A comprehensive review of tolerancing research

Ever since the plus/minus limits on dimensions first started to appear on engineering drawings in the early 1900s, tolerances have been one of the most important issues for every engineer involved in the product realization processes. In particular, with the advancement of computers and CAD/CAM techniques in the 1970s, the tolerance-related issues have continuously drawn the attention of many researchers since then. As a result, a tremendous number of research articles have been published over the last 30 years. This paper aims at a comprehensive state-of-the-art review on various tolerancing issues in design and manufacturing. However, due to the overwhelming number of existing research publications, any reviews on tolerancing issues could by no means be exhaustive. Rather, this review attempts to provide the reader with a view toward a balanced understanding of the various problems in tolerancing by presenting some typical research work for each of the classified fields, and tries to draw the potential research directions in the future.     

Tolerancing of board-level-free-space optical interconnects

For optical interconnects to become a mature technology they must be amenable to electronic packaging technology. Two main obstacles to including free-space optical interconnects are alignment and heat-dissipation issues. Here we study the issues of alignment tolerancing that are due to assembly and manufacturing variations (passive-element tolerancing) over long board-level distances (>10 cm) for free-space optical interconnects. We also combine these variations with active optoelectronic device variations (active-element tolerancing). We demonstrate a computer-aided analysis procedure that permits one to determine both active- and passive-element tolerances needed to achieve some system-level specification, such as yield or cost. The procedure that we employ relies on developing a detailed design of the system to be studied in a standard optical design program, such as code v. Using information from this model, we can determine the integrated power falling on the detector, which we term optical throughput, by performing Gaussian propagation or general Fresnel propagation (if significant vignetting occurs). This optical throughput can be used to determine system-level performance criteria, such as bit-error rate. With this computer-aided analysis technique, a sensitivity analysis of all the variations under study is made on a system with realistic board-level interconnect distances to find each perturbation's relative effects (with other perturbations set to 0) on the power falling on the detector. This information is used to set initial tolerances for subsequent tolerancing analysis and design runs. A tolerancing analysis by Monte Carlo techniques is applied to determine if the yield or cost (yield is denned as the percentage of systems that have acceptable system performance) is acceptable. With a technique called parametric sampling, a subsequent tolerancing design run can be applied to optimize this yield or cost with little increase in computation. We study a design example and show that most of the tolerances can be achieved with current technology.     

Tolerance analysis and synthesis for cam mechanisms

Tolerance is one of the most important parameters in product and process design, so tolerancing plays a key role in design and manufacturing. Tolerance synthesis is in a period of extensive study due both to increased demands for quality products and to increasing automation of machining and assembly. Optimum tolerance design and synthesis ensures good quality product at low cost. This paper presents an analytical methodology for tolerance analysis and synthesis for a disk cam-translating follower system. Both dimensional ( size) and geometric tolerances ( position and profile ) on the components are considered. Tolerance analysis is performed on individual tolerances as well as on total tolerance accumulation. With the lowest manufacturing cost as its objective function a nonlinear optimization model is formulated for tolerance synthesis and solved by a sequential quadratic programming ( SQP) algorithm. An example is provided to illustrate the optimization model and solution procedure.     

Simultaneous tolerancing for design and manufacturing

This paper presents a new tolerance design theory—simultaneous tolerancing— which works in the concurrent engineering context. After stating the need to develop a simultaneous tolerancing theory by showing the shortcomings of conventional tolerancing technique, the concept of simultaneous tolerancing is given, and its elements are briefly presented. Then we focus our attention on the development of a general mathematical model of optimal tolerancing supporting concurrent engineering. Two commonly used models, worst-case and statistical, are discussed in detail. Next, a method of ‘interim tolerances’, which help to determine an appropriate machining process without using functional tolerances, is proposed. The simultaneous tolerancing theory presented in this paper permits of determining directly optimal machining tolerances in product design, reducing the manufacturing cost and improving the quality of products. Finally, an example is given, showing that the proposed theory is feasible in practice.     

