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.     

Robust Tolerance Design by Response Surface Methodology

The Response Surface Methodology (RSM) approach is adopted for data analysis for an experimental model in order to determine the optimal component tolerances in an assembly. The response variable is the total cost, which consists of quality loss and tolerance cost. RSM is a combination of mathematical and statistical techniques, which provides designers, not only with the optimal tolerance values, but also with the critical components in an assembly. This feature is very important during the design activities since it enables designers to have feedback and suggestions for possible design improvement. Using RSM to optimise and analyse the experimental results, a tolerance design for quality improvement and cost reduction can be achieved at an early stage of design, particularly, in an uncertain design environment.     

The use of mixture experiments in tolerance allocation problems

The response surface methodology (RSM) approach can be used to determine the optimal component tolerances in an assembly. Frequently, response surface designs such as Box-Behnken design and central composite design are used in tolerance allocation problems. In this article, mixture experiments, which are essentially constructed for designing a blend composition, are proposed instead of response surface designs in order to observe the cost values. Also some advantages and disadvantages of mixture designs are discussed.     

<Emphasis Type="Bold">Simultaneous Parameter and Tolerance Design for an Electronic Circuit Via Computer Simulation and Statistical Optimization</Emphasis>

Conventionally, parameter design is carried out prior to tolerance design for economic reasons. However, a combined parameter and tolerance design may be completed in one stage for further cost reduction and quality improvement. In this study, a simultaneous optimization of component parameter and component tolerance for an electronic circuit design is realized via computer simulation and Response Surface Methodology (RSM). The approach first generates a set of outputs (experimental data) through computer simulation based on various combinations of parameter and tolerance levels as inputs. Then these outputs are converted into a total cost as a response value before applying RSM for statistical analysis and optimization. The response value(total cost) includes quality loss, tolerance cost, and failure cost, which reflect the combined effect of the assigned parameter and tolerance values being assigned. In this study, computer simulation can be carried out by two methods. The first method uses Monte Carlo Simulation to generate the outputs with the condition that the transfer function is known. The second method uses a Pspice simulation program to generate outputs without necessarily knowing the transfer function. The results provide designers with the optimal component parameter and tolerance values, and the critical components, and the response function – all of which are very important during the early design activities as they enable designers to have a repeated application, accurate feedback and appropriate suggestions, particularly under uncertain design conditions.     

Concurrent Optimisation of Parameter and Tolerance Design via Computer Simulation and Statistical Method

This study optimises component parameters and component tolerances simultaneously via computer simulation and response surface methodology (RSM). The approach first generates a set of experimental data through computer simulation, then the data are converted to a total cost as a response value before applying RSM for statistical analysis and mathematical optimisation. The response value (total cost) includes quality and related costs which reflect the combined effect of the parameter and tolerance values being assigned. The results provide designers with the optimal component design values, the critical components, and the response function of a product or process design, which are very important to know during design activities as they give designers information about repeated applications, accurate feedback and appropriate suggestions, particularly under uncertain design conditions. Three examples are provided: They are mechanical assembly design, machining process planning, and electronic circuit design.     

Optimal tolerance allocation using a multiobjective particle swarm optimizer

Particle swarm optimizers are routinely utilized in engineering design problems, but much work remains to take advantage of their full potential in the combined areas of sensitivity analysis and tolerance synthesis. In this paper, a novel Pareto-based multiobjective formulation is proposed to enhance the operations of a particle swarm optimizer and systematically distribute tolerances among various components of a mechanical assembly. The enhanced algorithm relies on nonlinear sensitivity analysis and the statistical root sum squares model to simultaneously optimize product performance criteria, the manufacturing cost, and the stack-up tolerance. It is shown that the proposed algorithm can accomplish its optimization task by successfully shifting nominal values of design parameters instead of the expensive tightening of component tolerances. Several numerical experiments for optimal design of a stepped bar assembly were conducted, which highlight the advantages of the proposed methodology.     

公差稳健优化设计的研究

为了解决产品加工成本与质量稳健的协调性问题,提出了一种新的公差稳健优化设计数学模型.依据公差稳健设计的思想,考虑产品质量的模糊性,以封闭环误差分布概率密度函数的方差和优质品概率之比为设计目标,建立了公差优化设计产品质量稳健性损失成本目标函数,并研究了优质品率和封闭环误差分布方差的确定方法.以加工成本和产品质量稳健性损失成本为目标,以模糊装配可靠度、可取公差极限范围为约束条件,建立了公差多目标优化数学模型.举例说明了文中所述的公差稳健优化设计方法的应用,采用遗传算法实现了公差的多目标优化设计.实例表明,该方法能够协调零件的加工成本和产品质量的稳健性损失成本,使优化指标的综合性能最佳.     

