Integrated tolerance optimisation with simulated annealing

Tolerance is one of the most important parameters in design and manufacturing. The allocation of design and machining tolerances has a significant impact on manufacturing cost and product quality. This article presents an analytical model for simultaneously allocating design and machining tolerances based on the least-manufacturing-cost criterion. In this study, tolerance allocation is formulated as a non-linear optimisation model based on the cost-tolerance relationship. A new global optimisation algorithm, simulated annealing, is employed to solve the non-linear programming problem. An example for illustrating the optimisation model and the solution procedure is provided.     

Concurrent tolerancing for design and manufacturing based on the present worth of quality loss

The quality loss function developed by Taguchi provides a monetary measure for the deviation of the product quality characteristic from the target value. Product use causes degradation on its quality characteristic, and since such a deviation can be changing over time, so can the quality loss. However, most studies on concurrent tolerancing theory do not consider the quality loss caused by the degradation. In this paper, the present worth of expected quality loss expressed as the function of the pertinent process tolerances in a concurrent tolerancing environment is derived to capture the quality loss due to product degradation over time as a continuous cash flow function under continuous compounding. A new tolerance optimization model, which is to minimize the summation of manufacturing cost and the present worth of expected quality loss, is established to realize the concurrent tolerance allocation for products with multiple quality characteristics. An example of the bevel gear assembly involving concurrent allocation of design and process tolerances is given, demonstrating that the proposed model is feasible in practice.     

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。     

基于多重相关特征质量损失函数的公差优化设计

在重点推导了具有多重相关特征产品的质量损失与尺寸公差的函数关系的基础上,提出了多重相关特征产品的公差优化设计方法,建立了基于制造成本一质量损失的公差优化设计的综合模型,建立该模型的目的是寻求制造成本和质量损失之间的平衡,实现旨在提高产品质量和降低成本的公差优化设计。应用实例验证了所提出方法的有效性。     

Optimal Parameter and Tolerance Design with a Complete Inspection Plan

The need to remain competitive has led manufacturing sectors to consider tolerances as the key to achieving low cost and high quality. To produce quality products at low cost in today’s manufacturing industry, an integration of product design and process planning is essential. Process tolerance is one of the most important parameters that link product design and process planning. The process mean is also a critical parameter for further quality improvement and cost reduction under the permissible process setting adjustment within design tolerance limits. This study discusses an approach to integrate the product and process design via the optimisation of process mean and process tolerance.     

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.     

Computer-aided geometrical dimensioning and tolerancing for process-operation planning and quality control

This paper describes an extension of a model which determines an optimum set of dimensions and tolerances for machining processes at minimum manufacturing cost. This optimisation minimizes the cost of scrap, which is a function of manufacturing tolerances, as the objective function. Requirements of design sizes, geometrical tolerances (including both form and position) and machining allowances are expressed mathematically as constraints for the optimisation. A computerised trace method has been extended to determine the relationships between geometrical tolerances and associated relevant manufacturing dimensions and tolerances. In addition to the manufacturing cost, the model takes into account manufacturing sequence, distribution of manufacturing dimensions, process capabilities, tolerances, design sizes, geometrical tolerances, machining allowances and optimum scrap level. The resulting computerized interactive system can be used not only in process planning, but also in quality control.     

Particle swarm optimization (PSO) algorithm for optimal machining allocation of clutch assembly

Nowadays tolerance optimization is increasingly becoming an important tool for manufacturing and mechanical design. This seemingly, arbitrary task of assigning dimension tolerance can have a large effect on the cost and performance of manufactured products. With the increase in competition in today’s market place, small savings in cost or small increase in performance may determine the success of a product. In practical applications, tolerances are most often assigned as informal compromises between functional quality and manufacturing cost. Frequently the compromise is obtained interactively by trial and error. A more scientific approach is often desirable for better performance. In this paper particle swarm optimization (PSO) is used for the optimal machining tolerance allocation of over running clutch assembly to obtain the global optimal solution. The objective is to obtain optimum tolerances of the individual components for the minimum cost of manufacturing. The result obtained by PSO is compared with the geometric programming (GP) and genetic algorithm (GA) and the performance of the result are analyzed .     