Process-oriented tolerancing for multi-station assembly systems

In multi-station manufacturing systems, the quality of final products is significantly affected by both product design as well as process variables. Historically, however, tolerance research has primarily focused on allocating tolerances based on the product design characteristics of each component. Currently, there are no analytical approaches to optimally allocate tolerances to integrate product and process variables in multi-station manufacturing processes at minimum costs. The concept of process-oriented tolerancing expands the current tolerancing practices, which bound errors related to product variables, to explicitly include process variables. The resulting methodology extends the concept of “part interchangeability” into “process interchangeability,” which is critical due to increasing requirements related to the selection of suppliers and benchmarking. The proposed methodology is based on the development and integration of three models: (i) the tolerance-variation relation; (ii) variation propagation; and (iii) process degradation. The tolerance-variation model is based on a pin-hole fixture mechanism in multi-station assembly processes. The variation propagation model utilizes a state space representation but uses a station index instead of a time index. Dynamic process effects such as tool wear are also incorporated into the framework of process-oriented tolerancing, which provides the capability to design tolerances for the whole life-cycle of a production system. The tolerances of process variables are optimally allocated through solving a nonlinear constrained optimization problem. An industry case study is used to illustrate the proposed approach.     

Concurrent optimization of assembly tolerances for quality with position control using scatter search approach

Tolerance allocation to individual parts in any assembly should be a vital design function with which both the design and manufacturing engineers are concerned. Generally design engineers prefer to have tighter tolerances to ensure the quality of their design, whereas manufacturing engineers prefer loose tolerances for ease of production and the need to be economical. This paper introduces a concurrent tolerance approach, which determines optimal product tolerances and minimizes combined manufacturing and quality related costs in the early stages of design. A non-linear multivariable optimization model is formulated here for assembly. A combinatorial optimization problem by treating cost minimization as the objective function and stack-up conditions as the constraints are solved using scatter search algorithm. In order to further explore the influence of geometric tolerances in quality as well as in the manufacturing cost, position control is included in the model. The results show how position control enhances quality and reduces cost.     

A genetic algorithm-based solution to optimal tolerance synthesis of mechanical assemblies with alternative manufacturing processes: focus on complex tolerancing problems

The product development process involves tolerance specification on the individual component dimensions. The impact of tolerance specification on manufacturing cost has drawn the attention of product designers towards economic tolerance synthesis using various optimization techniques. Simultaneous selection of manufacturing processes or machines from amongst the alternatives for producing a toleranced feature have also been considered. The solution surface for such a problem becomes a combinatorial and multi-modal function involving several local minima. Application of a genetic algorithm to the solution of this advanced tolerancing problem, together with benchmarking with the exact global solution obtained using Lagrange's multiplier-based exhaustive search method, has been reported in an earlier paper by the authors. The proposed algorithm is quite simple and straightforward, and automatically takes care of the process selection constraints. Application of the genetic algorithm has been demonstrated on complex tolerancing problems such as those involving interrelated dimension chains, complex stack-up conditions and complex cost functions, etc., where the use of traditional optimization techniques is not recommended.     

Quality design of tolerance allocation for sheet metal assembly with resistance spot weld

Tolerance directly influences the functionality of the products and the related manufacturing costs, and tolerance allocation is of great importance for improving the assembly quality. However, the information required to allocate tolerances for complex 3D assemblies is generally not available at the initial design stage. In this paper, a new quality design methodology is developed, which makes use of both original design data obtained by the response surface methodology and the extra interpolation data obtained by the Kriging method. The finite element modelling is presented for the sheet metal assembly process as no explicit relationship of the variations for key characteristic points are available. The robust tolerances can be allocated based on the quality design model. A case study with the typical assembly process of the rear compartment pan and the wheelhouse is carried out in the paper, the tolerance allocation results show that the developed quality design methodology is capable of determining the robust manufacturing tolerance before assembly, which satisfies the product requirements. This method enables a robust tolerancing scheme to be used in the sheet metal assembly process.     

