Optimal tolerance design for products with correlated characteristics by considering the present worth of quality loss

The quality loss function proposed by Taguchi provides a quantitative measurement of product quality when product’s quality characteristic value deviates from the ideal target at an arbitrary time. However, product use causes degradation on its quality characteristic, and since such a deviation can be changing over time, so can its quality loss. The quality loss caused by degradation on quality characteristic has not been considered in most research. In this paper, the time value of money for quality loss and product degradation over time is integrated into the total cost model, and a new optimization model for the tolerance design of products with correlated characteristics is established. The discussions focus on the multivariate quality loss function as an extension of the Taguchi loss function, which is used to model quality loss due to product degradation as a continuous cash flow function under continuous compounding. The optimal tolerance design is achieved by minimizing the total cost, which is the sum of manufacturing cost and the present worth of expected quality loss. An illustrative example is presented to demonstrate the effectiveness of the proposed model.     

基于制造成本与质量损失的并行公差设计

研究产品质量损失与产品尺寸公差的关系 ,提出具有多个相关装配尺寸产品的质量损失 ,并把它表示成工序公差的函数。利用成本 -公差函数和产品的质量损失函数 ,给出基于产品最低制造成本和多个相关装配尺寸产品质量损失的并行公差设计优化模型 ,实现公差的并行设计 ,最后通过工程实例验证所提出的方法     

Minimum-Loss Assembly Tolerance Allocation by Considering Product Degradation and Time Value of Money

In the optimisation of tolerance allocation for a mechanical assembly, much work has concentrated on the minimum cost– tolerance allocation without considering the quality of the final assembly. Cheng and Maghsoodloo combined the cost– tolerance function and quality loss function, to determine the optimal tolerances for individual components, so that the total assembly cost (including both tolerance cost and quality loss) might be minimised. The objective of this paper is to propose a model for optimal tolerance allocation by considering both tolerance cost and the present worth of quality loss such that the total assembly cost/loss is minimised. The proposed model takes into account the time value of money for quality loss and product degradation over time, and includes two new parameters: the planning horizon and the product user’s discount rate. From the result 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 yields a decrease in both tolerance cost and quality loss.     

基于工序加工能力的并行公差优化设计

提出一种基于工序加工能力的并行工序公差优化设计方法。在产品的初步结构设计阶段，通过相配零件的加工工艺规划把装配功能公差表示为零件的工序公差，建立以加权制造总成本最小为目标，以并行公差链、标准化的工序公差系数、机床最大经济极限公差为约束的非线性并行公差优化设计模型，求解该模型得到最佳的工序公差。最后给出了并行公差优化设计的一个工程实例，结果表明，所提的方法具有比传统串行等精度方法更合理、工序公差数值更大的优点。     

基于信噪比多元质量损失和制造成本的并行公差设计

质量损失是由于产品的功能波动所造成的,损失大小可由质量函数确定。在无量纲“标准化”多元质量损失函数的基础上,采用信噪比衡量各质量指标的波动,建立一般情形下的基于信噪比的多元质量损失模型,并把它表示成多个相关装配尺寸产品的工序公差函数。利用成本-公差函数和产品质量损失函数,给出基于产品最低制造成本和多个相关装配尺寸产品质量损失的并行公差设计优化模型,实现公差的并行设计,最后通过工程实例验证所提出的方法。     

