Robust optimal-parameter design approach for tolerance design problems

A robust optimal-parameter design method, henceforth called the TGAOA, to solve tolerance design problems has been investigated. Tolerance affects system performance and leads to violation of design constraints. The TGAOA approach conducts global exploration by using a genetic algorithm and exploits optimal offspring via the Taguchi method. It is able to effectively reduce the impact of parameter variations in reaching robust optimal-solutions as allowed by the tolerance. Two design examples are employed to evaluate the performance of the new method. The first is for a mixed H₂/H_∞ optimal PID controller under varying PID component specifications, plant uncertainty, and other external unknown disturbance. The second involves a 13-variable test function, which includes quadratic, linear, and polynomial forms to illustrate the general robustness and computational efficiency, for which comparisons are also made with its predecessors of genetic algorithm and hybrid Taguchi-genetic algorithm.     

Robust design of a polysilicon deposition process using split-plot analysis

The technique used for the analysis of experimental data must be appropriate for the design and treatment structures of the experiment; failure to take this into account can produce misleading results. This paper illustrates how split-plot designs can be used for the analysis of robust design experiments. In particular, the polysilicon deposition process data presented by Phadke¹ is analysed, and comparisons are made between the split-plot analysis of the raw data and the analysis conducted using signal-to-noise ratios. In addition, we demonstrate how the split-plot analysis provides information about interactions between control and noise factors, and how interaction plots can be used to assess the performance of the control factors across the levels of the noise factors. This information is particularly important to select the settings of the control factors that minimize the variation in the response induced by the noise factors.     

基于模拟试验法的离散公差稳健设计

 在分析计算机辅助公差设计技术的研究现状的基础上，提出了基于模拟试验法的离散公差稳健设计方法．在该方法中将加工成本与质量损失作为两个独立的目标函数，从而建立离散公差优化设计模型，采用试验设计法和CP方法相结合的技术实现公差的稳健设计，最后用实例证明该方法的可行性．     

Robust parameter design using a supersaturated design for a response surface model

Recently, the application of response surface methodology (RSM) to robust parameter design has attracted a great deal of attention. In some cases, experiments are very expensive and may require a great deal of time to perform. Central composite designs (CCDs) and Box and Behnken designs (BBDs), which are commonly used for RSM, may lead to an unacceptably large number of experimental runs. In this paper, a supersaturated design for RSM is constructed and its application to robust parameter design is proposed. A response surface model is fitted using data from the designed experiment and a stepwise variable selection. An illustrative example is presented to show that the proposed method considerably reduces the number of experimental runs, as compared with CCDs and BBDs. Numerical experiments are also conducted in which type I and II error rates are evaluated. The results imply that the proposed method may be effective for finding the effects (i.e. main effects, two‐factor interactions, and pure quadratic effects) of active factors under the ‘effect sparsity’ assumption. Copyright © 2010 John Wiley & Sons, Ltd.     

Robust design using a hybrid-cellular-evolutionary and interval-arithmetic approach: a reliability application

Sensitivity analysis is a technique by which one can determine, with good approximation, whether a system will work within the specification limits when the input vary between their limits.Although this type of analysis, from input to output, can provide useful information to the decision-maker, we present an approach based on the use of Cellular Evolutionary Strategies and Interval Arithmetic in which the inverse problem can be solved. That is, we move from output to input: starting with output range specifications, we infer the maximum uncertainty or variability that can be exhibited by the input parameters. The approach used is an indirect method based on optimisation instead of a direct method based on mapping from the output into the input space. The proposed approach is illustrated by computational examples applied to a reliability complex system. Results are compared with those obtained using a general non-linear optimisation package.     

Optimization strategies in robust engineering design and computer-aided design

This paper reviews the fundamental ideas involved in robust engineering design (RED), and how they relate to computer-aided design. There are several areas of RED that may be successfully resolved by the use of statistical methods or ideas. This paper gives a general overview of several popular statistical strategies in RED and discusses how these strategies approach the statistical problems involved.     

Robust design of a spindle motor: a case study

Robust design is the process of minimising variability of products and processes in order to improve their quality and reliability. In recent years, it has seen wide spread application in many fields such as mechanical [J Quality Technol 24 (1992) 22], chemical [Quality Engng 9 (1997) 391] and civil [Quality Engng 9 (1997) 441] engineering. As an additional application, this paper investigates the use of robust design techniques during early design process, in order to predict, analyse and improve quality and reliability of spindle motors found within hard disks. The spindle motor is an integral component of the hard disk that rotates the recording media. Constant speed rotation is an absolute necessity for accurate reading and recording of information. However, it is difficult to achieve this due to the effects of cogging torque. In this paper, the robust design of a spindle motor is attempted. The motor was modelled using multiple regression analysis. A dual response approach [J Quality Technol 22 (1990) 38] was then used. This performed a constrained minimisation of the mean of the cogging response, the constraint being that the variance be within a specified limit. The results showed that the current operating point was near optimal. However, the redesign was still able to achieve a 4% and 25% reduction in the mean and the variance of the cogging torque, respectively, before actual prototypes were made.     

