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     

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.     

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.     

基于双响应面法的公差设计研究

探讨了产品综合成本的构成,以此为基础提出了以产品综合成本最低为目标的公差稳健设计模型,并利用双响应面法(Response Surface Methodology ,RSM) 来实现公差的稳健优化设计。     

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.     

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.     

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.     

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.     

A robust cell formation approach for varying product demands

In cellular manufacturing environments, manufacturing cells are generally formed based on deterministic product demands. In this paper, we consider a system configuration problem with product demands expressed in a number of probabilistic scenarios. An optimization model integrating cell formation and part allocation is developed to generate a robust system configuration to minimize machine cost and expected inter-cell material handling cost. A two-stage Tabu search based heuristic algorithm is developed to find the optimal or near optimal solutions to the NP-hard problem. Numerical examples show that this model leads to an appropriate compromise between system configuration costs and expected material handling costs to meet the varying product demands. These example problems also show that the proposed algorithm is effective and computationally efficient for small or medium size problems.     

Robust optimal Robin boundary control for the transient heat equation with random input data

The problem of robust optimal Robin boundary control for a parabolic partial differential equation with uncertain input data is considered. As a measure of robustness, the variance of the random system response is included in two different cost functionals. Uncertainties in both the underlying state equation and the control variable are quantified through random fields. The paper is mainly concerned with the numerical resolution of the problem. To this end, a gradient‐based method is proposed considering different functional costs to achieve the robustness of the system. An adaptive anisotropic sparse grid stochastic collocation method is used for the numerical resolution of the associated state and adjoint state equations. The different functional costs are analysed in terms of computational efficiency and its capability to provide robust solutions. Two numerical experiments illustrate the performance of the algorithm. Copyright © 2016 John Wiley & Sons, Ltd.     

Tolerance allocation via simulation embedded sequential quadratic programming

Tolerance allocation is an important problem frequently encountered in the synthesis process, for designers as well as manufacturing engineers. Under the objective of minimizing the manufacturing cost while attaining an acceptable yield, the problem can be formulated as a stochastic program. Owing to the nonlinear nature of the stochastic program, a sequential quadratic programming algorithm is developed to solve the problem. The cumbersome multivariate integration in calculating the yield is approximated by a Monte Carlo simulation and the highly nonlinear yield constraint is supported by some auxiliary constraints. In limited experiments, the proposed method has performed efficiently and robustly. Compared with some previous studies, the designs solved in this paper have smaller manufacturing costs and higher yields, indicating that the proposed method is very promising in solving tolerance allocation problems.     

Inspection resource assignment in a multistage manufacturing system with an inspection error model

Producing high-quality products at low cost is always one concern for a multi-stage manufacturing system. That is, production costs and inspection efficiency should receive equal importance. Inspection planning to allocate inspection stations should then be performed to manage limited inspection resources during process planning. Product quality and the possible costs can then be concurrently considered when evaluating a manufacturing plan. Except for finite inspection station classes, the limited number of inspection stations of each inspection station class is considered to solve the inspection allocation problem in this research. Rather than utilizing a constant inspection error or a specified inspection error probability distribution determined by previous observations, the inspection allocation problem is solved using relative cost models in which the inspection error model is embedded. The inspection allocation problem can then be solved by practically reflecting the inspection error when tolerances are rapidly changed to satisfy customer requirements. Since determining the optimal inspection allocation plan seems impractical as the problem size becomes quite large, two heuristic methods have been developed by considering the defective rate, manufacturing cost and earliest stage priority in this research. The performance of each method is measured in comparison with the enumeration method that generates the optimal solution. A feasible manufacturing plan can then be determined and confirmed during process planning by concurrently solving the inspection allocation problem.     

Component choice for managing risk in engineered systems with generalized risk/cost functions

The constrained optimization of resource allocation to minimize the probability of failure of an engineered system relies on a probabilistic risk analysis of that system, and on ‘risk/cost functions’. These functions describe, for each possible improvement of the system's robustness, the corresponding gain of reliability given the considered component or management factor to be upgraded. These improvements can include, for example, the choice of components of different robustness levels (at different costs), addition of redundancies, or changes in operating and maintenance procedures. The optimization model is generally constrained by a maximum budget, a schedule deadline, or a maximum number of qualified personnel. A key question is thus the nature of the risk/cost function linking the costs involved and the corresponding failure-risk reduction. Most of the methods proposed in the past have relied on continuous, convex risk/cost functions reflecting decreasing marginal returns. In reality, the risk/cost functions can be simple step functions (e.g. a discrete choice among possible components), discontinuous functions characterized by continuous segments between points of discontinuity (e.g. a discrete choice among components that can be of continuously increasing levels of robustness), or continuous functions (e.g. exponentially decreasing failure risk with added resources).This paper describes a general method for the optimization of the robustness of a complex engineered system in which all three risk/cost function types may be relevant. We illustrate the method with a satellite design problem. We conclude with a discussion of the complexity of the resolution of this general type of optimization problem given the number and the types of variables involved.     

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.     

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.     

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     

Construction of closed-form equations and graphical representation for optimal tolerance allocation

The Lagrange multiplier method (LM) is currently used to allocate tolerance for optimum manufacturing cost. This is a tedious iterative process and sometimes it allocates a component's tolerance outside its process tolerance limits. The present work develops a graphical representation which can help the process engineer to visualize the minimum and maximum values for assembly tolerances. The graphical representation developed can also help the process engineer to determine the exact total manufacturing cost of the assembly and help to fix the tolerance, which would not fall outside the limits prescribed. A simple C program is developed to construct the closed-form equations (CFE), and a single EXCEL graphical representation is derived for assembly tolerance, allocated tolerance, and total manufacturing cost. The developed algorithm has been demonstrated on a two- to five-component linear assembly, to help the process/manufacturing engineer's visualization before determining the tolerance specification on component dimensions. The test results show a maximum percentage deviation of 0.09% of assembly tolerance and 0.33% of total manufacturing cost between the LM and the newly developed CFE 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.