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.     

Simultaneous tolerance synthesis for manufacturing and quality

Tolerance allocation affects product design, manufacturing, and quality. No existing technique has been found by the authors that takes product design, manufacturing, and quality into account simultaneously. This paper introduces a new concurrent engineering method for tolerance allocation. A nonlinear optimization model was constructed to implement the method. The model minimizes the combination of quality loss and manufacturing cost simultaneously in a single objective function by setting both process tolerances and design tolerances simultaneously. The purpose of the model is to balance manufacturing cost and quality loss to achieve near-optimal design and process tolerances simultaneously for minimum combined manufacturing cost and quality loss over the life of the product. Compared to other models, this model shows significant improvements. Electronic Publication     

Assembly tolerance allocation using a coalitional game method

Assembly tolerance allocation in modern manufacturing industries is important because it directly affects product quality and manufacturing cost. Loose tolerances may cause quality deficiency while tight tolerances can increase the cost. It is significant to develop a reasonable tolerance allocation strategy for every assembly component combining the cost and quality demands. Traditionally, designers often adopt the single objective optimization with some kind of constraint or establish a comprehensive evaluation function combining several optimization objectives with different weights to solve the tolerance allocation problem. These approaches may not be desirable as it is difficult to adequately consider the interaction and conflict between the cost and quality demands. In this article, an assembly tolerance allocation method using coalitional game theory is proposed in an attempt to find a trade-off between the assembly cost and the assembly quality. First, the assembly tolerance allocation problem is formulated as a multi-objective optimization problem and the concept of the Pareto-optimal solution is introduced. Then, how the assembly tolerance allocation model is transformed into a coalitional game model is discussed, and a key technique of transforming the tolerance design variables into the game strategies is presented. Further, the Shapley value method of coalitional game based on each player's contribution evaluation to the profit of the whole coalition is given. Finally, the feasibility of the procedure is demonstrated through an example of vehicle front structure assembly.     

Concurrent optimal allocation of geometric and process tolerances based on the present worth of quality loss using evolutionary optimisation techniques

Concurrent tolerancing becomes an optimisation problem to find out the optimum allocation of the process tolerances in the given design function constraints. In traditional optimisation methods, finding out the optimum solution for this advanced tolerance design problem is complex. The proposed algorithms (elitist non-dominated sorting genetic algorithm) and differential evolution extensively do better than the previous algorithms for attaining the optimum result. The aim of this paper is to suggest a model for optimal tolerance allocation by considering both tolerance cost and the present worth of quality loss such that the total manufacturing cost/loss is minimised. The suggested model takes into account the time value of money for quality loss and product degradation over time and consists of two new parameters: the planning horizon and the product user’s discount rate. From the outcome of this study, a longer planning horizon results in an increase in both tolerance cost and quality loss; however, a larger value of discount rate gives up a decrease in both tolerance cost and quality loss. Finally, a practical example is brought into reveal the effectiveness of the suggested method.     

Robust design of assembly and machining tolerance allocations

Tolerancing is one of the most important tasks in product and manufacturing process design. The allocation of design tolerances between the components of a mechanical assembly and manufacturing tolerances in the intermediate machining steps of component fabrication can significantly affect a product's quality and its robustness. This paper presents a methodology to maximize a product's robustness by appropriately allocating assembly and machining tolerances. The robust tolerance design problem is formulated as a mixed nonlinear optimization model. A simulated annealing algorithm is employed to solve the model and an example is presented to illustrate the methodology.     

Robust design of assembly and machining tolerance allocations

Tolerancing is one of the most important tasks in product and manufacturing process design. The allocation of design tolerances between the components of a mechanical assembly and manufacturing tolerances in the intermediate machining steps of component fabrication can significantly affect a product's quality and its robustness. This paper presents a methodology to maximize a product's robustness by appropriately allocating assembly and machining tolerances. The robust tolerance design problem is formulated as a mixed nonlinear optimization model. A simulated annealing algorithm is employed to solve the model and an example is presented to illustrate the methodology     

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.     

A complete tolerance charting system in assembly

The process of assigning tolerances in product design, more often than not, will affect the functionality and cost of the product. An unnecessarily tight tolerance leads to higher cost while an excessively loose tolerance may lead to malfunctioning of the product. Traditionally, this important phase of tolerance allocation is done intuitively to satisfy design constraints based on the skills and experience of the designers. In most cases, the result is not optimum and modification of the allocated tolerances is required. After the tolerances are allocated, the next step is to ensure that the assembly can be manufactured with the resources available. This can usually be verified by performing tolerance charting for the assembly. However, if the result shows that the blueprint (B/P) tolerances specified are too tight, the entire process of tolerance allocation and tolerance charting has to be repeated. This paper presents an efficient method of integrating the process of allocating optimum B/P tolerances in product design with the process of tolerance charting in process planning. This method ensures that the assembly is manufactured with optimum working dimensions and tolerances and yet satisfies all the assembly requirements. Two models are constructed. One based on process sequence and another based on the relationship between each component. Using the models coupled with unique algorithms, sets of linear equations are formulated based on the design constraints and assembly requirements. With another set of equations derived in terms of the process capabilities, the optimum B/P tolerances, working dimensions and tolerances can be determined by solving the equations.     