Generalization and evaluation of vectorial tolerances

In recent years, vectorial tolerancing has emerged as a new alternative for representing workpiece tolerances. In contrast to conventional geometric tolerances which originated from hard gauging practices, vectorial tolerancing follows the working principle of coordinate measuring machines and CAD/ CAM systems. Moreover it provides feedback from measurement directly to manufacturing process control. Many believe it is a better tolerancing method to tie design, manufacuturing, and measurement together. However, the current proposal of vectorial tolerancing has some limitations. First, the currently adopted orientation vector is not sufficient for representing true 3D orientations. As a result, the orientation of a free form surface cannot be properly established. Second, there is lacking a unified and consistent method for the evaluation of vectorial tolerances. This paper proposes a new orientation vector which provides a more general mathematical basis for representing vectorial tolerances. It enables true 3D orientation representation and relates tolerances to the functional requirement. With the improved mathematical definition, a systematic tolerance evaluation approach becomes possible for both analytical geometric elements and free-form surfaces. Computer simulations and real-world applications are studied to validate this new approach.     

Concurrent optimal allocation of design and process tolerances for mechanical assemblies with interrelated dimension chains

Concurrent tolerance allocation has been the focus of extensive research, yet very few researchers have considered how to concurrently allocate design and process tolerances for mechanical assemblies with interrelated dimension chains. To address this question, this paper presents a new tolerance allocation method that applies the concept of concurrent engineering. The proposed method allocates the required functional assembly tolerances to the design and process tolerances by formulating the tolerance allocation problem into a comprehensive model and solving the model using a non-linear programming software package. A multivariate quality loss function of interrelated critical dimensions is first derived, each component design tolerance is formulated as the function of its related process tolerances according to the given process planning, both manufacturing cost and quality loss are further expressed as functions of process tolerances. And then, the objective function of the model, which is to minimize the sum of manufacturing cost and expected quality loss, is established and the constraints are formulated based on the assembly requirements and process constraints. The purpose of the model is to balance manufacturing cost and quality loss so that concurrent optimal allocation of design and process tolerances is realized and quality improvement and product cost reduction is achieved. The proposed method is tested on a practical example.     

Tolerancing algebra: A building block for handling tolerance interactions in design and manufacturing Part 1: Concept and representation

As a fundamental building block for 3D tolerance transfer analysis, tolerancing algebra on a deviation space has been proposed and elaborated. Based on the investigation of the spatial characteristic and the propagation mechanism of geometric tolerances and manufacturing process dispersions, a set of primitives and operations has been defined, which forms the algebraic structure of tolerancing algebra. Some important algebraic properties have been derived that will be extensively used to establish a tolerance transfer technique for process planning. Companion papers present the idea of tolerancing algebra in two parts. Part 1 presents some important concepts and their representations. Part 2 endeavours to show the interactions between product tolerances and process dispersions by way of the basic algebraic operations on deviation volumes.     

Statistical tolerance synthesis using distribution function zones

Tolerance is one of the most important parameters in design and manufacturing. Tolerance synthesis has a significant impact on manufacturing cost and product quality. In the international standards community two approaches for statistical tolerancing of mechanical parts are being discussed: process capability indices and distribution function zone. The distribution function zone (DFZone) approach defines the acceptability of a population of parts by requiring that the distribution function of relevant values of the parts be bounded by a pair of specified distribution functions. In order to apply this approach to statistical tolerancing, one needs a method to decompose the assembly level tolerance specification to obtain tolerance parameters for each component in conjunction with a corresponding tolerance-cost model. This paper introduces an optimization-based statistical tolerance synthesis model based on the DFZone tolerance specifications. A new tolerance-cost model is proposed and the model is illustrated with an assembly example.     

