Asymmetric uniform designs based on mixture discrepancy

Efficient experimental design is crucial in the study of scientific problems. The uniform design is one of the most widely used approaches. The discrepancies have played an important role in quasi-Monte Carlo methods and uniform design. Zhou et al. [17] proposed a new type of discrepancy, mixture discrepancy ( $ {\mathcal {MD}} $ MD), and showed that $ {\mathcal {MD}} $ MD may be a better uniformity measure than other discrepancies. In this paper, we discuss in depth the $ {\mathcal{MD}} $ MD as the uniformity measure for asymmetric mixed two and three levels U-type designs. New analytical expression based on row distance and new lower bound of the $ {\mathcal {MD}} $ MD are given for asymmetric levels designs. Using the new formulation and the new lower bound as the benchmark, we can implement a new version of the fast local search heuristic threshold accepting. By this search heuristic, we can obtain mixed two and three levels U-type designs with low discrepancy.