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Many industrial experiments involve factors with levels more difficult to change or control than others, which leads to the development of two-level fractional factorial split-plot (FFSP) designs. Recently, mixed-level FFSP designs were proposed due to the requirement of different-level factors. In this paper, we generalize the Bayesian optimal criterion for mixed two- and four-level FFSP designs, and then provide Bayesian minimum aberration (MA) criterion to rank FFSP designs. Bayesian MA criterion can give a natural ordering for the effects involving two-level factors and three components of a four-level factor. We also discuss the relationship between the Bayesian optimal and Bayesian MA criteria. Furthermore, we consider the designs with both qualitative and quantitative factors.

The theory of uniform design has received increasing interest because of its wide application in the field of computer experiments. The generalized discrete discrepancy is proposed to evaluate the uniformity of the mixed-level factorial design. In this paper, the authors give a lower bound of the generalized discrete discrepancy and provide some construction methods of optimal mixed-level uniform designs which can achieve this lower bound. These methods are all deterministic construction methods which can avoid the complexity of stochastic algorithms. Both saturated mixed-level uniform designs and supersaturated mixed-level uniform designs can be obtained with these methods. Moreover, the resulting designs are also Χ 2 -optimal and minimum moment aberration designs.

While the orthogonal design of split-plot fractional factorial experiments has received much attention already, there are still major voids in the literature. First, designs with one or more factors acting at more than two levels have not yet been considered. Second, published work on nonregular fractional factorial split-plot designs was either based only on Plackett-Burman designs, or on small nonregular designs with limited numbers of factors. In this article, we present a novel approach to designing general orthogonal fractional factorial split-plot designs. One key feature of our approach is that it can be used to construct two-level designs as well as designs involving one or more factors with more than two levels. Moreover, the approach can be used to create two-level designs that match or outperform alternative designs in the literature, and to create two-level designs that cannot be constructed using existing methodology. Our new approach involves the use of integer linear programming and mixed integer linear programming, and, for large design problems, it combines integer linear programming with variable neighborhood search. We demonstrate the usefulness of our approach by constructing two-level split-plot designs of 16-96 runs, an 81-run three-level split-plot design, and a 48-run mixed-level split-plot design. Supplementary materials for this article are available online.

Curdlan production by Agrobacterium sp. IFO13140 immobilized on loofa sponge, alginate and loofa sponge with alginate was investigated. There was no statistically-significant difference in curdlan production when the microorganism was immobilized in different matrices. The loofa sponge was chosen because of its practical application and economy and because it provides a high stability through its continued use. The best conditions for immobilization on loofa sponge were 50 mg of cell, 200 rpm and 72 h of incubation, which provided a curdlan production 1.50-times higher than that obtained by free cells. The higher volumetric productivity was achieved by immobilized cells (0.09 g/L/h) at 150 rpm. The operating stability was evaluated, and until the fourth cycle, immobilized cells retained 87.40% of the production of the first cycle. The immobilized cells remained active after 300 days of storage at 4 °C. The results of this study demonstrate success in immobilizing cells for curdlan biosynthesis, making the process potentially suitable for industrial scale-up. Additional studies may show a possible contribution to the reduction of operating costs.

This research presents the increase of the Trichoderma harzianum production process in a biotechnology company. The NOBA (Near-Orthogonal Balanced arrays) method was used to fractionate a mixed-level factorial design to minimize costs and experimentation times. Our objective is to determine the significant factors to maximize the production process of this fungus. The proposed 213242 mixed-level design involved five factors, including aeration, humidity, temperature, potential hydrogen (pH), and substrate; the response variable was spore production. The results of the statistical analysis showed that the type of substrate, the air supply, and the interaction of these two factors were significant. The maximization of spore production was achieved by using the breadfruit seed substrate and aeration, while it was shown that variations in pH, humidity, and temperature have no significant impact on the production levels of the fungus.

Designs with full estimation capacity permit estimation of all main effects and all two-factor interactions. By allowing correlation among the effects, the run size of such designs can be smaller than required for a resolution of 5. To construct a design, one can either use commercial software for designs with optimized D-efficiencies or a catalog of orthogonal arrays. In the context of a wood construction experiment, we discuss how to choose between these approaches. We enumerate mixed-level and multilevel resolution-4 designs with run size up to 72 and with the maximum number of factors compatible with a full estimation capacity. Algorithmically constructed benchmark designs were generated with commercial software. Our study results in a list of recommended designs.

Screening designs for identifying active main effects among a group of factors are frequently used by experimenters. Methods for efficiently designing experiments that estimate main effects are needed, particularly if the experiment involves categorical factors at more than two levels and requires a nonstandard run size. We propose a new and powerful algorithm based on a sequential element-wise-column-wise strategy that focuses on maintaining balance within and between columns. We compare the new algorithm with a column-wise algorithm for finding some selected orthogonal and near-orthogonal arrays. We examine the performance of both algorithms using different criteria and provide a discussion of general strategies in searching for designs.

In order to precisely determinate absorbance of all kinds of heavy metals in soil,7- important factors were optimized by mixed-level orthogonal design.For example Pb,discuss the factors influence degree on digestion and determination of Pb.Pb-absorbance trends within the area is expressed by the change of two-dimensional planar contour.Pb-absorbance distribution was drawn by 3D visualization technology.In order to optimize the orthogonal design,taking the optimal value of decentralized points in orthogonal test convert into taking the optimal value on a continuous plane. Using Minitab16 software analyzed the experimental results.The results showed that affecting Pb digestion Primary and secondary order is B(HF)> C(HNO3)> A(HClO4)> I(Acetylene flow rate)>D(HCL)>G (Assistant combustion gas)>H(Lamp current)>F(Burner height),that was adding 9 mlHNO3, 3mLHF, 3 mlHClO4 in soil samples, after microwave digestion, got the optimal Pb-absorbance value 0.304 under the the working conditions of atomic absorption spectrophotometer for Pb 217.0nm, slit width 1.3nm, burner height 4nm, assistant combustion gas 2.4 L/min, the lamp current 4.0mA, propane flow rate 4.0 L/min.

The Lenth method is an objective method for testing effects from unreplicated factorial designs and eliminates the subjectivity in using a half-normal plot. The Lenth statistics are computed for the factorial effects and compared to corresponding critical values. Since the distribution of the Lenth statistics is not mathematically tractable, we propose a simple simulation method to estimate the critical values. Confidence intervals for the estimated critical values can also easily be obtained. Tables of critical values are provided for a large number of designs, and their use is demonstrated with data from three experiments. The proposed method can also be adapted to estimate critical values for other methods.

Running industrial experiments in strip-plot configurations can be a useful method of reducing cost. This article presents an effective procedure for constructing strip-plot arrangements of fractional factorial designs that can be used for m-level or mixed-level designs. The procedure consists of three steps: (1) Identify a suitable row design, (2) identify a suitable column design, and (3) select a latin-square fraction of the product of the designs in (1) and (2). Several examples are used to demonstrate the procedure.