Inspection Allocation Planning for a Multiple Quality Characteristic Advanced Manufacturing System

Producing products with multiple quality characteristics is always one of the concerns for an advanced manufacturing system. To assure product quality, finite automatic inspection systems should be used. Inspection planning to allocate inspection stations should then be performed to manage the limited inspection resource. Except for finite inspection station classes, in this work, the limited number of inspection stations, of each inspection station class, is considered for solving the inspection allocation problem in a multiple quality characteristic advanced manufacturing system. Since the product variety in batch production or job shop production increases to satisfy the changing requirements of the various customers, the tolerances specified will vary from time to time. This inspection allocation problem is solved using a unit cost model in which the manufacturing capability, inspection capability, and tolerance specified are concurrently considered for a multiple quality characteristic product. The situation of unbalanced tolerance design is also considered. The inspection allocation problem can then be solved according to customer requirements. Since determining the optimal inspection allocation plan seems to be impractical, as the problem size becomes large, two decision criteria (i.e. sequence order of workstation and tolerance interval) are employed separately to develop two different heuristic solution methods in this work. The performance of each method is measured in comparison with the enumeration method that generates the optimal solution. The result shows that a feasible inspection allocation plan can be determined efficiently. ID="A1"Correspondance and offprint requests to: Dr Yau-Ren Shiau, Department of Industrial Engineering, Feng-Chia University, 100 Wenhwa Road, Seatwen, PO Box 25–097, Taichung 407, Taiwan     

Simultaneous optimal selection of design and manufacturing tolerances with different stack-up conditions using scatter search

The allocation of design and manufacturing tolerances directly influences the manufacturing cost and functionality of a product. Traditionally, it is carried out in two phases; CAD and CAPP, in a sequential manner. This approach has the drawbacks of more lead-time and sub-optimality. The present work reports on a procedure to simultaneously allocate both design and manufacturing tolerances based on minimum total manufacturing cost. The optimization model for tolerance design is a non-linear multivariable problem. Since such a problem may have a noisy surface, there is a need to apply an efficient global optimization technique. In the present work, scatter search, one of the recently emerged optimization techniques in the area of metaheuristics, is applied successfully to determine the optimal tolerances at the minimum manufacturing cost. An example is taken from the literature to explain the proposed methodology. The results exhibit the effectiveness of the proposed algorithm over other methods.     

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。     

Optimum tolerance synthesis for complex assembly with alternative process selection using Lagrange multiplier method

Any part cannot be manufactured to the required nominal dimensions due to inherent variations in workmanship, material, and machine. The specification of tolerance on part dimensions plays a major role on performance, quality, and cost of the product. Distribution of tolerance among the components of an assembly is known as tolerance allocation. The selection of alternative processes for tolerance allocation also plays a vital role in reducing manufacturing cost. Near-optimal allocated tolerances are obtained using nontraditional optimization techniques in which the solutions are achieved randomly. Also, there is a chance for omitting the better process for allocation. The results of successive run of the program based on these techniques will not yield consistent results. An attempt has been made in this work to solve the above problem using Lagrange multiplier method for complex assemblies with univariate search method. The methodology has been demonstrated on wheel mounting assembly. The example product after implementing the proposed method would yield 1.4% savings in manufacturing cost as compared with the cost obtained by Singh.     

Tolerance design optimization of machine elements using genetic algorithm

An important problem that faces design engineers is how to assign tolerance limits. In practical applications, tolerances are most often assigned as an informal compromise between functionality, quality and manufacturing cost. Frequently, the compromise is obtained iteratively by trial and error. A more scientific approach is often desirable for better performance. In this paper, a genetic algorithm (GA) is used for the design of tolerances of machine elements to obtain the global optimal solution. The objective is to design the optimum tolerances of the individual components to achieve the required assembly tolerance, zero percentage rejection of the components and minimum cost of manufacturing. The proposed procedure using GA is described in this paper for two tolerance design optimization problems: gear train and overrunning clutch assemblies. Results are compared with conventional techniques and the performances are analyzed.     

