Optimal statistical tolerance allocation for reciprocal exponential cost–tolerance function

Abstract

Statistical tolerancing has been widely employed in industry as it is more practical compared with the worst-case tolerancing in achieving lower manufacturing cost while satisfying design specification. As reciprocal exponential function is one of the commonly employed cost–tolerance models in practice and current approach is difficult to allocate statistical tolerances for such a function, this article investigates a method for optimal statistical tolerance allocation with such a cost–tolerance function. The method is to minimize manufacturing cost subject to constraints on tolerance target and machining capabilities. The optimization problem is solved by applying the algorithmic approach. Particularly, a closed-form expression of the tolerance optimization problem is further derived based on the Lagrange multiplier method with integrating the Lambert W function, a multivalue complex function. In addition, for constrained minimization problems with only equality constraints, the optimal tolerance allocation can be obtained by solving simultaneous equations without the time-consuming computing on differentiating while keeping the solution accurate. An example is illustrated to demonstrate the application of this approach. Through comparisons with the regular Lagrange multiplier method applied to reciprocal exponential cost–tolerance type, the result reveals that tolerances can be allocated much faster using this proposed method.