Calculating derivatives in statistical tolerance analysis

Current statistical tolerance analysis techniques rely on Monte Carlo estimation. The typical Monte Carlo analysis does not provide derivative information however, so it is poorly suited to tolerance synthesis where derivatives are important in determining the next set of tolerances to analyze. Skowronski and Turner (Skowronski V.J. and Turner J.U., Using Monte Carlo variance reduction in statistical tolerance synthesis, Computer-Aided Design, 1997, 29, (1), 63–69) 1 proposed a method for estimating the derivatives of the Taguchi Quality Loss function with respect to tolerance values. This paper extends that derivative estimation technique to tolerance analysis problems that use the acceptance fraction criterion. The application of two variance reduction techniques to derivative estimation is also described. Finally, the accuracy of the techniques in this paper are compared to the finite difference calculation. It is concluded that the derivative estimation of Skowronski and Turner 1 provides equivalent or better accuracy than finite difference, and has the added advantage that the derivative of more than one tolerance can be calculated without a new analysis for each derivative.