Tolerance chart balancing with a complete inspection plan taking account of manufacturing and quality costs

This paper introduces a mathematical model of tolerance chart balancing for machining process planning under complete inspection. The criteria considered in this study are based on the combined effects of manufacturing cost and quality loss, under constraints such as process capability limits, product design specifications, and product quality requirements. Manufacturing costs include the machining cost, part cost, inspection cost, reworking cost, and replacement cost. The machining 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 the part measurement and the target value. An example is presented to demonstrate the proposed model. A comparison made with previous methods shows that the proposed model minimizes the total cost of manufacturing activities and quality-related issues in machining process planning, particularly in the early stages. Moreover, the applications are not be limited to machining process planning but can also be used in other forms of production planning.     

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.     

Optimal tolerance design for mechanical assembly considering thermal impact

In this paper, a cost–tolerance model based on neural network methods is proposed in order to provide product designers and process planners with an accurate basis for estimating the manufacturing cost. Tolerance allocation among the assembly components is carried out to ensure that the functionality and design quality are satisfied considering the effect of dimensional and geometric tolerance of various components of the assembly by developing a parametric computer aided design (CAD) model. In addition, deformations of various components of mechanical assembly due to inertia and temperature effects are determined and the same is integrated with tolerance design. The benefits of integrating the results of finite element simulation in the early stages of tolerance design are discussed. The proposed method is explained with an application example of motor assembly, where variations due to both dimensional and geometric tolerances are studied. The results show that the proposed methods are much effective, cost, and time saving than the ones considered in literature.     

Computer-Aided Tolerance Charting for Products with Angular Features

This paper describes a process by which computer-aided design methods are used for the tolerance charting of products with angular features. If a product contains one or more angular features, such as chamfers and tapered surfaces, radial or normal machining of the features will result in axial dimension changes. In this paper, basic trigonometric formulae are first presented to explain the phenomenon of tolerance accumulation. In the process of tolerance charting, dummy cuts are included to reflect the corresponding dimensional changes due to indirect machining. With the assistance of flags and linked lists, the system proposed can automatically identify all dimensional chains which are associated with either regular cuts or dummy cuts. Moreover, optimisation techniques are recommended to allocate the allowable tolerances as specified by blueprints. In the search for an optimal design, the total manufacturing cost defined by the working tolerances is the objective function to be minimised.     

Manufacturing environment-oriented robust tolerance optimization method

Tolerance design has a great impact on the cost and quality of a product. Previous research focused on process tolerances or robust tolerance design with little consideration on real manufacturing context. This paper presents a nonlinear method for robust tolerance design based on the real manufacturing context in three stages. The objective function to be minimized is the total manufacturing cost. The constraint equations for the optimization model are also deduced, which select suitable manufacturing processes based on the manufacturing environment. Simulation annealing (SA) is used for the nonlinear optimization. The approach is finally illustrated by a practical example. The results of the comparison with different models indicate that the proposed approach is more effective with the manufacturing resource. The robust and reliable tolerance can be obtained.     

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.     

Closed-form solutions for multi-objective tolerance optimization

Component tolerances have important influence on the cost and performance of products. In order to obtain suitable component tolerances, multi-objective tolerance optimization model is studied, in which the combined polynomial and exponential functions are used to model manufacturing cost. In this paper, analytical methods are proposed to solve the multi-objective optimization model. In this model, the objective function is not a monotone function, and it is possible that the assembly tolerance constraint, including worst-case method and root sum square method, is inactive. Therefore, two closed-form solutions are proposed for each component tolerance in terms of the Lambert W function. When the assembly tolerance constraint is not considered, the component tolerances are obtained and named as the initial closed-form solutions. If the initial solutions satisfy assembly tolerance constraint, it is the final value of optimal tolerances. Otherwise, constrained optimization model is established and Lagrange multiplier method is applied to obtain the new closed-form solution of component tolerances as the final value of optimal tolerances. Several simulation examples are used to demonstrate the proposed method.