Quality tolerancing and conjugate duality

This paper studies the relationship between product quality as defined by tolerances on the product to tolerances on the parts. All the standard sure-fit and the statistical-fit tolerancing models, which seek to find the least cost tolerances for the parts given the product specification, are found to be convex. Thus conjugate duality provides a unifying framework for studying the relationship between quality and tolerancing and the analysis and solution of such problems. Moreover this convexity is maintained even with the addition of quality loss functions. Quality loss is modeled on the basis of parts or the product. The analysis leads to an analytic solution to a number of models including the quality loss model.