Order Rotatable Designs Research Articles

Modified Slope Rotatability for Balanced Ternary Designs of Second Order Response Surfaces

Objective: To evaluate the best optimal solutions of modified Second Order Slope Rotatable Designs (SOSRD) using Balanced Ternary Designs (BTD) and to compare with different methods of Modified SOSRD for 3≤v≤8. Methods: To estimate the parameters of the response surface design of second order the principle of least squares was applied for design of experiments, we also evaluated modified slope rotatable value and variance of estimated responses. Findings: The outcome of this research reveals the influence of various design parameters on key aspects such as the variance of the estimated response, slope rotatable value, axial points, central points and also total number of design points of modified SOSRD framework. Novelty: The variance of the estimated responses in the new method of Modified SOSRD using BTD provides the best optimal solutions with minimum variances while compared to different methods of Modified SOSRD. Keywords: Response ­ Surface Experimental Designs, Second Order Rotatable Designs, Second ­ Order Slope ­ Rotatability, Second ­ Order Modified Slope ­ Rotatability, Balanced Ternary Designs

Construction of two New Second Order Rotatable Designs Using Trigonometric Functions

In this study, two new second order rotatable designs of thirty-five and forty-five points are constructed respectively using transformations of some trigonometric functions. The designs are s rotatable since they satisfy both the outlined moment conditions and the non-singularity conditions. Second-order rotatable designs are useful for optimizing these processes in chemical industries, agriculture and engineering by systematically varying the factors of interest and studying their combined effects on the response variables. This helps in identifying the optimal settings for maximum or minimum responses.

Construction of Twenty-Three Points Second Order Rotatable Design in Three Dimensions Using Trigonometric Functions

In this paper, a novel twenty-three-point second-order rotatable design is formulated utilizing trigonometric functions. Design of experiments plays a crucial role in various industries and research fields to investigate the relationship between multiple variables and their effects on a response variable. In particular, second order rotatable designs are widely used due to their ability to efficiently estimate the main, interaction, and curvature effects. Nevertheless, creating designs with a substantial number of points poses difficulties. This study concentrates on developing a second-order rotatable design with twenty-three points utilizing trigonometric functions. Trigonometric functions offer a systematic approach to distribute the points uniformly in the design space, thereby ensuring the optimal coverage of the experimental region. The proposed construction utilizes the properties of sine and cosine functions to generate a balanced and efficient design. The methodology involves dividing the design space into equidistant sectors and assigning the points using the trigonometric functions. By carefully selecting the starting angle and the angular increment, a complete and orthogonal design is achieved. The design is rotatable, meaning it can be rotated to any desired orientation without impairing the statistical properties of the design. Through this construction, the design effectively captures the main effects, interaction effects, and curvature effects. This enables reliable estimation of the model parameters, leading to accurate predictions and efficient optimization. Additionally, the design is efficient in terms of minimizing the number of experimental runs required, thereby reducing costs and time. The suggested second-order rotatable design comprising twenty-three points and employing trigonometric functions exhibits its superiority when compared to conventional designs. It offers a systematic and straightforward approach to construct a balanced and efficient design for studying the relationships between multiple variables. The design's rotatability ensures flexibility in experimental settings, making it a valuable tool for researchers and practitioners in various fields.

A Forty-Two PointsSecond Order Rotatable Design in Three Dimensions Constructed using Trigonometric Functions Transformations

In this study, a new forty-two point’ssecond order rotatable design is constructed usingtransformations of some trigonometric functions. This design permits a response surface to be fitted easily and provides spherical information contours besides the economic use of scarce resources in relevant production processes.

Slope Rotatable Central Composite Designs of Second Type

Central composite design (CCD) is the most commonly used fractional factorial design used in the response surface model. Kim [1] proposed second order rotatable designs (SORD) of second type using CCD, in which the positions of axial points are indicated by two numbers\(\left(a_{1}, a_{2}\right)\). Kim and\(K_{0}\) [2] introduced second order slope rotatable designs (SOSRD) of second type using CCD, in which the positions of axial points are indicated by two numbers \(\left(a_{1}, a_{2}\right)\). In this paper, second order slope rotatable central composite designs of second type with\(2 \leq n_{3} \leq 4\) (where \(n_{\text {s }}\) denotes the number of replications of axial points) are suggested for \(2 \leq \mathrm{v} \leq 17\) (v-stands for number of factors). It is observe that the value of level \(\mathrm{a}_{2}\) (taking \(\mathrm{a}_{1}=1\) ) for the axial points in CCD required for slope rotatability for second type is appreciably larger than the value required for SORD of second type using CCD. And also noted that if we replicate axial points \(\left(n_{9}\right)\)in SOSRD of second type using \(\mathrm{CCD}\) then the value of \(\mathrm{a}_{2}\) (taking \(\mathrm{a}_{1}=1\) ) is approximately nearer to\(\mathrm{SORD}\) value \(\mathrm{a}_{2}\) of second type using CCD.

