Tolerancing for an Apple Pie: A Fundamental Theory of Tolerances

Abstract

Tolerancing began with the notion of limits imposed on the dimensions of realized parts both to maintain functional geometric dimensionality and to enable cost-effective part fabrication and inspection. Increasingly, however, component fabrication depends on more than part geometry as many parts are fabricated as a result of a “recipe” rather than dimensional instructions for material addition or removal. Referred to as process tolerancing, this is the case, for example, with IC chips. In the case of tolerance optimization, a typical objective is cost minimization while achieving required functionality or “quality.” This article takes a different look at tolerances, suggesting that rather than ensuring merely that parts achieve a desired functionality at minimum cost, a typical underlying goal of the product design is to make money, more is better, and tolerances comprise additional design variables amenable to optimization in a decision theoretic framework. We further recognize that tolerances introduce additional product attributes that relate to product characteristics such as consistency, quality, reliability, and durability. These important attributes complicate the computation of the expected utility of candidate designs, requiring additional computational steps for their determination. The resulting theory of tolerancing illuminates the assumptions and limitations inherent to Taguchi’s loss function. We illustrate the theory using the example of tolerancing for an apple pie, which conveniently demands consideration of tolerances on both quantities and processes, and the interaction among these tolerances.