Minimum Aberration Research Articles

Construction on large four-level designs via quaternary codes

In this paper, two simple and effective construction methods are proposed to construct four-level design with large size via quaternary codes from some small two-level initial designs. Under the popular criteria for selecting optimal design, such as generalized minimum aberration, minimum moment aberration and uniformity measured by average Lee discrepancy, the close relationships between the constructed four-level design and its initial design are investigated, which provide the guidance for choosing the suitable initial design. Moreover, some lower bounds of average Lee discrepancy for the constructed four-level designs are obtained, which can be used as a benchmark for evaluating the uniformity of the constructed four-level designs. Some numerical examples show that the large four-level designs can be constructed with high efficiency.

Generalised minimum moment aberration for designs with both qualitative and quantitative factors

The minimum moment aberration and the minimum Lee‐moment aberration criteria are two popular conceptually simple and computationally cheap criteria for selecting good designs. However, the minimum moment aberration is suitable for qualitative factors, and the minimum Lee‐moment aberration cannot distinguish some designs with high‐level quantitative factors. In this paper, the minimum absolute‐moment aberration criterion is proposed to compare and select designs with multi‐level quantitative factors. We validate the statistical justifications of this criterion from theoretical and numerical aspects. Furthermore, we extend the minimum absolute‐moment aberration criterion into screening designs with both qualitative and quantitative factors, naming the new criterion as the minimum mixed‐moment aberration criterion. Then we utilise a numerical study to compare and evaluate the performance of some popular designs with both qualitative and quantitative factors in computer experiments.

Comparative Study of Multi-Objective and Single-Objective Optimisation of Colour Matching for Opaque Products

Opaque products usually refer to all types of products with opaque packaging. In order to improve the market competitiveness of opaque products, it is crucial to determine the optimal colour scheme for the products. In this paper, we study the colour matching scheme for opaque products, taking different concentrations of red, yellow and blue dye as the research variables, taking the minimum chromatic aberration, the lowest cost, and the type of dye as the objective function, establishing the mathematical model of the optimal colour matching scheme, adopting the single-objective particle swarm algorithm and multi-objective hierarchical sequential method respectively, and using specific examples to carry out a comparative analysis to check the two methods. The results show that the single-objective method can provide the colour matching scheme with the minimum chromatic aberration from the ideal colour, and the multi-objective method can consider the cost and price of dyes as well as the types of dyes to provide a more scientific colour matching scheme. The computational results also verify the validity of the model and show that the model can provide decision-making and support for colour matching schemes for opaque products.

Bayesian minimum aberration mixed-level split-plot designs

Bayesian minimum aberration mixed-level split-plot designs

Construction of mixed-level fractional factorial split-plot designs with combined minimum aberration

Running agricultural experiments in split-plot configurations is an efficient way to estimate main-effect contrasts with higher precision. In a fractional factorial split-plot design, the hard-to-vary factors are called whole-plot (WP) treatment factors and the easy-to-vary factors are called subplot treatment factors. Yan and Zhao (2022) proposed the combined minimum aberration criterion of WP (WP-MAc) for mixed-level split-plot designs. This article studies the construction methods of the WP-MAc 2 ( n 1 + n 2 ) − ( k 1 + k 2 ) 4 1 designs with four-level factor in subplot. Some examples are given to illustrate the theories. Finally, the results of 8, 16, 32, 64 runs WP-MAc 2 ( n 1 + n 2 ) − ( k 1 + k 2 ) 4 s 1 designs are tabulated for possible employ by practitioners.

Theory of [formula omitted]-characteristics of four-level designs under quaternary codes

The J-characteristics play an instrumental role in the development of generalized resolution and minimum aberration criteria for two-level designs. In this paper, the concept of J-characteristics is naturally generalized to four-level designs via quaternary codes, which maps the four-level designs to two-level designs. Based on the relationship between the minimum G2-aberration criterion of two-level design and the generalized minimum aberration criterion of its projection designs, the properties of J-characteristics of four-level designs are explored. The relationship between J-characteristics of four-level design and J-characteristics of corresponding effective two-level sub-designs is built. The generalized resolution, confounding frequency vector and B-vector of four-level design are respectively defined based on the J-characteristics and their upper bounds, and minimum G-aberration and minimum G2-aberration criteria of four-level design are proposed, which are useful to assess the goodness of four-level designs.