Relative Efficiency Criteria for the Six Specific Calculus Optimum Values Second Order Rotatable Designs with a Practical Example on Twenty-Four Points

In this study we focus on the existing six specific calculus optimum second order rotatable designs in three dimensions denoted by M1, M2, M3, M4, M5, and M6 as illustrated in the text form. A design matrix X is developed from the designs and the information matrices C1, C2, C3, C4, C5, and C6 are obtained and hence the relative efficienciesA-, D-, E-, T and I/IV- for the six specific Calculus optimum values designs are evaluated. From the results it is evident that the determinant criterion (Deff) for the six designs has the highest relative efficiency average. is found to be the most efficient design as compared to the rest.

New Series for Construction of Second Order Rotatable Designs

Second order Rotatable designs have good significance in response surface methodology. In this paper, two new seriesfor the construction the same using Binary Ternary Designs were presented with illustrated examples.

Run order consideration for sequential third order rotatable designs

Random execution of run sequences is employed in response surface designs to avoid bias in the response. Since the randomization of run sequences may induce frequent changes in the factor levels, which makes it difficult or expensive for the experimenter, especially when hard-to-change factors are involved in the experiment. Mostly, second order response surface designs (SORDs) are being used to explore the functional relationship between the response and the input variables. But when the lack of fit of the model becomes significant, it is desirable to explore the relationship with a polynomial model of order three. This requires third order rotatable designs (TORDs), which can be sequentially implemented by adding few more runs to the SORDs. In this paper, the construction of sequential TORDs with minimum level changes in the run sequences has been discussed.

Combinatorial Construction of Second Order Rotatable Designs

In this paper, we constructed a new series for the construction of Second Order Rotatable Design using Partially Balanced Incomplete Block Designs (PBIBD).

Second Order Rotatable Designs of Second Type Using Pairwise Balanced Designs

In this paper, second order rotatable designs of second type using pairwise balanced designs is suggested. This design is compared with second order rotatable designs of first type using pairwise balanced designs [1] on the basis of efficiency.

Second Order Rotatable Designs of Second Type using Symmetrical Unequal Block Arrangements with Two Unequal Block Sizes

Second Order Rotatable Designs of Second Type using Symmetrical Unequal Block Arrangements with Two Unequal Block Sizes

Second Order Rotatable Designs of Second Type Using Central Composite Designs

Kim [1] introduced rotatable central composite designs of second type with two replications of axial points for 2≤v≤8 (v: number of factors). In this paper we have extended the work of Kim [1] for second order rotatable designs of second type using central composite designs for 9≤v≤17.

Optimal Sequential Third Order Rotatable Designs in Three, Four and Five Dimensions

Response surface methodology (RSM) is a statistical technique used to evaluate the relationship between multiple input variables and one or more response variables with the aim of optimizing the response variables. Sequential experiments are very economical and useful in practice. Therefore, rotatable designs such as the third order rotatable design (TORD) may be run sequentially in three stages with three or four blocks depending on the model adequacy. Normally, the first section consisting of first order is run and the response function is approximated using a first order model. If the first order model is found to be adequate, as the representation of the unknown function by noting the evidence of the goodness of fit, the experiment may be stopped at this stage. However, if the first model is found to be unfit, the trials of the second order are run and ultimately, proceed to fit a third order if a second order model is also found to be inadequate. In this paper, two sets of second order rotatable designs are combined to form sequential third order rotatable designs (TORD) in three, four and five dimensions. The TORDs are then evaluated on their alphabetic optimality criteria with the aim of reducing the costs of experimentation. The classical optimality criteria includes; D-criterion, A-criterion, T-criterion and E- criterion.

Second order rotatable designs of second type using balanced incomplete block designs

In this paper second order rotatable designs of second type using balanced incomplete block designs is suggested. This design is compared with second order rotatable designs of first type using balanced incomplete block designs (cf. Das and Narasimham [6]) on the basis of efficiency.

