Tolerance Optimization: A Decomposition Scheme, Variance Reduction and Fractional Approximation

Abstract Development of involved optimization algorithms is not an easy task for several reasons: First, every analyst is interested in a specific problem; Second, the capabilities of these methods may not be fully understood a priori; Third, coding of multi-purpose and more involved algorithms is not an easy job. In this paper, the optimization problem employing the near to global optimum algorithm is studied (Gadallah, M.H., 2000). The focus is to exploit 2 ideas: First, the algorithm can be modified to act as a variance reduction technique; Second, the algorithm can be modified to tackle the problem of system decomposition. Both ideas are novel within the context of statistical design of experiments. The first, if fully proved experimentally could yield the simultaneous integration of nominal and variance optimization possible. The second, can be extended to deal with multi-dimensional highly constrained systems with ease. These two ideas are explained wife the use of a simple example to illustrate the idea. An algorithm is developed that deal with the problem in several stages according to a predetermined decomposition scheme. The original objective and constraint functions are dealt with to suit each stage. Accordingly, all NP hard problems can ideally be transformed into NP complete ones with a consequence on the number of stages resulting from decomposition. Several decomposition scenarios are used and their results are compared numerically. Two orthogonal arrays and four composite arrays are used to plan experimentation; these are L27OA and L54OA and their subfamilies. These arrays are compared with respect to their statistical measures. The algorithm as such, is very promising optimization tool, especially for coupling system decomposition and variance reduction. Past work focused on either decomposition or statistical optimization. This work offers both capabilities. Several studies are reviewed and conclusions are drawn.