Uniform design has been widely applied in many fields, such as manufacturing, system engineering, pharmaceutics and natural sciences. The measure of uniformity plays a key role in constructing uniform design. There are many measures to assess the uniformity. Among them, the wrap-around and the centered discrepancy have been regarded more reasonable and practicable. It is an important issue to find good lower bounds for the discrepancy, because lower bounds can be used as bench marks not only in searching for uniform (or optimal) designs but also in helping to validate that some good designs are in fact uniform (or optimal). A design whose discrepancy value achieves a strict lower bound is a uniform design with respect to this discrepancy. In this book, i investigate new analytically expressions, which gives new lower bounds of the centered and wrap-around discrepancy for symmetric two, three and four-level and asymmetric mixed two and three-level U-type designs. Furthermore, the results in this book can be used to characterize lower discrepancy or uniform designs and to use this characterization for a guided selection of the search direction in optimization heuristics.

Department of Mathematics, Faculty of Science, Zagazig University, Zagazig 44519, Egypt. Elsawah's research interests: Optimal Uniform Designs, Design and Modeling for Computer Experiments and Order Statistics with It's Application. For more information, please visit Dr. Ahmed Elsawah web page: http://www.staff.zu.edu.eg/a.elsawah