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.     

一种设计和工序公差计算机辅助并行设计方法的研究

提出一种计算机辅助设计公差和工序公差并行设计的数学模型，以成本公差函数作为目标函数，以装配功能要求、加工方法、加工余量、工序制造公差范围作为约束条件，并用蒙特卡洛法模拟尺寸装配、模拟退火算法用于优化求解，实现了设计公差和工序公差并行设计，缩短了设计周期。     

基于DFA和DFC的综合公差优化

公差是机械设计和制造过程中最重要的参数之一,其对加工成本和产品质量有很大的影响。本文采用 DFA和DFC的综合方法对制造过程中的装配精度进行分析和综合,建立了以公差为主的成本信息模型,对设计公 差和加工公差进行了系统分析,以最低制造成本为目标,根据成本 公差间的函数关系,建立了非线性的公差优化模 型,并采用遗传算法使这一问题得到了很好的解决。实例证明该方法是可行的。并为并行工程实施中其他的DFX 与DFC结合实现制造成本最低提供了良好的解决方案。     

A contribution on the optimization strategies based on moving least squares approximation for sheet metal forming design

Computer-aided procedures to design and optimize forming processes are, nowadays, crucial research topics since industrial interest in costs and times reduction is always increasing. Many researchers have faced this research challenge with various approaches. Response surface methods (RSM) are probably the most known approaches since they proved their effectiveness in the recent years. With a peculiar attention to sheet metal forming process design, RSM should offer the possibility to reduce the number of numerical simulations which in many cases means to reduce design times and complexity. Actually, the number of direct problems (FEM simulations) to be solved in order to reach good function approximations by RSM is a key aspect of their application in sheet metal forming operations design. In this way, the possibility to build response surfaces basing on moving least squares approximations (MLS) by utilizing a moving and zooming region of interest can be considered a very attractive methodology. In this paper, MLS is utilized to solve two optimization problems for sheet metal forming processes. The influence on the optimization results was analyzed basing on MLS peculiarities. The idea is to utilize these peculiarities and make the MLS approximation as flexible as possible in order to reduce the computational effort of an optimization strategy. An innovative optimization method is proposed and the results show it is possible to strongly reduce the computational effort of sheet metal forming processes optimization. In particular, the advantages, in terms of computational effort reduction, with respect to classical RSM approaches have been demonstrated and quantified.     

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.     

A multivariate mean square error optimization of AISI 52100 hardened steel turning

Hardened steel turning has received special attention in recent years due to its many applications in modern industries. The characteristics that define its machinability—expressed in terms of multiple response problems—are usually represented by experimental model building strategies like response surface methodology (RSM). Such strategies, however, have a particular drawback when multiple correlated regression functions are present. The optimization of multiple process characteristics without considering the variance–covariance structure among the responses may lead to an inadequate optimum. To deal with this constraint, this paper presents a novel multiobjective optimization method; it correctly focuses the multiple correlated characteristics of the AISI 52100 hardened steel, based on the concept of multivariate mean square error. This novel approach combines principal component analysis with RSM focusing a multidimensional nominal-the-best problem. In this kind of optimization, all the characteristics (tool life, cutting time, cost, material removal rate, and surface roughness) have a specific target while maintaining a strong correlation structure. Transforming the original responses and respective targets to the plane of a multivariate principal component scores, an optimization routine is capable of finding out a compromise solution that attends all the established targets. The following AISI 52100 turning process variables were considered in this study: cutting speed, feed rate, and depth of cut. Theoretical and experimental results were convergent and confirmed in a case study.     