Tolerance optimization of a mobile phone camera lens system

In the manufacturing process for the lens system of a mobile phone camera, various types of assembly and manufacturing tolerances, such as tilt and decenter, should be appropriately allocated. Because these tolerances affect manufacturing cost and the expected optical performance, it is necessary to choose a systematic design methodology for determining optimal tolerances. In order to determine the tolerances that minimize production cost while satisfying the reliability constraints on important optical performance indices, we propose a tolerance design procedure for a lens system. A tolerance analysis is carried out using Latin hypercube sampling for evaluating the expected optical performance. The tolerance optimization is carried out using a function-based sequential approximate optimization technique that can reduce the computational burden and smooth numerical noise occurring in the optimization process. Using the proposed design approach, the optimal production cost was decreased by 28.3% compared to the initial cost while satisfying all the constraints on the expected optical performance. We believe that the tolerance analysis and design procedure presented in this study can be applied to the tolerance optimization of other systems.     

Scheduling manufacturing systems for delayed product differentiation in agile manufacturing

Production of customized products to respond to changing markets in a short time and at a low cost for agile manufacturing can be implemented with delayed product differentiation in a manufacturing system. The successful implementation of delayed product differentiation lies in efficient scheduling of the manufacturing system. Scheduling problems in implementing delayed product differentiation in a general flexible manufacturing system are defined, formulated and solved here. The manufacturing system consists of two stages: machining and assembly. At the machining stage, a single machine is used to produce standard component parts for assembly products. These parts are then assembled at the assembly stage by multiple identical assembly stations to form customized products. The products to be produced in the system are characterized by their assembly sequences represented by digraphs. The scheduling problem is to determine the sequence of products to be produced in the system so that the maximum completion time (makespan) is minimized for any given number of assembly stations at the assembly stage. Based on the representation of assembly sequence of the products, three production modes are defined: production of a single product with a simple assembly sequence ; production of a single product with a complex assembly sequence ; and production of N products . According to the three defined production modes, the associated scheduling problems are defined as G s scheduling problems, G c scheduling problems and N-product scheduling problems, respectively. Optimal and heuristic methods for solving the scheduling problems are developed. The computational experiment shows that the heuristics provide good solutions to the scheduling problems.     

Statistical Analysis of Geometrical Tolerances: A Case Study

Tolerancing is one of the most important but complex activities in design. Tolerance information takes place at every phase of design activity. It represents the fundamental link between the theoretical model of the mechanical product and the actual one. During the two previous decades, engineering projects and scientific researches demonstrated that ongoing miniaturization increased the influence of geometric tolerances. They also admit that mass production is mainly based on statistical techniques. On the other hand, the decomposition of the global tolerancing process into functional-level, assembly-level, part-level, and manufacturing-level, reduces dramatically the domain of the solution. In this paper, the global tolerancing process is described and a novel method for statistical analysis of geometrical tolerances is discussed. Then the statistical approach is introduced and its performance is evaluated on a best case study. The analysis of the technical drawing of a part is given in order to highlight the advantages of the statistical approach.     

Computer-aided production management issues in the engineer-to-order production of complex capital goods explored using a simulation approach

This paper describes the exploration of manufacturing planning and control issues in the capital goods industry using a simulation approach. The companies produce products which have deep and complex product structures and are produced in low volume on an engineer- or make-to-order basis (ETO, MTO). The work reported here draws on the results of surveys of companies involved in the manufacture of capital goods which identified their characteristics of ETO and MTO capital goods companies and their strategic issues. The planning and control approaches adopted in the manufacturing facilities and the difficulties experienced in the application of computer-aided production management (CAPM) systems were also examined. The simulation model developed enables complex manufacturing systems to be modelled and was configured to represent a typical ETO/MTO facility using industrial data. A series of full factorial experiments were performed to explore a number of production management problems identified in surveys including capacity planning, assembly planning and scheduling strategies. Conclusions are drawn on the effects on performance and capacity of: applying minimum set-up and processing times for both major and minor activities; using different data update periods and assembly lead times; and adopting various scheduling and despatching approaches. These results are compared with those obtained by other workers who used survey techniques alone, and have implications for the manufacturers of capital goods.     

