New non-isomorphic detection methods for orthogonal designs

Abstract

Two fractional factorial designs are called isomorphic if one can be obtained from the other by relabeling the factors, reordering the runs and switching the levels of factors. Given a set of all orthogonal designs (ODs) with n runs, q levels and s factors, it may have several non-isomorphic subclasses. Once a new OD with this design size is generated, it is interesting to know which subclass it belongs to. In this paper, we review several existing methods, which can classify newly generated ODs to the correct non-isomorphic subclass. We also propose two new non-isomorphic detection methods. They can be utilized for the design classification purpose and take some advantages over the existing methods in terms of computation efficiency and classification capability.

Keywords:

• Hamming distance
• Isomorphism
• Level permutation
• Orthogonal design
• Uniformity

Notes

1 Throughout this paper, the computation time (or CPU time) refers to the total execution time of programmes built in MATLAB (R2014a) with CPU: Intel Core (TM) i7-2620M 2.70 G Hz, RAM: 8GB and OS: Microsoft Windows 64 bit.

2 The total number of level permutations of $L_n(3^s)$ is ${(3!)}^{13}.$ For 3-level designs, some level permutations do not change the MD value of a design because they are the “mirror image” operation (Tang, Xu, and Lin 2012). For this reason, the effective permutations in ${\mathcal{L}}_n(3^s)$ can be reduced from ${(3!)}^s$ to $3^s.$

Additional information

Funding

This work was partially supported by the UIC Grant (R201810, R201912, R202010) and the Zhuhai Premier Discipline Grant.