Optimal centering, tolerancing, and yield determination via updated approximations and cuts

This paper presents a new approach to optimal design centering, the optimal assignment of parameter tolerances and the determination and optimization of production yield. Based upon multidimensional linear cuts of the tolerance orthotope and uniform distributions of outcomes between tolerance extremes in the orthotope, exact formulas for yield and yield sensitivities, with respect to design parameters, are derived. The formulas employ the intersections of the cuts with the orthotope edges, the cuts themselves being functions of the original design constraints. Our computational approach involves the approximation of all the constraints by low-order multidimensional polynomials. These approximations are continually updated during optimization. Inherent advantages of the approximations which we have exploited are that explicit sensitivities of the design performance are not required, available simulation programs can be used, inexpensive function and gradient evaluations can be made, inexpensive calculations at vertices of the tolerance orthotope are facilitated during optimization and, subsequently, inexpensive Monte Carlo verification is possible. Simple circuit examples illustrate worst case design and design with yields of less then 100 percent. The examples also provide verification of the formulas and algorithms.