Real options approach for a mixed-model assembly line sequencing problem

Mixed-model assembly lines are widely used in manufacturing. This can be attributed to increased product variety and potential just-in-time (JIT) benefits obtained by applying mixed-model assembly lines. Because of market demand volatility, the flexibility of such a line is increasingly becoming more important and, consequently, determining an accurate sequence is becoming more complex. In this paper, first, we use the real options approach to evaluate one specific type of flexibility, i.e., product-mix flexibility. This methodology is applied to determine the products’ quantity that must be satisfied by the mixed-model assembly line. Then, in order to determine a desired sequence, we consider three objectives simultaneously: (1) total utility work cost, (2) total production rate variation cost, and (3) total set-up cost. A nonlinear zero–one model is developed for the problem whose objective function is a weighted sum of the above-mentioned objectives. Moreover, two efficient metaheuristics, i.e., a genetic algorithm (GA) and a memetic algorithm (MA), are proposed. These solution methods are compared with the optimal solution method using Lingo 6 software over a set of randomly generated test problems. The computational results reveal that the proposed memetic algorithm performs better than the proposed genetic algorithm.     

A response surface based sequential approximate optimization using constraint-shifting analogy

When a traditional response surface method (RSM) is used as a meta-model for inequality constraint functions, an approximate optimal solution is sometimes actually infeasible in a case where it is active at the constraint boundary. The paper proposes a new RSM that ensures the constraint feasibility with respect to an approximate optimal solution. Constraint-shifting is suggested in order to secure the constraint feasibility during the sequential approximate optimization process. A central composite design is used as a tool for design of experiments. The proposed approach is verified through a mathematical function problem and engineering optimization problems to support the proposed strategies.     

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.     

A new algorithm for optimum tolerance allocation of complex assemblies with alternative processes selection

Allocating tolerance to sub-components of a complex assembly with alternative processes selection by using Lagrange’s multiplier method is tedious as well as difficult. The present work is aims to solve the problem with simple effort in three stages. In the first stage, the maximum of two processes are selected from the alternative processes of each component and these two processes correspond to the smaller sum of difference in manufacturing cost. A hybrid optimum tolerance allocation method is developed in a second and third stage by combining Tabu search (TS) and heuristic approach. Application of the proposed algorithm is demonstrated on complex tolerancing products like knuckle joint and wheel mounting assembly. For the same manufacturing conditions, compared with tolerance synthesis by Singh method, the proposed method saved nearly $74,880 and $479,520, respectively, per year in manufacturing costs of knuckle joint and wheel mounting assembly.     

基于宽容分层序列法的飞机装配公差稳健设计技术

飞机制造中由于工序能力指数和公差等设计变量存在变差,可能导致无法满足飞机装配质量要求以及制造成本波动较大的问题。运用稳健设计方法建立了飞机装配公差的可行稳健性和敏感稳健性两类设计模型。可行稳健设计考虑了公差等设计变量对装配可行性的影响,使其变差不影响装配功能的实现;敏感稳健性设计则考虑了公差等设计变量对制造成本和装配质量的影响,在目标成本较小的前提下使得目标成本和装配质量受设计变量变差的影响最小。提出了针对飞机装配公差可行敏感稳健设计的多目标优化问题的宽容分层序列求解算法。将公差设计中制造成本、装配质量波动、制造成本波动多个设计目标按照重要性依次排序。首先求解成本最小情况下的公差一般优化解,然后在成本最小值的宽容约束下,求得具有装配质量波动最小值的公差,最后在上述两个优化目标的宽容约束下,求得成本波动最小的最优公差。应用实例和分析结果表明了方法的有效性。       

Concurrent tolerance design for manufacture and assembly with a game theoretic approach

This paper proposes a game theoretic approach for concurrent tolerance design considering the practical manufacturing process and assembly process, to achieve lower manufacturing cost and good product assemblability. The optimization objectives and constraints for concurrent tolerance design for manufacture and assembly are discussed, and the key technologies for concurrent tolerance design with game theory are investigated, including the strategy set division, payoff calculation, and Nash equilibrium evolution. An optimization algorithm is proposed to achieve concurrent tolerance design with a game theoretic approach, and the genetic algorithm is used to conclude the Nash equilibrium of the game for manufacture and assembly. Finally, a case study is given to verify the proposed approach.     

Multi-objective optimization for optimum tolerance synthesis with process and machine selection using a genetic algorithm

This paper presents a new approach to the tolerance synthesis of the component parts of assemblies by simultaneously optimizing three manufacturing parameters: manufacturing cost, including tolerance cost and quality loss cost; machining time; and machine overhead/idle time cost. A methodology has been developed using the genetic algorithm technique to solve this multi-objective optimization problem. The effectiveness of the proposed methodology has been demonstrated by solving a wheel mounting assembly problem consisting of five components, two subassemblies, two critical dimensions, two functional tolerances, and eight operations. Significant cost saving can be achieved by employing this methodology.