Dimensional and geometrical tolerance balancing in concurrent design

In conventional design, tolerancing is divided into two separated sequential stages, i.e., product tolerancing and process tolerancing. In product tolerancing stage, the assembly functional tolerances are allocated to BP component tolerances. In the process tolerancing stage, the obtained BP tolerances are further allocated to the process tolerances in terms of the given process planning. As a result, tolerance design often results in conflict and redesign. An optimal design methodology for both dimensional and geometrical tolerances (DGTs) is presented and validated in a concurrent design environment. We directly allocate the required functional assembly DGTs to the pertinent process DGTs by using the given process planning of the related components. Geometrical tolerances are treated as the equivalent bilateral dimensional tolerances or the additional tolerance constraints according to their functional roles and engineering semantics in manufacturing. When the process sequences of the related components have been determined in the assembly structure design stage, we formulate the concurrent tolerance chains to express the relations between the assembly DGTs and the related component process DGTs by using the integrated tolerance charts. Concurrent tolerancing which simultaneously optimizes the process tolerance based on the constraints of concurrent DGTs and the process accuracy is implemented by a linear programming approach. In the optimization model the objective is to maximize the total weight process DGTs while weight factor is used to evaluate the different manufacturing costs between different means of manufacturing operations corresponding to the same tolerance value. Economical tolerance bounds of related operations are given as constraints. Finally, an example is included to demonstrate the proposed methodology.     

A tolerancing optimisation method for product design

Although extensive research has been carried out in the area of tolerancing techniques for product design, concurrent engineering is still very seldom used in this context. This paper introduces a unique tolerancing method which applies the concept of concurrent engineering. The proposed method essentially allocates the required functional assembly tolerances to the component tolerances by formulating the tolerancing problem into a mathematical model and solving the model using a linear programming approach. The component tolerances are first represented in terms of the process tolerances, assigned by process planners at an early stage of the product design. The objective function of the mathematical model, which is to maximise the residual tolerances of the processes, is then established and the constraints formulated based on the assembly requirements and process constraints. The model is subsequently solved using a linear programming approach. Finally, the proposed method is tested on a practical example.     

基于多重相关特征质量损失函数的并行公差设计

基于多变量质量损失函数,推导了多重相关特征产品的质量损失与有关零件工序公差的函数关系,建立了基于制造成本-质量损失的并行公差设计的优化综合模型,其目的是寻求制造成本和质量损失之间的平衡,以实现相关特征产品的设计公差与工序公差并行优化分配,达到提高产品质量和降低成本的目的。圆锥齿轮装配的并行公差设计实例验证了所提方法的有效性。     

产品概念结构设计中的公差进化模型和算法

在产品概念结构设计自动化技术基础上，研究了产品公差设计过程及其特点，提出了四层产品公差结构模型。应用局部公差分配模型与公差设计过程建立了全局公差优化理论模型，提出了以保证产品总精度需求为目标的概念结构设计阶段公差的同步设计算法。实例表明，提出的公差进化模型和算法，避免了人为建模工作，大大提高了公差设计效率和质量，为快速完成多方案的并行设计奠定了基础。     

Economic-based design of engineering systems with degrading components using probabilistic loss of quality

The allocation of means and tolerances to provide quality, functional reliability and performance reliability in engineering systems is a challenging problem. Traditional measures to help select the best means and tolerances include mean time to failure and its variance; however, they have some shortcomings. In this paper, a monetary measure based on present worth is invoked as a more inclusive metric. We consider the sum of the production cost and the expected loss of quality cost over a planned horizon at the customer’s discount rates. Key to the approach is a probabilistic loss of quality cost that incorporates the cumulative distribution function that arises from time-variant distributions of system performance measures due to degrading components. The proposed design approach investigates both degradation and uncertainty in component. Moreover, it tries to obviate problems of current Taguchi’s loss function-based design approaches. Case studies show the practicality and promise of the approach.     

A continuous tolerance system

This paper analyzes the current ISO tolerance systems (ISO 286-1¹ and ISO 286-22) and presents new formulas that allow the aggregation of the tolerance system in a simple way. The approach used makes it possible to compute the tolerances in a continuous manner with respect to both dimension and grade of tolerance (quality). The results are always within the tolerances tabulated in the ISO 286-1' and 286-22 Standards. This continuous approach is suitable for optimization in design and automation in tolerancing on computer-assisted design (CAD) systems. The paper proposes a tolerancing approach closer to the modern tolerancing philosophy as given by the Taguchi methods.     

