Level Permutation Method For Constructing Mixed Level Uniform Designs Under The Centered L₂-discrepancy

In the practical application of uniform design,it is hard to guarantee that the levels of factors are the same,the existing literature including uniform design homepage has mostly discussed equal-level uniform design,so it is necessary to study mixed level design.In this paper,an efficient method for constructing mixed level uniform designs under the centered L₂-discrepancy by permuting the levels of factors is proposed.We first give the definition of the average centered L₂-discrepancy.Due to the particularity of the centered L₂-discrepancy,we obtain the relationship between average centered L₂-discrepancy and generalized wordlength pattern according to the parity of the number of factors.Relevant conclusions indicate that uniformity and minimum aberration criterion are approximately equivalent.Following it,we employ an orthogonal arry as the starting design,then conduct all the level permutations,calculate the centered L₂-discrepancy,and obtain the best design.Finally,we utilize special property of the centered L₂-discrepancy to make the algorithm much more efficient.Numerical results show that the designs constructed via this approach perform well under both the generalized wordlength pattern and the centered L₂-discrepancy.