Second Type Second Order Slope Rotatable Designs Utilizing Balanced Incomplete Block Designs with Unequal Block Sizes

Abstract

Kim and Ko proposed second type second order slope rotatable designs (SOSRD) utilizing central composite designs (CCD) wherein the two digits denotes the position of star points. In this study, we propose SOSRD of second type utilizing balanced incomplete block designs (BIBD) with unequal block sizes. In specific cases, the recommended procedure results in fewer design points than SOSRD of second type acquired through pairwise balanced designs (PBD), symmetrical unequal block arrangements (SUBA) with two unequal block sizes and balanced incomplete block designs (BIBD).

Introduction

Response surface methodology (RSM) involves statistical and mathematical models analyzing how multiple explanatory variables affect a response variable. Box and Hunter introduced the concept of rotatability in central composite designs (CCD). Subsequent researchers developed various second order rotatable designs (SORD) using different block design methods such as balanced incomplete block designs (BIBD), pairwise balanced designs (PBD), incomplete block designs (IBD), and symmetrical unequal block arrangements (SUBA).

Slope rotatable central composite designs (SRCCD) and second order slope rotatable designs (SOSRD) extend these ideas, focusing on designs that maintain slope rotatability. Various studies have refined these designs for different numbers of factors, using combinations of BIBD, PBD, SUBA, and CCD.

This paper proposes a new method for constructing second type SOSRD using BIBD with unequal block sizes. This approach often yields SOSRDs with fewer design points compared to previous methods using PBD, SUBA, or standard BIBD. The theoretical foundations rely on specific quadratic polynomial models satisfying conditions for slope rotatability, including symmetry and moment constraints.

The design construction uses BIBD matrices with unequal block sizes, forming incidence matrices with elements representing treatment appearances. By defining the design points, axial points, and central points carefully, the authors prove the existence of these designs through a biquadratic equation. Examples for 6- and 8-factor cases show the new method produces designs with fewer points while satisfying non-singularity and rotatability criteria.

Conclusion

In this paper, second type of SOSRD utilizing BIBD with unequal block sizes is suggested. It is observed that the proposed procedure can generate designs with fewer design points than SOSRD of second type acquired utilizing PBD, SUBA with two unequal block sizes and BIBD.

References

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Copyright

Copyright © 2025 B Venkata Ravikumar, B. RE. Victorbabu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.