Absolute deviance method for symmetrical uniform designs

Uniform design is a kind of space filling designs which is widely used in various field due to its great advantages. There are two types of uniform design; symmetrical and asymmetrical. Measure of uniformity and construction methods are essentials for construction of uniform designs. Uniform designs can be achieved by minimizing a discrepancy where the discrepancy is a measure of uniformity. From the various discrepancies that have been suggested, centered L2 discrepancy and mixture discrepancy are employed in our research. In this research, we focused on the type of symmetrical uniform designs, - - s n U n which the factors have same number of levels and the number of experimental runs equal to the number of levels. There are numbers of construction methods of uniform designs or nearly uniform designs in the literature. A design with low discrepancy or a good approximation to uniform design is a nearly uniform design.The existing construction methods such as good lattice point method, optimization searching method and the cutting method exhibited their advantages. However, there are still having areas which need to improve. Moreover, there is no development of new construction methods in the recent years. Therefore, two of the existing construction methods of uniform design; the optimization method and the cutting method are analyzed and modified to a better approach in terms of computation time and uniformity. The optimization method is modified by proposing the absolute difference equivalence (ADE) approach which coordinate with ruin and recreate (R&R) approach in reducing the size of neighborhood and decreasing the computational load. Ultimately, the size of the neighborhood and the computational time are decreased and the global solution can be obtained. We have shown that ADE approach effectively reduce the size of neighborhood which is determined by the R&R approach. Furthermore, an optimization part is added to the cutting method to find an appropriate number for experimental runs of the initial design. Conclusively, suggestion tables are given on the number of experimental runs for initial design which results in uniform designs with more stable uniformity. It shows that choosing the suggested number of experimental runs of initial design produces uniform designs with lower uniformity. Besides, we proposed a new method called absolute deviance method (ADM) for construction of symmetrical nearly uniform designs. The concept of ADM is from the idea of uniform design which uniformly scattered the points in the experimental domain. The uniformity of the uniform design can be achieved by setting specific pattern of the absolute differences between points. It shows that this new method is an efficient method in constructing symmetrical uniform designs with better uniformity.

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