Order Response Surface Designs Research Articles

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.

Measure of Slope Rotatability for Second Order Response Surface Designs Under Tri-Diagonal Correlation Error Structure Using Symmetrical Unequal Block Arrangements with Two Unequal Block Sizes

In this paper, measure of slope rotatability for second order response surface designs using symmetrical unequal block arrangements with two unequal block sizes under tri-diagonal correlation error structure is suggested and illustrated with examples.

Measure of Modified Rotatability for Second Order Response Surface Designs using Pairwise Balanced Designs

Measure of Modified Rotatability for Second Order Response Surface Designs using Pairwise Balanced Designs

Measure of modified rotatability for second order response surface designs

A measure enables us to assess the degree of rotatability for a given response surface designs. In this paper, a new measure of modified rotatability for second order response surface designs is suggested. The method is illustrated using central composite designs for 2≤v≤17.

Measure of slope rotatability for second order response surface designs under intra-class correlated structure of errors using central composite designs

In the design of experiments for estimating the slope of the response surface, slope rotatability is a desirable property. In this paper, measure of slope rotatability for second order response surface designs using central composite designs under intra-class correlated structure of errors is suggested and illustrated with examples.

Measure of Slope Rotatability for Second Order Response Surface Designs under Intra-class Correlated Structure of Errors Using Pairwise Balanced Designs

In this paper, measure of slope rotatability for second order response surface designs using pairwise balanced designs under intra-class correlated structure of errors is suggested and illustrated with examples.

An empirical study on modified second order response surface and Taguchi designs for optimizing the process parameters

In this paper, Following the works of Rajendar et al. (2004) and Prabhakar et al. (2014), we combine the methods of modified second order response surface design and Taguchi design of experiments to determine optimum level of process parameters. Here we mainly focused on the statistical analysis of these two methods and obtain the optimum response with a smaller number of design points by using modified second order response surface design. The results in our study showing that modified second order response surface designs required a smaller number of design points to obtain the optimum value. The optimal process parameters of surface roughness are determined using Modified response surface combinations and Taguchi approach along with response graphs.

Second Order Slope Rotatable Designs under Tri-diagonal Correlation Structure of Errors Using a Pair of Incomplete Block Designs

Box and Hunter [1] introduced the concept of rotatability for response surface designs. The concept of slope-rotatability was introduced by Hader and Park [2] as an analogous to rotatability property, which is an important design criterion for response surface design. Slope-rotatable design is that of which the variance of partial derivative is a function of distance from the design (d). Recently, a few measures of slope-rotatability for a given response surface design was introduced. In this paper, a new method of slope rotatability for second order response surface designs under tri-diagonal correlation structure of errors using a pair of symmetrical unequal block arrangements with two unequal block sizes is studied. Further, a study on the dependence of variance function of the second order response surface at different design points for different values of tri-diagonal correlation coefficient ρ which lies between -0.9 to 0.9 and the distance from centre (d) is suggested.

An empirical study on modified second order response surface and Taguchi designs for optimizing the process parameters

In this paper, Following the works of Rajendar et al. (2004) and Prabhakar et al. (2014), we combine the methods of modified second order response surface design and Taguchi design of experiments to determine optimum level of process parameters. Here we mainly focused on the statistical analysis of these two methods and obtain the optimum response with a smaller number of design points by using modified second order response surface design. The results in our study showing that modified second order response surface designs required a smaller number of design points to obtain the optimum value. The optimal process parameters of surface roughness are determined using Modified response surface combinations and Taguchi approach along with response graphs.

Some sequential third order response surface designs

Sequential third order designs are very important due to their nice sequential property. These designs become even more important in case of second order model lack of fit. There is possibility that an experimenter uses a second order design to explore the second order response surfaces, however, lack of fit problem may arise. If one is using the sequential third order design, then there is an opportunity to use the next portion of that design to estimate the third order terms and remove the second order model lack of fit. Das and Narasimham (1962) and Arshad, Akhtar, and Gilmour (2012) have proposed sequential third order designs that can be used to estimate a complete third order model, in case we face second order model lack of fit. However, these designs have very large sizes. Sometime, it becomes difficult for an experimenter, in real world situation, to use a design with such a large size due to different issues like high cost of experimental material. Arshad, Akhtar, and Gilmour (2012) augmented the very popular designs of Box and Behnken (1958 and 1960) and made them third order. Those designs, called augmented Box–Behnken designs (ABBDs), are sequential in nature and can be used for estimation of a complete third order model in case of second order model lack of fit. These designs have been developed by augmenting the Box–Behnken designs using different combinations of factorial points, axial points and complementary design points. In this research paper, we have also selected the Box–Behnken designs due to their popularity and augmented them using the concept of doubly balanced incomplete block design provided by Calvin (1954). We have developed 9 designs one for each from 4 to 12 number of factors (design for 3 factors was not achievable). These third order designs are also sequential in nature, and can be used for estimation of complete third order model in case the second order model lack of fit is exhibited. The newly proposed designs have smaller sizes as compared to Das–Narasimham designs (DNDs) in all cases, and smaller than ABBDs in many cases. In addition to the designs sizes, some other design optimality criteria have also been explored for these designs, and compared with the DNDs and ABBDs. Mostly the new designs have been found better than DNDs and ABBDs.

