Robust tolerance design by computer experiment

Tolerance design affects the quality and cost of a product cycle time. Most of the literature on tolerance design problems has focused on developing exact methods to minimize manufacturing cost or quality loss. The inherent assumption in this approach is that the assembly function is known before a tolerance design problem is analysed. With the current development in CAD (Computer-Aided Design) software, design engineers can proceed with the tolerance design problems, without knowing assembly functions in advance. In this study, the Monte Carlo simulation is employed using VSA-3D/Pro software to obtain experimental data. Then the design of experiments (DOE) approach is adopted for data analysis in order to select critical components for cost reduction and quality improvement. Implementing the discussed computer experiments, a tolerance design analysis which improves quality and reduces cost can be performed for any complex assembly via computer during the early stage of design.     

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.     

Robust product development for multiple quality characteristics using computer experiments and an optimization technique

Conventional parameter or tolerance designs focus on developing exact methods to minimize quality loss or manufacturing cost. The inherent assumption is that the response functions which represent the link between controllable variables and response values of quality characteristics are known before a design is developed. Moreover, parameter and tolerance values are assumed to be independent controllable variables in previous works; namely, they are determined separately in design activities. Currently, advanced computer software, such as computer-aided engineering, can help engineers to handle design problems with unknown response functions, at the stage of product design and process planning. Therefore, in this study, the software ANSYS was employed to obtain simulation data which represent the response values of quality characteristics. These response values will be used to fit a set of response functions for later analysis. However, previous works in computer simulation for design and planning usually lack consideration of the noise impact from an external design system. To approximate a realistic design environment, various levels of controllable variables, in conjunction with artificial noises created from uncontrollable variables, are used to generate simulated data for statistical analysis via Response Surface Methodology (RSM). Then, an optimization technique, such as mathematical programming, is adopted to integrate these response functions into one formulation so that optimal parameter and tolerance values are concurrently determined, with multiple quality characteristics taken into consideration. A bike-frame design was used to demonstrate the presented approach, followed by multiple quality characteristics of interest: material cost, bike-frame weight, structure reliability, and rigidity dependability. The goal is to minimize material cost and bike frame weight and to maximize structure reliability and rigidity dependability. This approach is useful for solving any complex design problems in the early stages, while providing enhanced functionality, quality, economic benefits, and a shorter design cycle.     

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.     

Robust computer-aided parameter and tolerance determination for an electronic circuit design

Conventionally, parameter design precedes tolerance design in the course of product design or process planning. To lower the production costs, as well as to improve quality, this study proposes the simultaneous determination of parameter and tolerance values when designing an electronic circuit. With the current development of CAD (Computer-Aided Design) software for electronic circuit design, engineers can determine parameter and tolerance values without providing transfer functions for circuit analysis. In this study, a computer experiment is performed by using CAD software (PSpice) to obtain outputs that will be converted into the total cost, which includes the quality loss, the tolerance cost and the failure cost. Then, Response Surface Methodology (RSM) is employed to minimize the total cost and to find the optimal parameter and tolerance values statistically. Consequently, a parameter and tolerance design for quality improvement and cost reduction can be achieved for any complex electronic circuit during the early stages of design.     

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.     

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.     

Nonconvex optimization of desirability functions

Desirability functions (DFs) are commonly used in optimization of design parameters with multiple quality characteristic to obtain a good compromise among predicted response models obtained from experimental designs. Besides discussing multi-objective approaches for optimization of DFs, we present a brief review of literature about most commonly used Derringer and Suich type of DFs and others as well as their capabilities and limitations. Optimization of DFs of Derringer and Suich is a challenging problem. Although they have an advantageous shape over other DFs, their nonsmooth nature is a drawback. Commercially available software products used by quality engineers usually do optimization of these functions by derivative free search methods on the design domain (such as Design-Expert), which involves the risk of not finding the global optimum in a reasonable time. Use of gradient-based methods (as in MINITAB) after smoothing nondifferentiable points is also proposed as well as different metaheuristics and interactive multi-objective approaches, which have their own drawbacks. In this study, by utilizing a reformulation on DFs, it is shown that the nonsmooth optimization problem becomes a nonconvex mixed-integer nonlinear problem. Then, a continuous relaxation of this problem can be solved with nonconvex and global optimization approaches supported by widely available software programs. We demonstrate our findings on two well-known examples from the quality engineering literature and their extensions.     

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 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.     

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     

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.     