Economic tolerance design for quality

This paper provides a few general mathematical models for determining product tolerances which minimize the combined manufacturing costs and quality loss. The models contain quality cost with a quadratic loss function and represent manufacturing costs with geometrical decay functions. The models are also formulated with multiple variables which represent the set of characteristics in a part. Applications of these models include minimizing the total cost with effective tolerance allocation in product design.     

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.     

Optimization of machining datum selection and machining tolerance allocation with genetic algorithms

In machining process planning, selection of machining datum and allocation of machining tolerances are crucial as they directly affect the part quality and machining efficiency. This study explores the feasibility to build a mathematical model for computer aided process planning (CAPP) to find the optimal machining datum set and machining tolerances simultaneously for rotational parts. Tolerance chart and an efficient dimension chain tracing method are utilized to establish the relationship between machining datums and tolerances. A mixed-discrete nonlinear optimization model is formulated with the manufacturing cost as the objective function and blueprint tolerances and machine tool capabilities as constraints. A directed random search method, genetic algorithm (GA), is used to find optimum solutions. The computational results indicate that the proposed methodology is capable and robust in finding the optimal machining datum set and tolerances. The proposed model and solution procedure can be used as a building block for computer automated process planning.     

ROBUST TOLERANCE ALLOCATION USING STOCHASTIC PROGRAMMING

Abstract

Tolerance allocation in manufacturing is a prominent industrial task for enhancing productivity and reducing manufacturing costs. The classical tolerance allocation problem can be formulated as a stochastic program to determine the assignment of component tolerances such that the manufacturing cost is minimized. However, tolerance design is a prerequisite to the overall quality and cost of a product; robust tolerance design is particularly important and should be considered. In this paper, robustness is considered in formulating the tolerance allocation problem by minimizing the manufacturing cost's sensitivity. Moreover, from a practical perspective, the process capability index for each component and the upper bound of the manufacturing cost are also considered. To effectively and efficiently resolve the robust tolerance allocation problem, a sequential quadratic programming algorithm embedded with a Monte Carlo simulation is developed. To demonstrate this design method's robustness, two commonly used test problems are solved. The designs devised in this paper have lower manufacturing costs and smaller variations in manufacturing costs than those in previous studies, indicating that the proposed method is highly promising in the robust tolerance design.     

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.     

Tolerance allocation for compliant beam structure assemblies

This paper presents a tolerance allocation methodology for compliant beam structures in automotive and aerospace assembly processes. The compliant beam structure model of the product does not require detailed knowledge of product geometry and thus can be applied during the early design phase to develop cost-effective product specifications. The proposed method minimizes manufacturing costs associated with tolerances of product functional requirements (key product characteristics, KPCs) under the constraint(s) of satisfying process requirements (key control characteristics, KCCs). Misalignment and fabrication error of compliant parts, two critical causes of product dimensional variation, are discussed and considered in the model. The proposed methodology is developed for stochastic and deterministic interpretations of optimally allocated manufacturing tolerances. An optimization procedure for the proposed tolerance allocation method is developed using projection theory to considerably simplify the solution. The non-linear constraints, that ellipsoid defined by τ(stochastic case) or rectangle defined by T x (deterministic case) lie within the KCC region, are transformed into a set of constraints that are linear in σ(or T x )-coordinates. Experimental results verify the proposed tolerance allocation method.     

Tolerance chart optimization for quality and cost

This paper introduces a mathematical model for tolerance chart balancing during machining process planning. The criteria considered in this study are based on the combined effects of manufacturing cost and quality loss, under the constraints of process capability limits, design functionality restrictions, and product quality requirements. Manufacturing cost is expressed in geometrical decreasing functions, which represent tolerances to be assigned. Process variability is expressed in quadratic loss functions, which represent the deviation between part measurement and the target value. Application of this model minimizes the total cost of manufacturing activities and quality issues relating to machining process planning, particularly in the early stages of planning.     

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.     

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.     

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.     

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.     

A new tolerance design method for a secondary rechargeable battery using design of experiments with mixture

Mixing errors in the manufacturing process of a mixture may cause a sizeable variation in the performance of the product, leading to the need for the tolerance design. Even though a variety of procedures have been proposed for the optimal tolerance design based on quality loss and manufacturing costs, there are no available tolerance design methods when mixing errors exist in the manufacturing process of a mixture. In this article, we propose a new tolerance design method for the case where mixing errors are involved in massive manufacturing process of a secondary rechargeable battery. Using an approximation method, we derive quality loss function, reflecting the effects of mixing errors on the product performances. Statistical design of mixture experiments is applied to build empirical models of performances as functions of component proportions in the corresponding quality loss function. A real‐life case study on the tolerance design of a secondary battery is provided for the illustration of the proposed method. The results show the efficiency of the proposed method in designing the tolerances to minimize the quality loss and manufacturing costs. Copyright © 2008 John Wiley & Sons, Ltd.