Sequential asymmetric third order rotatable designs (SATORDs)

Rotatable designs that are available for process/ product optimization trials are mostly symmetric in nature. In many practical situations, response surface designs (RSDs) with mixed factor (unequal) levels are more suitable as these designs explore more regions in the design space but it is hard to get rotatable designs with a given level of asymmetry. When experimenting with unequal factor levels via asymmetric second order rotatable design (ASORDs), the lack of fit of the model may become significant which ultimately leads to the estimation of parameters based on a higher (or third) order model. Experimenting with a new third order rotatable design (TORD) in such a situation would be expensive as the responses observed from the first stage runs would be kept underutilized. In this paper, we propose a method of constructing asymmetric TORD by sequentially augmenting some additional points to the ASORDs without discarding the runs in the first stage. The proposed designs will be more economical to obtain the optimum response as the design in the first stage can be used to fit the second order model and with some additional runs, third order model can be fitted without discarding the initial design.

Calculus Optimum Values Optimality Criteria for Twenty Four Points Specific Second Order Rotatable Design in Three Dimensions

This study uses the existing second order rotatable design to obtain optimum design based on the known classical optimality criteria that is the determinant criterion, the average-variance criterion, the smallest-Eigen value criterion and the trace criterion. These criteria measure the desirability of a design, D-optimum design minimizes the content of the ellipsoidal confidence region for the parameters of the linear model, A-optimum design minimizes the sum (or average) of the variances of the parameter estimates, E-criterion reduces the variance of each individual parameter estimate and T-criterion is one that has not enjoyed much use because of the linearity aspect of T-criterion. This study considers the already existing twenty four points three dimensional specific rotatable design of order two. The information matrices C1, for this design is obtained from the moment matrix M, for the second order model for three factors using the relation C=(K1M-1K)-1, where M=1/N(X1X), is the moment matrix, K is the coefficient matrix of the parameter sub system of interest. Our parameter system of interest is that of the linear and pure quadratic factors only. The optimality criteria for the design with the corresponding information matrix C1, is determined as A-optimal.

A Study on Second Order Rotatable Designs under Tri-diagonal Correlated Structure of Errors Using a Pair of Balanced Incomplete Block Designs

In this paper, a study on second order rotatable designs under tri-diagonal correlated structure of errors using a pair of balanced incomplete block designs is suggested. Further, the variance function of the estimated response for different values of tri-diagonal correlated coefficient and distance from centre for ( - factors) are studied.

Modified non-sequential third order rotatable designs constructed using Pairwise Balanced Design

The technique of fitting a response surface design is useful in modelling of experimental designs. Response surface is used in situations where the response of interest is influenced by several experimental variables. The objective of fitting a response surface design is to reduce cost of experimentation and to obtain optimal designs. The property of rotatability is a desirable quantity of experimental design and requires the variance of the fitted design to be constant on circles or spheres about the centre of the design. In this article, a construction technique of fitting modified non-sequential third order rotatable design (TORD) using Pairwise Balanced Design (PBD) is presented. The variance function of a third order response surface design and the properties of Pairwise Balanced Design are utilised for the construction.

Construction of twenty-six points specific optimum second order rotatable designs in three dimensions with a practical example

This quadratic response surface methodology focuses on finding the levels of some (coded) predictor variables x = (x1u, x2u, x3u)' that optimize the expected value of a response variable yu from natural levels. The experiment starts from some best guess or “control” combination of the predictor variables (usually coded to x = 0 for this case x1u=30, x2u=25 and x3u =40) and experiment is performed varying them in a region around this center point.We go further to construct a specific optimum second order rotatable design of three factors in twenty-six points. The achievement of this is done with estimation of the free parameters using calculus in an existing second order rotatable design of twenty-six points. Such a design permits a response surface to be fitted easily and provides spherical information contours besides the realizations of optimum combination of ingredients in Agriculture, horticulture and allied sciences which results in economic use of scarce resources in relevant production processes. The expected second order rotatable design model in three dimensions is available where the responses would then facilitate the estimation of the linear and quadratic coefficients. An example involving Phosphate (x1u), Nitrogen (x2u) and Potassium (x3u) is used to represent the three factors in the coded level and converted into natural levels. Â

First order rotatable designs incorporating differential neighbour effects from experimental units up to distance 2

In this paper, first-order response surface model incorporating neighbour effects from experimental units up to distance 2 has been discussed where experimental plots experience the neighbour effects from immediate left and right neighbouring units as well as from units at distance 2 assuming the plots are placed linearly without gaps. Conditions have been derived for the orthogonal estimation of the parameters of the model. A method of constructing designs for fitting first-order response surface model which ensures the constancy in the prediction variance after incorporating neighbor effects up to distance 2 has been developed and named as FORDDNE-2.