Integrating the LambertW Function to a Tolerance Optimization Problem

Abstract

This paper explores the integration of the Lambert W function to a tolerance optimization problem with the assessment of costs incurred by both the customer and a manufacturer. By trading off manufacturing and rejection costs, and a quality loss, this paper shows how the Lambert W function, widely used in physics, can be efficiently applied to the tolerance optimization problem, which may be the first attempt in the literature related to tolerance optimization and synthesis. Using the concept of the Lambert W function, a closed-form solution is derived, which may serve as a means for quality practitioners to make a quick decision on their optimal tolerance without resorting to rigorous optimization procedures using numerical methods. A numerical example is illustrated and a sensitivity analysis is performed.

Design verification and validation in product lifecycle

“…The economic factor is often expressed in a quality loss function [111] and in most applications the Taguchi loss function is used. Govindaluri et al [97] consider the quality loss from the perspective of the customer and the manufacturing and rejection costs by the manufacturer.…”

Achieving cost robustness in processes with mixed multiple quality characteristics and dynamic variability

“…The form of P(Y, τ) can vary widely depending on the manufacturing application. Chase and Parkinson (1991), Shin et al (2005), as well as Chen and Huang (2011), used linear functions to represent the cost of achieving tighter tolerances in various production systems. Many researchers, such as Michael and Siddall (1981), Fang and Wu (2000), Shin and Cho (2007), and Peng et al (2008), have employed a form of the exponential cost function to serve this same purpose.…”

Economic tolerance design for folded normal data

“…Cao et al (2004) presented a robust tolerance design that considered the process capability constraint. Shin et al (2005) integrated the Lambert W function into the tolerance optimisation problem; a closed-form solution was derived efficiently by a computer program. Moskowitz et al (2001) and Plante (2002) conducted parametric and non-parametric studies in multivariate design cases.…”