Comparative Analysis of Performance of Water-Based Coatings Prepared by Two Kinds of Anti-Bacterial Microcapsules and Nano-Silver Solution on the Surface of Andoung Wood

A nano-silver solution, urea-formaldehyde resin-coated nano-silver solution microcapsules (AgNPS@UF), and melamine-formaldehyde resin-coated chitosan-modified nano-silver solution microcapsules (CS-AgNPS@MF) were added into the coatings at different contents for comparative tests to explore an anti-bacterial agent with the best anti-bacterial properties in the water-based coatings on Andoung wood (Monopetalanthus spp.). As the content of anti-bacterial agents increased, AgNPS@UF had the best anti-bacterial property towards Staphylococcus aureus and Escherichia coli, with anti-bacterial rates of 79.0% and 82.1%, respectively. The optical and mechanical properties of the coating quickly worsened as the content of anti-bacterial agents increased. The anti-bacterial coatings with AgNPS@UF have the minimum chromatic aberration change of 6.5. The anti-bacterial coating with 5% content of AgNPS@UF decreased the aging rate. This coating had good optical properties, and its liquid-resistant level is 2. When the 5% content of AgNPS@UF was added, after high-temperature accelerated aging and ultraviolet (UV) aging, the anti-bacterial rates of the coating with AgNPS@UF decreased toward Escherichia coli from 82.1% to 62.2% and decreased toward Staphylococcus aureus from 79.3% to 61.1%, respectively. This shows that the coating had a long-lasting anti-bacterial property. The anti-bacterial property of the coating after high-temperature accelerated aging was superior to that after the UV aging. The incorporation of AgNPS@UF in water-based coatings protects people from the risks of Escherichia coli and Staphylococcus aureus more effectively with long-lasting property. The research results provide a reference for the preparation of anti-bacterial water-based coatings on the surface of the Andoung wood.

Bayesian-inspired minimum contamination designs under a double-pair conditional effect model

In two-level fractional factorial designs, conditional main effects can provide insights by which to analyze factorial effects and facilitate the de-aliasing of fully aliased two-factor interactions. Conditional main effects are of particular interest in situations where some factors are nested within others. Most of the relevant literature has focused on the development of data analysis tools that use conditional main effects, while the issue of optimal factorial design for a given linear model involving conditional main effects has been largely overlooked. Mukerjee, Wu and Chang [Statist. Sinica 27 (2017) 997–1016] established a framework by which to optimize designs under a conditional effect model. Although theoretically sound, their results were limited to a single pair of conditional and conditioning factors. In this paper, we extend the applicability of their framework to double pairs of conditional and conditioning factors by providing the corresponding parameterization and effect hierarchy. We propose a minimum contamination-based criterion by which to evaluate designs and develop a complementary set theory to facilitate the search of minimum contamination designs. The catalogues of 16- and 32-run minimum contamination designs are provided. For five to twelve factors, we show that all 16-run minimum contamination designs under the conditional effect model are also minimum aberration according to Fries and Hunter [Technometrics 22 (1980) 601–608].

Optimal fractions of three-level factorials under a baseline parameterization

The paper proposes the minimum aberration (MA) criterion for three-level designs under a baseline parameterization. An algorithm is put forward to search the MA designs. A catalogue of MA 18-run and 27-run three-level baseline designs is tabulated.

Minimum aberration 412 n designs via secondary complementary sets

Mixed-level designs are widely used in various experiments. This paper considers how to construct minimum aberration 4 1 2 n mixed-level designs. Based on finite projective geometry, secondary complementary set is defined as part of the design which plays a crucial role in constructing the optimal designs. Then, algebraic connection between the wordlength pattern of a 4 1 2 n design and that of its secondary complementary set is established. According to the connection, some general rules for identifying type 0 minimum aberration 4 1 2 n mixed-level designs are proposed via their secondary complementary sets. Those rules can help construct minimum aberration 4 1 2 n designs conveniently when the secondary complementary set contains fewer elements than the complementary set. The type 0 minimum aberration 4 1 2 n designs with large n are tabulated via secondary complementary sets.

A study on average Lee discrepancy measure

In this paper, we proposed the average Lee discrepancy to measure the uniformity of designs and studied its connection with the standard optimality measures used in the context of design of experiments. Lower bounds to such measure are provided as a benchmark to obtain uniform designs. We also defined the minimum average Lee-moment aberration criterion used to compare and screen the optimal design, and some lower bounds to such criterion are also provided.

Analysis and design of a microlens array scanning system based on spherical aberration.

The microlens array (MLA) scanning system is a relatively new solution for beam scanning. The excrescent light generated by overlapping divergent beams from adjacent microlenses increases with the scanning angle, and as a result, the rays do not fill the clear aperture of the MLA. The effect on the detection distance and imaging resolution of the MLA system are analyzed. Based on the principle of minimum spherical aberration, a design method for initial configuration of the MLA system is proposed. The findings of this paper can be beneficial for the design and evaluation of MLA systems.