Design and optimization of concurrent tolerance in mechanical assemblies using bat algorithm

Concurrent designing of tolerance has become a vital concern in product and process development due to the relationship between quality, functionality and product cost. It is one of the well explored areas in combinatorial optimization. In this paper, a recently developed optimization algorithm, called Bat algorithm (BA), is used for optimizing the tolerance based on concurrent objectives to minimize the manufacturing cost, present worth of expected quality loss and quality loss. The mechanical assemblies such as Bevel gear assembly (A), Gear box assembly (B) and Suction union assembly (C) are considered to demonstrate the proposed algorithm. It is found that the BA has produced better results than other methods in initial generations for concurrent tolerance problems.     

Fuzzy-small degrees of freedom representation of linear and angular variations in mechanical assemblies for tolerance analysis and allocation

Tolerances naturally generate an uncertain environment for design and manufacturing. In this paper, a novel fuzzy based tolerance representation approach for modeling the variations of geometric features due to dimensional tolerances is presented. The two concepts of fuzzy theory and small degrees of freedom are combined to introduce the fuzzy-small degrees of freedom model (F-SDOF). This model is suitable for tolerance analysis of mechanical assemblies with linear and angular tolerances. Based on the fuzzy concept, a new index (called the assemblability index) is introduced which signifies the fitting quality of parts in the assembly. Graphical and numerical representations of tolerance allocation by this method are presented. The goal of tolerance allocation is to adjust the tolerances assigned at the design stage so as to meet a functional requirement at the assembly stage. The presented method is compatible with the current dimensioning and tolerancing standards. The application of the proposed methodology is illustrated through presenting an example problem.     

Optimal target selection for unbalanced tolerance design

In many manufacturing processes, unbalanced tolerance design is a common occurrence. It occurs when the deviation of a quality characteristic in one direction is more harmful than in the opposite direction. The failure mode in these two directions is usually different. Furthermore, automatic inspection and measurement technology are widely used by the industries. The non-conforming part will be detected automatically. Thus, a truncated asymmetrical quadratic loss function is assumed for the unbalanced tolerance design. Traditionally, the manufacturer would either choose the smaller tolerance as the tolerance for both sides, or would set the process mean at the middle of the tolerances. Both methods fail to minimise the expected quality loss. The purpose of this paper is to find out the optimal manufacturing setting such that the expected quality loss is minimised. The results show that the process mean should be shifted a little from the target value.     

Concurrent process tolerance design based on minimum product manufacturing cost and quality loss

In a concurrent design environment, a robust optimum method is presented to directly determine the process tolerances from multiple correlated critical tolerances in an assembly. With given distributions of multiple critical assembly dimensions, the Taguchi quadric quality loss function is first derived. The quality loss is then expressed as the function of pertinent process tolerances. A nonlinear optimal model is established to minimize the summation of manufacturing costs and product quality loss. An example illustrates the proposed model and the solution method .     

Optimal tolerance design of assembly for minimum quality loss and manufacturing cost using metaheuristic algorithms

Tolerance allocation is a design tool for reducing overall cost of manufacturing while meeting target levels for quality. An important consideration in product design is the assignment of design and manufacturing tolerances to individual component dimensions so that the product can be produced economically and functions properly. The allocation of tolerances among the components of a mechanical assembly can significantly affect the resulting manufacturing costs. In this work, the tolerance allocation problem is formulated as a non-linear integer model by considering both the manufacturing cost of each component by alternate processes and the quality loss of assemblies so as to minimise the manufacturing cost. Metaheuristics techniques such as genetic algorithm and particle swarm optimisation are used to solve the model and obtain the global optimal solution for tolerance design. An example for illustrating the optimisation model and the solution procedure is provided. Results are compared with conventional technique and the performances are analysed.     

