for both the initial design and the combined design derived through the foldover. In this paper we give some additional foldover designs which have desirable properties that may provide viable alternative options to some of the plans provided in McLeod and Brewster (Technometrics 46:135–146, 2008).

A note on lower bound of centered L 2-discrepancy on combined designs

This note provides a theoretical justification of optimal foldover plans in terms of uniformity. A new lower bound of the centered L2-discrepancy values of combined designs is obtained, which can be used as a benchmark for searching optimal foldover plans. Our numerical results show that this lower bound is sharper than existing results when more factors reverse the signs in the initial design.

Semifold plans for mixed‐level designs

AbstractThis article presents a more efficient method for sequential augmentation of mixed‐level designs. The proposed approach reduces the optimal foldover plan of a mixed‐level design to a semifold plan by selecting half of the treatment combinations of the foldover fraction using exhaustive search and the criterion of general balance metric. The resulting design is a more economic run size augmented fraction that possesses good balance and orthogonality properties for main effects and two‐factor interactions. Three efficient arrays consisting of 20, 24 and 30 runs were selected for the analysis. Efficient arrays composed of a higher number of runs can be semifolded in a similar manner. Copyright © 2011 John Wiley & Sons, Ltd.

A note on the selection of optimal foldover plans for 16- and 32-run fractional factorial designs

Abstract The minimum aberration criterion has been advocated for ranking foldovers of 2 k − p fractional factorial designs ( Li and Lin, 2003 ); however, a minimum aberration design may not maximize the number of clear low-order effects. We propose using foldover plans that sequentially maximize the number of clear low-order effects in the combined (initial plus foldover) design and investigate the extent to which these foldover plans differ from those that are optimal under the minimum aberration criterion. A small catalog is provided to summarize the results.