Optimum Selection of Discrete Tolerances

Tolerancing involves considerations from all phases of the life cycle of a product including design, manufacturing, assembly, and inspection. Along with minimum cost and maximum functionality and interchangeability, the practice of tolerancing urges a designer to choose an appropriate manufacturing (or inspection) process as well. This situation is formalized as a discrete optimization problem. For an optimum selection of tolerances from a given discrete model involving various manufacturing processes, minimization of manufacturing cost is achieved under the constraint of tolerance stack-up. A random variable and its standard deviation are associated with a dimension and its tolerance. This probabilistic approach enables a trade-off between performance and tolerance (cost). But it also suggests probabilistic optimization. With the aid of a notion called the reliability index [8], tolerance selection is formulated as an integer programming problem. A branch and bound algorithm for ensuring optimum selection is developed by exploiting the special structure of the constraints. To make the enumeration tree small, monotonic relations among the reliability index, cost, and tolerance are examined. The algorithm is tested with examples.