Two-level Factorials Research Articles

The Effect of Different Extraction Conditions on the Physicochemical Properties of Novel High Methoxyl Pectin-like Polysaccharides from Green Bell Pepper (GBP)

Green peppers are massively produced all over the world; however, substantial quantities of peppers are wasted. Functional polysaccharides can be produced from pepper waste. A conventional acid extraction method was used to obtain pectin-like materials from green bell pepper (GBP). A 23 experimental design (two-level factorials with three factors: temperature, pH, and time) was used to study the relationship between the extraction conditions and the measured physicochemical properties. The extracted polysaccharides were further analysed regarding their physicochemical and functional properties. The yields were in the range of (11.6–20.7%) and the highest yield value was extracted at pH 1. The polysaccharides were classified as “pectin-like”, as the galacturonic acid content was lower than 65%. Glucose and galactose were the major neutral sugars, and their relative amounts were dependent on the extraction conditions. The degree of esterification (DE) of the pectin-like extracts was greater than 50% and they were therefore classified as high methoxyl regardless of the extraction conditions. Also, important levels of phenolic materials (32.3–52.9 mg GAE/g) and proteins (1.5–5.4%) were present in the extract and their amounts varied depending on the extraction conditions. The green bell pepper polysaccharides demonstrated antioxidant and emulsifying activities and could also be used adequately to stabilise oil/water emulsion systems. This finding shows that green bell pepper could be used as an alternative source of antioxidants and an emulsifier/stabilising agent, and furthermore, the extraction conditions could be fine-tunned to produce polysaccharides with the desired quality depending on their application.

Construction on two-level factorials with flexible partially replicated runs via quarter foldover

Fractional factorial designs with partially replicated runs are desirable, since they not only save experimental cost and but also estimate the experimental error variance. In this paper, a simple technique called as quarter foldover, which can construct two-level factorial designs with flexible partially replicated runs, is firstly proposed. For an unreplicated initial design with resolution III (IV) or higher, whether regular or not, the sufficient and necessary conditions for the constructed design to be a resolution III (IV) or higher design with flexible partially replicated runs are investigated by the tool of indicator function. Theoretical results and examples show that the constructed designs are more flexible than most of the existing results in terms of the replicated runs. For initial designs of 12, 16, 20 and 24 runs with s factors, where 3≤s≤6, a catalog of optimal plans to construct some designs with highest resolution and flexible partially replicated runs is also tabulated for practice.

Joint Identification of Location and Dispersion Effects in Unreplicated Two-Level Factorials

Most procedures that have been proposed to identify dispersion effects in unreplicated factorial designs assume that location effects have been identified correctly. Incorrect identification of location effects may impair subsequent identification of dispersion effects. We develop a method for joint identification of location and dispersion effects that can reliably identify active effects of both types. A normal-based model containing parameters for effects in both the mean and variance is used. Parameters are estimated using maximum likelihood, and subsequent effect selection is done using a specially derived information criterion. An exhaustive search through a limited version of the space of possible models is conducted. Both a single-model output and model averaging are considered. The method is shown to be capable of identifying sensible location-dispersion models that are missed by methods that rely on sequential estimation of location and dispersion effects. Supplementary materials for this article are available online.

Two-level minimum aberration designs in N =2 (mod 4) runs

For two-level factorials, we consider designs in N=2 (mod 4) runs as obtained by adding two runs, with a certain coincidence pattern, to an orthogonal array of strength two. These designs are known to be optimal main effect plans in a very broad sense in the absence of interactions. Among them, we explore the ones having minimum aberration, with a view to ensuring maximum model robustness even when interactions are possibly present. This is done by sequentially minimizing a measure of the bias caused by interactions of successively higher orders.

Minimum Aberration Designs for Two-Level Factorials in N =1 (Mod 4) Runs

Minimum Aberration Designs for Two-Level Factorials in N =1 (Mod 4) Runs

Creating clear designs: a graph-based algorithm and a catalog of clear compromise plans

A graph-based algorithm is proposed for creating regular fractional factorial designs with two-level factors such that a pre-specified set of two-factor interactions is clear of aliasing with any main effects or two-factor interactions (clear design). The Clear Interactions Graph (CIG) used in the algorithm is unique for each design and different in nature from the well-known Taguchi linear graph. Based on published catalogs of two-level fractional factorials, enhanced by the CIG, a search algorithm finds an appropriate clear design or declares its non-existence. The approach is applied to the creation of a catalog of minimum aberration clear compromise plans, which is also of interest in its own right. [Supplementary materials are available for this article. Go to the publisher’s online edition of IIE Transactions for additional discussions on run times and implementation of the algorithm for larger designs.]

