Closed-form solutions for multi-objective tolerance optimization

Component tolerances have important influence on the cost and performance of products. In order to obtain suitable component tolerances, multi-objective tolerance optimization model is studied, in which the combined polynomial and exponential functions are used to model manufacturing cost. In this paper, analytical methods are proposed to solve the multi-objective optimization model. In this model, the objective function is not a monotone function, and it is possible that the assembly tolerance constraint, including worst-case method and root sum square method, is inactive. Therefore, two closed-form solutions are proposed for each component tolerance in terms of the Lambert W function. When the assembly tolerance constraint is not considered, the component tolerances are obtained and named as the initial closed-form solutions. If the initial solutions satisfy assembly tolerance constraint, it is the final value of optimal tolerances. Otherwise, constrained optimization model is established and Lagrange multiplier method is applied to obtain the new closed-form solution of component tolerances as the final value of optimal tolerances. Several simulation examples are used to demonstrate the proposed method.