Use of Doehlert Designs for Second-order Polynomial Models

A review about the application of response surface methodology in the optimization of food technology is presented. The theoretical principles of response surface methodology and steps for its application are described. The response surface methodologies : three-level full factorial, central composite, Box-Behnken, and Doehlert designs are compared in terms of characteristics and efficiency. Furthermore, recent references of their uses in food technology are presented. A comparison between the response surface designs (three-level full factorial, central composite, Box-Behnken and Doehlert design) has demonstrated that the Box-Behnken and Doehlert designs are slightly more efficient than the central composite design but much more efficient than the three-level full factorial designs.

The aim of this study was to find the best design that is suitable for optimizing the recovery of the representatives of the 2nd, 3rd and 4th generation of fluoroquinolones. The following designs were applied: Central Composite Design, Box–Behnken Design and Doehlert Design. The recovery, which was a dependent variable, was estimated for liquid–liquid extraction. The time of shaking, pH, and the volume of the extracting agent (dichloromethane) were the independent variables. All results underwent the statistical analysis (ANOVA), which indicated Central Composite Design as the best model for evaluation of the recovery. For each analyte, an equation was generated that enabled to estimate the theoretical value for the applied conditions. The graphs for these equations were provided by the Response Surface Methodology. The statistical analysis also estimated the most significant factors that have an impact on the liquid–liquid extraction, which occurred to be pH for ciprofloxacin and moxifloxacin and the volume of an extracting solvent for levofloxacin.

Two types of symmetry can arise when the proportions of mixture components are constrained by upper and lower bounds. These two types of symmetry are shown to be useful for blocking first-order designs, as well as for finding the centroid of the experimental region. Orthogonal blocking of first-order mixture designs provides a method of including process variables in the mixture experiment, with the mixture terms orthogonal to the process factors. Symmetric regions are used to develop spherical and rotatable response surface designs for mixtures. The central composite design and designs based on the icosahedron and the dodecahedron are given for four-component mixtures. The uniform shell designs are three-level designs when applied to mixture experiments.

Biodiesel was produced from the waste cotton-seed cooking oil with methanol in presence of the calcium oxide catalyst by the ultrasound-assisted transesterification method. The ultrasonication approach was used for biodiesel production as it significantly lower reaction time and is highly energy-efficient. This paper-work focused on the optimization of biodiesel production parameters methanol: oil molar ratio (A), CaO amount (B), process temperature (C) using the two different response surface methodology (RSM) such as box-behnken design and central composite design (CCD). Also a comparison of the results of the box-behnken design (BBD) and central composite design (CCD) has been carried out. Quadratic polynomial equations are obtained by investigating the experimental yield of the transesterification process. The impact of process parameters on biodiesel yield is discussed by various plots. It was found that the foremost influential parameter on biodiesel production was the CaO amount based on both BBD and CCD methods. A critical relationship with experimental results in BBD method R2 value is 97.73%, while within the case of the CCD method R2 value is 99.90%. The Optimized process parameters for the CaO catalyst were determined as Methanol: oil molar ratio: 12:1; CaO amount (w/w)%: 1%; reaction temperature: 50°C; and the corresponding yield: 96.45% for BBD and CCD methods, respectively.

This study investigated the effect of replicating axial points in two-factor second-order experimental designs with a single centre point. The designs considered include the Doehlert design (DD), Inscribed Central Composite Design (ICCD), and Face-Centred Central Composite Design (FCCCD), all configured with a common radial distance of 1.0. Among these, the DD and ICCD exhibit spherical geometry, while the FCCCD does not. The objective was to evaluate how replication of axial points influences the statistical properties of these designs compared to their un-replicated counterparts. To assess the impact, various design evaluation metrics were employed, including D-optimality, T-optimality, prediction variance, and G-efficiency. The results showed that the un-replicated FCCCD exhibited both D- and T-optimality. The un-replicated Doehlert and ICCD designs were only D-optimal, whereas their replicated versions were T-optimal. This indicates a consistent shift from D-optimality to T-optimality upon replication of the axial points across all designs. Additionally, the findings suggested that the Doehlert design shares more structural and statistical similarities with the ICCD than with the FCCCD. Based on these results, the study recommended replicating axial distance of Doehlert and Central Composite designs in order to obtain T-optimal designs.

