MODIFIED SECOND ORDER SLOPE ROTATABLE DESIGNS USING SUPPLEMENTARY DIFFERENCE SETS

In this paper, following the methods of constructions of Mutiso et al. [7-8], Chiranjeevi and Victorbabu [2-3], a new method of construction of modified second order slope rotatable designs using supplementary difference sets is suggested. Some illustrative examples are also presented.


INTRODUCTION
The concept of rotatability, which is very important in response surface designs, was proposed by Box and Hunter [1]. Das and Narasimham [4] developed second and third order rotatable designs and constructed rotatable designs using balanced incomplete block designs (BIBD). Seberry [9] studied some remarks on supplementary difference sets (SDS) and applications of SDS.
Koukouvinos et al. [15] suggested a general construction method for five level second order 678 P. CHIRANJEEVI, B. RE. VICTOR BABU rotatable designs (SORD) using SDS. Hader and Park [5] introduced slope rotatability for second order response surface designs and constructed slope rotatable central composite designs.
Victorbabu [10] constructed SOSRD using symmetrical unequal block arrangements (SUBA) with two unequal block sizes. Victorbabu [11][12] suggested a new restriction 4 2 2 iu iu ju x =N x x  to get modified slope rotatability for second order response surface designs. Further, they have constructed modified SOSRD using central composite designs and BIBD. Victorbabu [13][14] suggested reviews on second order rotatable and sloped rotatable designs. Victorbabu and Surekha [21] constructed a new method of three level SOSRD using BIBD. Victorbabu [15] suggested a bibliography on slope rotatable designs. Victorbabu [16] suggested a review on SOSRD over axial directions. Victorbabu [17] suggested a note on SOSRD using a pair of partially balanced incomplete block designs. Specially, Mustio et al. [7][8] suggested five level second order rotatable and modified second order rotatable designs using SDS. Chiranjeevi and Victorbabu [2] studied measure of slope rotatability for second order response surface designs and constructed measure of SOSRD using SDS. Chiranjeevi and Victorbabu [3] suggested a method of construction of SOSRD using SDS.
In this paper, following the methods constructions of Mutiso et al. [7][8], Chiranjeevi and Victorbabu [2][3], a new method of construction of modified second order slope rotatable designs using supplementary difference sets is suggested. Some illustrative examples are also presented.

Conditions for second order slope rotatable designs
Suppose we want to use the second order response surface designs D= ((xiu)) to fit the surface, where xiu denotes the level of the i th factor (i=1,2,…,v) in the u th run (u=1,2,…,N) of the experiment and the eu's are uncorrelated random errors with mean zero and variance σ 2 .
A second order response surface design D is said to be SOSRD if the design points satisfy the following conditions (cf. Hader and Park [5], Victorbabu and Narasimham [18]).
Where c, λ2 and λ4 are constants and the summation is over the design points.
The variances and co-variances of the estimated parameters are, and other covariances vanish. (2.7) Therefore the conditions (2.2) to (2.7) give a set of conditions for slope rotatability in any general second order response surface design.

CONDITIONS FOR MODIFIED SECOND ORDER SLOPE ROTATABLE DESIGNS
A second order response surface design D is said to be modified SOSRD that if the design points are satisfy the conditions (2.2) to (2.6) are met (cf. Hader and Park [5], Victorbabu and Narasimham [18] and further we have Victorbabu [11] suggested the conditions of modified variance and covariances of the estimated parameters are also satisfied. The usual method of construction of SOSRD is to take combinations with unknown constants, • Consider the first (v-1) 2 columns of A. An array with e rows and e columns, where is obtained, whose every column has one zero element and e-1 elements equal to 1.
• Superimpose a e-r 2 factorial fraction onto the units of each row of the array, while onto the zero elements superimpose e-r 2 ×1vector with all elements zero. In this way, a three level design with e factors and e-r (e-1)×2 runs is obtained.
• Add an axial point ±b in every column of the design in order to attain the rotatability of the design; b must be equal to 1/4 a , where e-r-1 a = (2e-5)×2 .
Further, Koukouvinos et al. [15] stated that, it was convenient to choose to use the smallest fraction of e 2 factorial, so the resulting design has the minimum possible number of runs.
However, for more than three factors, it is necessary to use fractions of resolution V in order to attain the rotatability of the design. Then, Koukouvinos et al. [15] suggested SORD with (v-1) m= 2 factors at five levels t(m) a (±1,0,±b) and N =m2 +n design points, where t(m) 2 denotes resolution-V fractional factorial design replicate of m 2 in ±1 levels, and a n is number of axial points. Here, we suggest to construct a SOSRD with (v-1) m= 2 factors at five levels (±1,0,±b) and  The illustrate examples of modified SOSRD using SDS are given bellow in the following table.