A general model for process-setting with an asymmetrical linear loss function

In many industries, unbalanced tolerance design is a common occurrence. It occurs when the deviation of a quality characteristic in one direction is more harmful than the deviation in the opposite direction. The failure mode in the two directions is usually different. Taguchi redefines quality using Gauss’s quadratic function as the loss that a product imparts to society from the time the product is shipped. However, using a quadratic loss function when the actual loss function is nonquadratic may yield incorrect input parameter levels. In certain situations, a linear loss function is more appropriate in industrial applications. Furthermore, automatic inspection and measurement technology is widely used by many industries; the nonconforming quality characteristic would be detected automatically. Thus, rather than use a quadratic loss function, we assume a truncated asymmetrical linear loss function to describe unbalanced tolerance design. The purpose of this paper is to find out the optimal setting of the process mean such that the expected quality loss is minimised. The results show that the process mean should be slightly offset from the target value.     

多工位装配过程夹具系统公差和维护综合优化设计

提出一种面向二维多工位装配过程、综合考虑装配夹具系统全寿命周期成本、产品零件孔制造成本和产品质量损失成本的公差和维护综合优化方法。分析多工位装配尺寸偏差传递关系,建立多工位装配过程产品质量损失模型。然后根据4-2-1夹具定位原则,构建考虑夹具磨损过程损失的夹具定位销副偏差统计数字特征模型。继而发展了以夹具系统全寿命周期成本、零件孔制造成本和和产品质量损失成本为装配总成本最小化的定位销公差、零件孔公差与更换周期优化模型。以汽车侧围装配过程为例,分别研究定位销公差、零件孔公差、定位销更换周期、配合间隙、平均磨损率和磨损率方差对装配总成本的影响,并优化设计定位销公差、零件孔公差和定位销更换周期。所提出的综合优化设计方法比采用定位销等公差设计、零件孔等公差设计、定位销与零件孔等公差设计和定周期更换设计的装配总成本分别减少了16.25%、11.31%、39.93%和13.54%。该方法为产品装配夹具系统高质量低成本设计提供了一种新的途径。     

Concurrent optimization of machining process parameters and tolerance allocation

With the advent use of sophisticated and high-cost machines coupled with higher labor costs, concurrent optimization of machining process parameters and tolerance allocation plays a vital role in producing the parts economically. In this paper, an effort is made to concurrently optimize the manufacturing cost of piston and cylinder components by optimizing the operating parameters of the machining processes. Design of experiments (DoE) is adopted to investigate systematically the machining process parameters that influence product quality. In addition, tolerance plays a vital role in assembly of parts in manufacturing industries. For the selected piston and cylinder component, improvements efforts are made to reduce the total manufacturing cost of the components. By making use of central composite rotatable design method, a module of DoE, a mathematical model is developed for predicting the standard deviation of the tolerance achieved by grinding process. This mathematical model, which gives 93.3% accuracy, is used to calculate the quality loss cost. The intent of concurrent optimization problem is to minimize total manufacturing cost and quality loss function. Genetic algorithm is followed for optimizing the parameters. The results prove that there is a considerable reduction in manufacturing cost without violating the required tolerance, cutting force, and power.     

Optimum Tolerance Allocation in Assembly

Dimensioning and tolerancing are both important phases of product design. Although dimensions can usually be assigned based on design constraints and aestheticism without much difficulty, tolerancing is often a major problem to designers. Traditionally, tolerances are assigned intuitively followed by an analysis to check for any violation of the assembly requirements. Based on the analysis results, modifications are made manually by a "trial and error" method. This method relies on the experience of the designers and the results may not be optimal. This paper presents a new approach to optimum tolerancing of components in an assembly such that all interaction requirements are met. The requirements may be for unilateral tolerance for control of clearance and interference or they may be for bilateral tolerance control. A model showing the relationship between components is constructed directly from the design. Using the model, coupled with a unique algorithm, a set of linear equations is formulated based on the design constraints and assembly requirements. The linear equations are then solved to determine the optimum tolerances of the assembly.