Optimal fractions of two-level factorials under a baseline parameterization

Two-level fractional factorial designs are considered under a baseline parameterization. The criterion of minimum aberration is formulated in this context and optimal designs under this criterion are investigated. The underlying theory and the concept of isomorphism turn out to be significantly different from their counterparts under orthogonal parameterization, and this is reflected in the optimal designs obtained. Copyright 2012, Oxford University Press.

A trigonometric approach to quaternary code designs with application to one-eighth and one-sixteenth fractions

The study of good nonregular fractional factorial designs has received significant attention over the last two decades. Recent research indicates that designs constructed from quaternary codes (QC) are very promising in this regard. The present paper shows how a trigonometric approach can facilitate a systematic understanding of such QC designs and lead to new theoretical results covering hitherto unexplored situations. We focus attention on one-eighth and one-sixteenth fractions of two-level factorials and show that optimal QC designs often have larger generalized resolution and projectivity than comparable regular designs. Moreover, some of these designs are found to have maximum projectivity among all designs.

A Bayesian Analysis of Unreplicated Two-Level Factorials Using Effects Sparsity, Hierarchy, and Heredity

ABSTRACT This article proposes a Bayesian procedure to calculate posterior probabilities of active effects for unreplicated two-level factorials. The results from a literature survey are used to specify individual prior probabilities for the activity of effects and the posterior probabilities are then calculated in a three-step procedure where the principles of effects sparsity, hierarchy, and heredity are successively considered. We illustrate our approach by reanalyzing experiments found in the literature.

Application of minimum projection uniformity criterion in complementary designs

In this article, we consider the characterization problem in design theory. The objective is to characterize minimum projection uniformity for two-level designs in terms of their complementary designs. Here, the complementary design means a design in which all the Hamming distances of any two runs are the same, which generalizes the concept of a pair of complementary designs in the literature. Based on relationships of the uniformity pattern between a pair of complementary designs, we propose a minimum projection uniformity (MPU) rule to assess and compare two-level factorials.

Lower Bounds for the Uniformity Pattern of Asymmetric Fractional Factorials

The objective of this article is to study the issue of the projection discrepancy (Fang and Qin, 2005; Ma et al. 2003), which has wide application to the field of fractional factorials. Fang and Qin (2005) and Zhang and Qin (2006) derived some properties of the projection properties of two-level factorials. Here we extend their results to asymmetric factorials. Some lower bounds of projection discrepancy of asymmetrical factorials are also obtained.

Computer experiments: application to the case of a recovery boiler

Computer experiments are increasingly used to generate useful information about process and product, which could hardly be possible otherwise. This article explores the applicability of two-level fractional factorials for computer experiments with the aim at identifying the operating parameters of a black liquor recovery boiler, which have the strongest effect on the mass of carryover particles by flue gases. Six analysis methods, which can be easily used by practitioners for factor screening are reviewed and illustrated. The results show that scree plot is a practical and effective tool for screening by using deterministic computer models. Regarding the results, this study allows reducing the mass of carryover particles.

Applications of design of experiments in engineering

Applications of design of experiments in engineering

A generalization of the alias matrix

The investigation of aliases or biases is important for the interpretation of the results from factorial experiments. For two-level fractional factorials this can be facilitated through their group structure. For more general arrays the alias matrix can be used. This tool is traditionally based on the assumption that the error structure is that associated with ordinary least squares. For situations where that is not the case, we provide in this article a generalization of the alias matrix applicable under the generalized least squares assumptions. We also show that for the special case of split plot error structure, the generalized alias matrix simplifies to the ordinary alias matrix.