Graphical and computer-aided approaches to plan experiments

Design of experiments is a quality technology to achieve product excellence, that is to achieve high quality at low cost. It is a tool to optimize product and process designs, to accelerate the development cycle, to reduce development costs, to improve the transition of products from R & D to manufacturing and to troubleshoot manufacturing problems effectively. It has been successfully, but sporadically, used in the United States. More recently, it has been identified as a major technological reason for the success of Japan in producing high-quality products at low cost. In the United States, the need for increased competitiveness and the emphasis on quality improvement demands a widespread use of design of experiments by engineers, scientists and quality professionals. In the past, such widespread use has been hampered by a lack of proper training and a lack of availability of tools to easily implement design of experiments in industry. Three steps are essential, and are being taken, to change this situation dramatically. First, simple graphical methods, to design and analyse experiments, need to be developed, particularly when the necessary microcomputer resources are not available. Secondly, engineers, scientists and quality professionals must have access to microcomputer-based software for design and analysis of experiments.¹ Availability of such software would allow users to concentrate on the important scientific and engineering aspects of the problem by computerizing the necessary statistical expertise. Finally, since a majority of the current workforce is expected to be working in the year 2000, a massive training effort, based upon simple graphical methods and appropriate computer software, is necessary.² The purpose of this paper is to describe a methodology based upon a new graphical method called interaction graphs and other previously known techniques, to simplify the correct design of practically important fractional factorial experiments. The essential problem in designing a fractional factorial experiment is first stated. The interaction graph for a 16-trial fractional factorial design is given to illustrate how the graphical procedure can be easily used to design a two-level fractional factorial experiment. Other previously known techniques are described to easily modify the two-level fractional factorial designs to create mixed multi-level designs. Interaction graphs for other practically useful fractional factorial designs are provided. A computer package called CADE (computer aided design of experiments), which automatically generates the appropriate fractional factorial designs based upon user specifications of factors, levels and interactions and conducts complete analyses of the designed experiments is briefly described.¹ Finally, the graphical method is compared with other available methods for designing fractional factorial experiments.     

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.     

Combined parameter and tolerance design optimization with quality and cost

The need to remain competitive for survival in the current world market has led manufacturing sectors to consider the low cost and high quality of product and process design. Production of quality products at low cost in today's manufacturing industry requires simultaneous consideration of product design and process planning, particularly in the early stage of design and planning. During product design, parameter design determines the design target (design setting) and tolerance design determines design tolerance. During process design, the parameter design determines the process mean (process setting) and tolerance design determines process tolerance. This study provides a mathematical relationship to link the elements of design target, design tolerance, process mean and process tolerance in one equation. By following this equation, manufacturability for all possible combinations of product and process design can be ensured, which increases the flexibility of both product design and process planning. With this in mind, an analysis model that includes manufacturing cost and quality loss simultaneously has been developed to determine the optimal values of design tolerance, process mean and process tolerance. The proposed model provides a method of combining the optimization of parameter and tolerance design over product/process in the early stage of design.     

TOLERANCE SYNTHESIS FOR NONLINEAR SYSTEMS BASED ON NONLINEAR PROGRAMMING

Automatic assignment of tolerances to dimensioned mechanical assemblies is studied as an optimization problem; the objective of which is to minimize the (manufacturing) cost, subject to the constraints of (design) functionality and (assembly) interchange-ability. By associating a nominal dimension and a tolerance to the variance, a probabilistic approach is adopted.

Trigonometric functions relating the component geometries give rise to the nonlinearity in the system. Estimating an n-dimensional nonlinear integral by a polytope converts the probabilistic optimization formulation to a deterministic one. It also allows rapid evaluation of tolerance analysis embedded in tolerance synthesis.

Local optimality is ensured by analysis of convexity and quasi-concavity of the objective function and some of the constraints. Sensitivity analysis is performed to provide search directions for global optimality. An implementation is reported with an example.     

Statistical tolerance synthesis using distribution function zones

Tolerance is one of the most important parameters in design and manufacturing. Tolerance synthesis has a significant impact on manufacturing cost and product quality. In the international standards community two approaches for statistical tolerancing of mechanical parts are being discussed: process capability indices and distribution function zone. The distribution function zone (DFZone) approach defines the acceptability of a population of parts by requiring that the distribution function of relevant values of the parts be bounded by a pair of specified distribution functions. In order to apply this approach to statistical tolerancing, one needs a method to decompose the assembly level tolerance specification to obtain tolerance parameters for each component in conjunction with a corresponding tolerance-cost model. This paper introduces an optimization-based statistical tolerance synthesis model based on the DFZone tolerance specifications. A new tolerance-cost model is proposed and the model is illustrated with an assembly example.     

Modeling and implementation of a neurofuzzy system for surface mount assembly defect prediction and control

A high-speed surface mount assembly can reduce both production cost and time; however, it could allow an enormous number of boards to be built before a problem is detected. Therefore, early detection and assessment of a surface mount assembly problem is critical for cost-effective manufacturing. This paper proposes a neurofuzzy system for surface mount assembly defect prediction and control. Hybrid data from both in-process quality control database and from a fractional factorial experimental design are collected for neurofuzzy learning and modeling. Customized programming codes are generated for rule retrieval and for graphical user interface modeling. The proposed system is successfully implemented at a surface mount assembly plant, ll significantly improves plant throughput by the downtime reduction that is a result of a better defect prediction and control.     

Optimal robust and tolerance design for computer experiments with mixture proportion inputs

Computer experiments often have inputs that are proportions/fractions of components in a mixture. In these mixture computer experiments, it can be of interest to perform robust and tolerance design on the mixture proportions since the proportions are subjected to noise variations. Traditionally, manufacturing of mixture products is controlled via interval tolerances for mixture amounts. In this paper, an optimal tolerance region for proportions, which gives optimal quality cost among all possible tolerance regions for mixture proportions with the same acceptance probability, is proposed for integrated parameter and tolerance design in mixture computer experiments. Real examples are given to demonstrate the improvements that can be achieved with the optimal tolerance region.