The Adaptive Branch and Bound Method of Tolerance Synthesis Based on the Reliability Index

Tolerance synthesis is a stochastic optimisation problem. It can be converted into deterministic optimisation by replacing stochastic design function constraints with deterministic reliability index constraints. This problem was first approached and solved with both a heuristic algorithm TOL-M and a feasible directional method. In this paper, we propose an adaptive branch and bound method, which is a coarse to fine positioning technique, to resolve it. The proposed heuristic algorithm significantly outperforms the previous algorithms for the optimum solution. Although the adaptive branch and bound algorithm is heuristic and does not guarantee a global optimum, we give a theoretical reason why it provides a superior local minimum to both the TOL-M and the feasible directional method. Issues of the relationship between yield and reliability index are addressed. Various related results in the literature are compared with ours to show that some reliability indices are less important than others and can be violated without suffering too much loss of yield.