Orthogonal uniform composite designs

This book on experimental design is somewhat similar in flavor to the classic book by G.E

: The central composite design and designs based on the icosahedron and the dodecahedron are given for four component mixture systems. A uniform shell design is applied to the development of an obscurant smoke. Oils, cements, gasolines, and perfumes, are mixtures of components. The development of better products of these types involves experimentation with different formulations, or blends, of the components. Keywords: Surface methodology, Shell design, Geometry, Review, Central composite.

Some response surface designs for finding optimal conditions

The performance of a number of response surface designs for estimating a quadratic response surface model in symmetric experimerltal regions, k-spheres and hypercubes, is compared. The designs compared are composite designs, Box-Behnken designs, Uniform Shell designs, Hoke designs, Pesotchinsky designs, and Box-Draper designs. The performance criteria for the designs are their D-efficiency and their G-efficiency. All of the compared designs have high efficiencies. For large numbers of factors designs hnving higher efficiencies do exist; however, these designs have not yet been discovered.

: New three-level designs for fitting second-order response surfaces are obtained by three methods. New designs for 9, 10, and 13 factors are obtained by combining two-level factorial designs and incomplete block designs. New designs for 6, 8 and 10 factors are obtained by design rotation; these designs are rotations of central composite designs. Two sequences of design (for 7, 11 and 15 factors) are obtained by using a Hadamard matrix to reduce v+1 linearly dependent variables to a design for v factors. One of these sequences of design corresponds to a rotation of uniform shell designs; the other sequence is complementary to the rotated uniform shell designs in the sense that the O's and + or - 1's of the design points are interchanged.

Doehlert uniform shell designs and chromatography

Hybrid designs were created to achieve the same degree of orthogonality as central composite or regular polyhedral designs, to be near-minimum-point, and to be near-rotatable. They resemble central composite designs which have been augmented with at extra variable column. Eight designs are presented covering 3, 4, and 6 variables. All of these are at or within one point of minimum. Characteristics relevant to choice of design are discussed. Efliciencies are compared to central composite or polyhedral designs on n-spheres. A 46 point 7 variable design is also presented which, although it is not near-minimum, is an economical alternative to a 79 point central composite design.

Hybrid designs were created to achieve the same degree of orthogonality as central composite or regular polyhedral designs, to be near-minimum-point, and to be near-rotatable. They resemble central composite designs which have been augmented with at extra variable column. Eight designs are presented covering 3, 4, and 6 variables. All of these are at or within one point of minimum. Characteristics relevant to choice of design are discussed. Efliciencies are compared to central composite or polyhedral designs on n-spheres. A 46 point 7 variable design is also presented which, although it is not near-minimum, is an economical alternative to a 79 point central composite design.

Introduction: The use of response surface designs for drug formulation is highly warranted nowadays. Such smart designs reduce the number of required experiments compared to full-factorial designs, while providing highly accurate and reliable results. Aim: This study compares the effectiveness of two of the most commonly used response surface designs—Central Composite Design (CCD) and D-optimal Design (DOD)—in modeling a polymer-based drug delivery system. The performance of the two designs was further evaluated under a challenging scenario where a central point was deliberately converted into an outlier. Methods: Both methods were assessed using ANOVA, R-squared values, and adequate precision, and were assessed through an experimental validation point. Results: Both models demonstrated statistical significance (p-value < 0.05), confirming their ability to describe the relationships between formulation variables and critical quality attributes (CQAs). CCD achieved higher R-squared and predicted R-squared values compared to DOD (0.9977 and 0.9846 vs. 0.8792 and 0.7858, respectively), rendering it as the superior approach in terms of modeling complex variables’ interactions. However, DOD proved to be more predictive as it scored a lower percentage relative error. Conclusion: The demonstrated resilience of both models, despite the introduction of an outlier, further validates their utility in real-world applications, instead of the exhaustive full-factorial design.