Minimum projection uniformity criterion and its application

Based on a so-called uniformity pattern, Fang and Qin [2005. Uniformity pattern and related criteria for two-level factorials. Sci. China Ser. A 48, 1–11] proposed the minimum projection uniformity criterion to assess and compare two-level factorials. In present paper, we study applications of minimum projection uniformity criterion in aberration designs, orthogonal arrays and supersaturated designs. Relationship among minimum projection uniformity, generalized minimum aberration [Ma, C. X., Fang, K. T., 2001. A note on generalized aberration factorial designs. Metrika 53, 85–93; Xu, H., Wu, C. F. J., 2001. Generalized minimum aberration for asymmetrical fractional factorial designs. Ann. Statist. 29, 549–560], V-criterion [Tang, B., 2001. Theory of J-characteristics for fractional factorial designs and projection justification of minimum G 2 -aberration. Biometrika 88, 401–407], nearest balance criterion [Fang, K. T., Lu, X., Winker, P., 2003. Lower bounds for centered and wrap-around L 2 -discrepancies and construction of uniform designs by threshold accepting. J. Complexity 19, 692–711] and E ( s 2 ) criterion [Booth, K. H. V., Cox, D. R., 1962. Some systematic supersaturated designs. Technometrics 4, 489–495] is made explicit. Noting that minimum projection uniformity criterion is related to uniform designs, our results should raise the hope of improving the connection between uniform design theory and factorial design theory.

Uniformity pattern and related criteria for two-level factorials

In this paper, the study of projection properties of two-level factorials in view of geometry is reported. The concept of uniformity pattern is defined. Based on this new concept, criteria of uniformity resolution and minimum projection uniformity are proposed for comparing two-level factorials. Relationship between minimum projection uniformity and other criteria such as minimum aberration, generalized minimum aberration and orthogonality is made explict. This close relationship raises the hope of improving the connection between uniform design theory and factorial design theory. Our results provide a justification of orthogonality, minimum aberration, and generalized minimum aberration from a natural geometrical interpretation.

Irregular Four-level Response Surface Designs

Four-level response surface designs based on regular two-level fractional factorial designs were introduced by Edmondson (1991). Here, the methods are extended to include designs based on irregular two-level fractional factorials. These designs allow orthogonal blocking and require fewer experimental units than the regular designs.

Application of factorial designs and Doehlert matrix in optimization of experimental variables associated with the preconcentration and determination of vanadium and copper in seawater by inductively coupled plasma optical emission spectrometry

In the present paper a procedure for preconcentration and determination of vanadium and copper in seawater using inductively coupled plasma optical emission spectrometry (ICP OES) is proposed, which is based on solid-phase extraction of vanadium (IV), vanadium (V) and copper (II) ions as 1-(2-pyridylazo)-2-naphthol (PAN) complexes by active carbon. The optimization process was carried out using two-level full factorials and Doehlert matrix designs. Four variables (PAN mass, pH, active carbon mass and shaking time) were regarded as factors in the optimization. Results of the two-level full factorial design 2 4 with 16 runs for vanadium extraction, based on the variance analysis (ANOVA), demonstrated that the factors pH and active carbon mass, besides the interaction (pH×active carbon mass), are statistically significant. For copper, the ANOVA revealed that the factors PAN mass, pH and active carbon mass and the interactions (PAN mass×pH) and (pH×active carbon mass) are statistically significant. Doehlert designs were applied in order to determine the optimum conditions for extraction. The procedure proposed allowed the determination of vanadium and copper with detection limits (3σ/ S) of 73 and 94 ng l −1, respectively. The precision, calculated as relative standard deviation (R.S.D.), was 1.22 and 1.37% for 12.50 μg l −1 of vanadium and copper, respectively. The preconcentration factor was 80. The recovery achieved for determination of vanadium and copper in the presence of several cations demonstrated that this procedure improved the selectivity required for seawater analysis. The procedure was applied to the determination of vanadium and copper in seawater samples collected in Salvador City, Brazil. Results showed good agreement with other data reported in the literature.

Analytic Use of Two-Level Factorials in Incomplete Blocks to Examine the Stability of Factor Effects

The traditional purpose of the randomized block design is to remove the effects that distinguish blocks in order to make the presence of other effects more apparent. However, in industrial experime...

Experimental designs when there are one or more factor constraints

In response surface methodology, designs of orders one or two are often needed such that some or all the factor levels satisfy one or more linear constraints. A method is discussed for obtaining such designs by projection of a standard design onto the constraint hyperplane. It is shown that a projected design obtained from a rotatable design is also rotatable, and for a rotatable design that is also orthogonal (in particular any orthogonal first-order design) a least squares analysis carried out on the generating design supplies a least squares solution for the constrained design subject to the constraints. Some useful properties of the generating design, such as orthogonal blocking and fractionation are retained in the projected design. Some second-order mixture designs generated by two-level factorials are discussed.