General minimum lower-order confounding split-plot designs with important whole-plot factors

Fractional factorial split-plot (FFSP) designs have been widely used in industrial experiments. Whole-plot (WP) factors and subplot (SP) factors play different roles in FFSP designs. This paper proposes a new optimality criterion for selecting FFSP designs, which is the general minimum lower-order confounding of type WP (WP-GMC), under the circumstances that the experimenter has prior information that the WP factors are more important than the SP factors. Comparison with minimum aberration FFSP designs of type WP is given. Some optimal FFSP designs under the WP-GMC criterion are constructed.

Minimum Aberration Split-Plot Designs When the Whole Plot and Subplot Factors Do Not Have the Same Importance

In practical factorial experiments, we sometimes find that complete randomization of the order of the runs is infeasible because it is more difficult to change the levels of some factors than the others, especially in some engineering experiments. Then, fractional factorial split-plot (FFSP) designs represent a practical option in such situations. The difficult-to-change factors are called whole plots (WP) factors, and the other factors are called subplot (SP) factors. The WP and SP factors do not have the same importance in many experiments. Then, the popular minimum aberration criterion is not suitable any more for choosing FFSP designs. This paper proposes two criteria for selecting FFSP designs. Algorithms for constructing the optimal FFSP designs under the two criteria are proposed. Some optimal designs under the two criteria are tabulated.

Minimum Aberration Split-Plot Designs Focusing on the Whole Plot Factors

In some experiments, the levels of some factors are difficult to change; then fractional factorial split-plot (FFSP) designs are suitable for selection. In an FFSP design, the factors are divided into two classes—the whole plot (WP) and subplot (SP) factors. In some experiments, the selection of the levels of the WP factors can affect that of the SP factors, which attracts much attention to the WP factors. Paying more attention to the WP factors, a new optimality criterion for selecting such FFSP designs is proposed. The robustness of the proposed method is discussed. The construction method of the optimal designs under the new criterion is studied.

Constructing minimum aberration split-plot designs via complementary designs when the subplot factors are more important

Robust parameter design (RPD) is a statistical engineering method for reducing the sensitivity of a system (i.e., a product or process) to noise variations by selecting a treatment combination of control factors, with the aim of reducing performance fluctuations of the system. Then the control factors are more important in RPD. The fractional factorial split-plot (FFSP) designs can be used when the noise and control factors are considered as the whole plot (WP) factors and subplot (SP) factors respectively in the design. Based on the above situation, this paper considers the FFSP design when the SP factors are more important. A criterion called minimum aberration of type SP (SP-MA) is proposed and the SP-MA FFSP designs are constructed by using complementary designs. The superiority of the SP-MA criterion in seeking FFSP designs when the SP factors are more important than the WP factors is illustrated by comparison.

Uniformity pattern of mixed two- and three-level factorials under average projection mixture discrepancy

The issue of the projection uniformity of mixed two- and three-level factorials measured by mixture discrepancy is investigated in this paper. The average projection mixture discrepancy is defined to measure the uniformity for low-dimensions projection designs. Moreover, the relationship between uniformity pattern and generalized minimum aberration is established. Two numerical examples are provided to illustrate the theoretical results.

Utilizing individual clear effects for intelligent factor allocations and design selections

Extensive studies have been conducted on how to select efficient designs with respect to a criterion. Most design criteria aim to capture the overall efficiency of the design across all columns. When prior information indicated that a small number of factors and their two-factor interactions (2fi’s) are likely to be more significant than other effects, commonly used minimum aberration designs may no longer be the best choice. Motivated by a real-life experiment, we propose a new class of regular fractional factorial designs that focus on estimating a subset of columns and their corresponding 2fi’s clear of other important effects. After introducing the concept of individual clear effects (iCE) to describe clear 2fi’s involving a specific factor, we define the clear effect pattern criterion to characterize the distribution of iCE’s over all columns. We then obtain a new class of designs that sequentially maximize the clear effect pattern. These newly constructed designs are often different from existing optimal designs. We develop a series of theoretical results that can be particularly useful for constructing designs with large run sizes, for which algorithmic construction becomes computationally challenging. We also provide some practical guidelines on how to choose appropriate designs with respect to different run size, the number of factors, and the number of 2fi’s that need to be clear.

Mixed two- and four-level split-plot designs with combined minimum aberration

Mixed two- and four-level split-plot designs with combined minimum aberration

Optimization and Distance Function Proce-dures in Aberration Criteria of Efficient Frac-tional Factorial Design

An efficient orthogonal array was constructed with near balance and near the orthogonal property for the lowest common multiples of runs, using the balance coefficient criteria for determining near balance and J2 optimality criteria for orthogonal properties. The optimization and distance function forms of balance coefficient criteria were used for the classification of the designs. The Minimum Moment Aberration (MMA) and Minimum Aberration Projection (MAP) are compared using the optimization and distance function to determine the near balance criteria. The result indicated that, the MMA and MAP criteria was efficient using the optimization procedure of the balance coefficient.