From tolerance allocation to tolerance-cost optimization: a comprehensive literature review

Abstract

It is widely acknowledged that the allocation of part tolerances is a highly responsible task due to the complex repercussions on both product quality and cost. As a consequence, since its beginnings in the 1960s, least-cost tolerance allocation using optimization techniques, i.e. tolerance-cost optimization, was continuously in focus of numerous research activities. Nowadays, increasing cost and quality pressure, availability of real manufacturing data driven by Industry 4.0 technologies, and rising computational power result in a continuously growing interest in tolerance-cost optimization in both research and industry. However, inconsistent terminology and the lack of a classification of the various relevant aspects is an obstacle for the application of tolerance-cost optimization approaches. There is no literature comprehensively and clearly summarizing the current state of the art and illustrating the relevant key aspects. Motivated to overcome this drawback, this article provides a comprehensive as well as detailed overview of the broad research field in tolerance-cost optimization for both beginners and experts. To facilitate the first steps for readers who are less familiar with the topic, the paper initially outlines the fundamentals of tolerance-cost optimization including its basic idea, elementary terminology and mathematical formulation. These fundamentals serve as a basis for a subsequent detailed discussion of the key elements with focus on the different characteristics concerning the optimization problem, tolerance-cost model, technical system model and the tolerance analysis model. These aspects are gathered and summarized in a structured mind map, which equips the reader with a comprehensive graphical overview of all the various facets and aspects of tolerance-cost optimization. Beside this, the paper gives a retrospect of the past fifty years of research in tolerance cost-optimization, considering 290 relevant publications. Based thereon, current issues and future research needs in tolerance-cost optimization were identified.

Integrated Tolerance and Fixture Layout Design for Compliant Sheet Metal Assemblies

“…While tolerance types and references are chosen for the functional relevant geometry elements in tolerance specification, tolerance allocation is used to assign suitable tolerance values to them [7]. In the early years of tolerancing, experience, handbooks or best practices [8], graphical and simple analytical approaches, such as equal or proportional scaling [9,10], as well as the repetitive application of tolerance and sensitivity analysis [11], could meet the former requirements [12,13].…”

“…However, the steadily increasing cost and quality pressure required the development of more sophisticated approaches to automatically find the combination of tolerance values that lead to minimum costs, but can also assure high product quality [12,13]. Tolerance-cost optimization-also known under the terms optimal/least-cost tolerance allocation/design/synthesis-accelerates the usually manual, trial-and-error based search by turning the task into a mathematical optimization problem in order to find the optimal compromise between cost and quality [12]. In the beginning, its applicability was mainly limited by the low computing powers and the lack of powerful optimization algorithms [12].…”

“…Tolerance-cost optimization-also known under the terms optimal/least-cost tolerance allocation/design/synthesis-accelerates the usually manual, trial-and-error based search by turning the task into a mathematical optimization problem in order to find the optimal compromise between cost and quality [12]. In the beginning, its applicability was mainly limited by the low computing powers and the lack of powerful optimization algorithms [12]. The development of soft-computing, nature-inspired meta-heuristic algorithms, in combination with the significantly increased computer technology, has finally paved the way for its practical application to complex engineering systems [12,13].…”

Concurrent optimization of dimensions and tolerances on structures and mechanisms

“…Introductions to tolerance allocation and classic application examples are given in [29,30]. A recent review on allocation methods [31] cites about 250 studies, classifying them in a detailed taxonomy. This includes the definition of the optimization problem (objective functions, constraints), the costtolerance models, and the assumptions on assembly requirements, tolerances (dimensional or geometric), and stackup criteria (worst-case, statistical).…”

“…Two common choices for the quality measure are the quality loss and the nonconforming rate. In the first approach (design for quality), the objective function is a sum of the manufacturing cost and the quality loss; this defines an unconstrained optimization problem, which has been solved with several search algorithms (see again [31][32][33] for reviews) and applied to some special formulations of the tolerancing problem [44][45][46][47][48][49][50][51][52]. In the second approach (design for reliability), several formulations have been proposed: minimize cost with a constraint on the nonconforming rate [53][54][55][56][57], maximize a process capability index [58], or optimize a function of cost and various indices related to the nonconforming rate [59][60][61].…”