Simultaneous optimal selection of design and manufacturing tolerances with different stack-up conditions using genetic algorithms

Tolerance design is one of the most critical aspects of product design and development process as it affects both the product's functional requirements and manufacturing cost. Unnecessarily tight tolerances lead to increased manufacturing cost, while loose tolerances may lead to malfunctioning of the product. Traditionally, this important phase of product development is accomplished intuitively to satisfy design constraints, based on handbooks' data and/or skill and experience of the designers. Tolerance design carried out in this manner does not necessarily lead to an optimum design. Research in this area indicates that, in general, tolerance design is carried out sequentially in two steps; (1) tolerance design in CAD to obtain design or functional tolerances and (2) tolerance design in CAPP to obtain manufacturing tolerances. Such a sequential approach to tolerance design suffers from several drawbacks, such as more time consumption, suboptimality and unhealthy working atmosphere. This paper reports on an integrated approach for simultaneous selection of design and manufacturing tolerances based on the minimization of the total manufacturing cost. The nonlinear multivariable optimization problem formulated in this manner may result in a noisy solution surface, which can effectively be solved with the help of a global optimization technique. A solution methodology using genetic algorithms and applying penalty function approach with proper normalization of the penalty terms for handling the constraints is proposed. The application of the proposed methodology is demonstrated on a simple mechanical assembly with different tolerance stack-up conditions.     

Tolerancing algebra: A building block for handling tolerance interactions in design and manufacturing

Part 1 of two companion papers was concerned with the elementary concepts for tolerancing algebra, such as deviation space, deviation volume and tolerance primitives. Part 2 deals with the interactions between those primitives. The tolerance transfer problem is discussed first to figure out the relationship between the the process planning decisions and the tolerances. Then the manufacturing process dispersions are modelled in a deviation space so that they can be handled with the tolerance primitives on a common basis. As a means of describing the interactions between the deviation primitives--either tolerances or process dispersions, or both-four basic tolerancing operations are defined: composition, decomposition, transfer, and aggregation. An example is presented to illustrate how the proposed algebra can be used for various types of tolerance interactions.     

Optimum tolerance allocation in mechanical assemblies using an interval method

Tolerance allocation methods serve as effective tools for design engineers to reduce the overall manufacturing cost of products. In every mechanical design, it is the design engineer’s task to assign tolerances to all dimensions and clearances to all joints in an assembly. This paper presents an optimum allocation method, based on interval analysis, for finding the optimum values of tolerances and clearances in mechanical assemblies that will not only minimize a stated objective function, but also satisfy the required functional and design constraints. The design constraints include dimensional requirements that the related parts must match relative to each other with a specified precision. Given a set of trial values of component tolerances and joint clearances, the present method utilizes the sequential quadratic programming method, Broyden–Fletcher–Goldfarb–Shanno quasi-Newton method and line search approaches to find the optimum values of tolerances and clearances. The effects of different cost function models on the manufacturing cost are also compared and discussed. Numerical examples are presented to illustrate the application of the method.     

Modelling and analysis of selective assembly using Taguchi's loss function

An assembly is the integrative process of joining components to make a completed product. It brings together the upstream process of design, engineering and manufacturing processes. The functional performance of an assembled product and its manufacturing cost are directly affected by the individual component tolerances. But, the selective assembly method can achieve tight assembly tolerance through the components manufactured with wider tolerances. The components are segregated by the selective groups (bins) and mated according to a purposeful strategy rather than being at random, so that small clearances are obtained at the assembly level at lower manufacturing cost. In this paper, the effect of mean shift in the manufacturing of the mating components and the selection of number of groups for selective assembly are analysed. A new model is proposed based on their effect to obtain the minimum assembly clearance within the specification range. However, according to Taguchi's concept, manufacturing a product within the specification may not be sufficient. Rather, it must be manufactured to the target dimension. The concept of Taguchi's loss function is applied into the selective assembly method to evaluate the deviation from the mean. Subsequently, a genetic algorithm is used to obtain the best combination of selective groups with minimum clearance and least loss value within the clearance specification. The effect of the ratio between the mating part quality characteristic's dimensional distributions is also analysed in this paper.