Blocking first order response surface designs with interaction under correlated error

Blocking first order response surface designs with interaction under correlated error

On the generation of cost effective response surface designs

Run order consideration for mixed factorial, fractional factorial and confounded factorial have been studied by several authors in depth, but is lacking for Response Surface Designs (RSDs) except the results obtained by Quinlan and Lin (2015) for Plackett-Burman Design, a commonly used first order response surface design for screening purpose. Second Order Response Surface Designs (SORDs) are used to explore relationship between the response variable and the input variables and to find out the optimum input combinations to achieve a desired response. In this paper, we aim to find out optimal run orders with respect to minimizing level changes using a computer programme. Minimizing the level changes implies the minimization of experimental cost. Generation of four classes of designs viz., Plackett-Burman Design, Cost-effective Central Composite Design (CCD) with full factorial as well as fractional factorial points and Cost-effective Box Behnken Design (BBD) have been described through Macros developed using SAS IML.

On the Efficiencies of Second Order Response Surface Designs for the Estimation of Responses and Slopes Using Symmetrical Unequal Block Arrangements with Two Unequal Block Sizes

In this paper, the efficiencies of various second order response surface designs, like second order rotatable designs (SORD), second order slope rotatable designs (SOSRD), SORDs with an equi-spaced doses design, and SOSRDs with an equispaced doses design using symmetrical unequal block arrangements (SUBA) with two unequal block sizes, are studied for the estimation of responses and slopes at different points (central, axial, cube corner points) on second order response surface designs. The study is done because in some cases a SORD, and SOSRD constructed by using a SUBA with two unequal block sizes have fewer design points compared to those constructed by other methods.

A graphical method for evaluating slope-rotatability in axial directions for second order response surface designs

Draper and Ying (J. Statist. Plann. Inference 41 (1994) 113–119) showed that the surface of the averaged slope variance of second order response surface function is an ellipsoid. Using this idea, a graphical method for evaluating slope-rotatability in axial directions with respect to second order response surface designs is proposed.

A measure of rotatability for second order response surface designs

The concept of rotatability introduced by Box and Hunter (1957,Ann. Math. Statist.,28, 195–241) is an important design criterion for response surface design. Recently, a few measures of rotatability that enable us to assess the degree of rotatability for a given response surface design have been introduced. In this paper, a new measure of rotatability for second order response surface designs is suggested, and illustrated for 3 k factorial design and central composite design. Also a short comparison is made between the proposed measure with the previously suggested measures.

Some Second Order Response Surface Designs in Equal Sized Blocks

Application of response surface designs in field plot experiments is limited for want of suitable layout in manageable design points split into blocks of equal and practicable size. Keeping this end in view, some second order response surface (SORS) designs in suitable blocks have been suggested in this article. In the end, a list of SORS block designs for v factors (5⩽ v⩽10) having k⩽16 is appended.

Center Points in Second—Order Response Surface Designs

Previous approaches for selecting the number of center points in a second-order response surface design are explained, and an integrated variance criterion is offered. Calculations extending results in the literature are made for various composite and Box-Behnken designs for 2 ≤ k ≤ 8. The results are compared and discussed and indicate that fewer center points than initially recommended are generally appropriate.

On Some Mixed Order Response Surface Designs Useful in Sequential Experiments

In this paper we have introduced some mixed order response surface designs like FORD­SORD, FORD­TORD, SORD­TORD and discussed their analysis and construction. We have also explained how we can improve upon a FORD­SORD where p factors are first order and ( v- p) factors are second order to get one in which p1 factors are first order, p2 factors are seeond order and other ( v - p) factors are second order so that the design becomes a FORD­GDSORD. We can then improve it further whenp, factors are first order and ( v- p+ p2) are second order.

Slope-Rotatable Central Composite Designs

An analogue of the Box-Hunter rotatability property for second order response surface designs in k independent variables is presented. When such designs are used to estimate the first derivatives with respect to each independent variable the variance of the estimated derivative is a function of the coordinates of the point at which the derivative is evaluated and is also a function of the design. By choice of design it is possible to make this variance constant for all points equidistant from the design origin. This property is called slope-rotatability by analogy with the corresponding property for the variance of the estimated response, ŷ. For central composite designs slope-rotatability can be achieved simply by adjusting the axial point distances (α), so that the variance of the pure quadratic coefficients is one-fourth the variance of the mixed second order coefficients. Tables giving appropriate values of α have been constructed for 2 ≤ k ≤ 8. For 5 ≤ k ≤ 8 central composite designs involving fractional factorials are used. It is also shown that appreciable advantage is gained by replicating axial points rather than confining replication to the center point only.

On Practical Use of the Concept of D-Optimality

The purpose of this article is to review the possibility of practical use of the concept of D-optimality in response surface design. Second order quasi-D-optimum designs on a cube have been built and their high efficiency proved. Using the computer, a number of different second order response surface designs, previously built, have been compared. It has been proved that the D-optimum concept, being more general than the rotatable one, can be used as the theoretical basis for building and comparing response surface designs in use.