Sculptured surface tolerance verification with design datums

Sculptured or free-form surfaces are widely used in many fields with extensive applications. Once such surfaces are manufactured, surface inspection compares the manufactured surfaces with the surface design specifications to verify conformance. Although significant research and development efforts have been devoted to the design and manufacturing of products consisting of partial or sole free-form surfaces, the inspection of these surfaces is still a difficult task. For many engineering applications, a free-form surface is assigned a profile tolerance with reference to design datums for assembly, functionality and other manufacturing requirements. The paper discusses developments of surface inspection techniques for profile tolerance of free-form surfaces. The concept of datum direction frame is proposed to find the transformation information that localizes measurement data to design model. The technique consists of two major steps: localization of measurement data to the design system, based on the datum reference information; and further localization based on the information from free-form surfaces. Testing examples were carried out to validate the developed techniques. The new method does not need corresponding points from the datums of the design model and measured surfaces. Therefore, it is simpler, yet more robust. It can also be used conveniently in manufacturing processes.     

Tolerance design: Choosing optimal tolerance specifications in the design of machined parts

This paper provides a general approach for determining tolerances for machined parts which minimizes the combined machining cost and quality cost. We derive a mathematical model which takes into consideration both machining cost and quality cost, which depend on tolerance specifications and the distribution of the machined part dimension produced by a chosen machining process. The model represents the quality cost with a loss function and represents the machining cost with a fitted curve based on some original data. The distribution of the machined part characteristic is taken to be either normal or triangular, depending on model assumptions. By applying this model, designers can determine the tolerance which provides the minimum expected total cost.     

Robust design optimization of nonlinear energy sink under random system parameters

Generally, in designing nonlinear energy sink (NES), only uncertainties in the ground motion parameters are considered and the unconditional expected mean of the performance metric is minimized. However, such an approach has two major limitations. First, ignoring the uncertainties in the system parameters can result in an inefficient design of the NES. Second, only minimizing the unconditional mean of the performance metric may result in large variance of the response because of the uncertainties in the system parameters. To address these issues, we focus on robust design optimization (RDO) of NES under uncertain system and hazard parameters. The RDO is solved as a bi-objective optimization problem where the mean and the standard deviation of the performance metric are simultaneously minimized. This bi-objective optimization problem has been converted into a single objective problem by using the weighted sum method. However, solving an RDO problem can be computationally expensive. We thus used a novel machine learning technique, referred to as the hybrid polynomial correlated function expansion (H-PCFE), for solving the RDO problem in an efficient manner. Moreover, we adopt an adaptive framework where H-PCFE models trained at previous iterations are reused and hence, the computational cost is less. We illustrate that H-PCFE is computationally efficient and accurate as compared to other similar methods available in the literature. A numerical study showcasing the importance of incorporating the uncertain system parameters into the optimization procedure is shown. Using the same example, we also illustrate the importance of solving an RDO problem for NES design. Overall, considering the uncertainties in the parameters have resulted in a more efficient design. Determining NES parameters by solving an RDO problem results in a less sensitive design.     

Optimization of product and process design for quality and cost

Robust product and process design is an important technique for achieving high quality at low cost. It involves making the product's function much less sensitive to various sources of noise such as manufacturing variation, environmental variation and deterioration. This is a problem in optimization involving minimization of the mean square loss resulting from the deviation of the product's function from its target. Here we show that the optimization can be carried out in two steps: first maximize a quantity called signal-to-noise ratio (S/N) and then bring the performance on target by special adjustment parameters. The two-step procedure works for a wide variety of product functions and makes the optimization process more efficient and practical compared to the direct minimization of the quadratic loss function.     

Robust design with dynamic characteristics using stochastic sequential quadratic programming

Robust design with dynamic characteristics is an important off-line quality engineering technique for improving product quality over a range of input conditions by reducing variations caused by uncontrolled factors. Since several studies have indicated that there are important limitations to Taguchi's S/N ratio analysis, the solution procedure for dynamic systems deserves further investigation. This paper proposes a stochastic optimization modeling procedure to overcome the difficulty in Taguchi's method to accommodate dynamic characteristics. The main idea underlying the proposed method is to minimize the total variations on quality characteristics while attaining the target performance over a range of input conditions. Due to the nonlinear nature of the stochastic optimization model, two stochastic versions of sequential quadratic programming respectively embedded with a Monte Carlo simulation and numerical approximations are devised to solve the problem. In the robust design of a temperature control circuit often discussed in dynamic problems, the proposed method performs efficiently and effectively. Compared with the Taguchi method, the design solved in this paper has smaller variations, indicating that the proposed method is a promising technique for dynamic-characteristic robust design.     

Stochastic Subset Optimization for optimal reliability problems

Reliability-based design of a system often requires the minimization of the probability of system failure over the admissible space for the design variables. For complex systems this probability can rarely be evaluated analytically and so it is often calculated using stochastic simulation techniques, which involve an unavoidable estimation error and significant computational cost. These features make efficient reliability-based optimal design a challenging task. A new method called Stochastic Subset Optimization (SSO) is proposed here for iteratively identifying sub-regions for the optimal design variables within the original design space. An augmented reliability problem is formulated where the design variables are artificially considered as uncertain and Markov Chain Monte Carlo techniques are implemented in order to simulate samples of them that lead to system failure. In each iteration, a set with high likelihood of containing the optimal design parameters is identified using a single reliability analysis. Statistical properties for the identification and stopping criteria for the iterative approach are discussed. For problems that are characterized by small sensitivity around the optimal design choice, a combination of SSO with other optimization algorithms is proposed for enhanced overall efficiency.     

A robust cellular manufacturing system design for dynamic part population using a genetic algorithm

As changing conditions prevail in the manufacturing environment, the design of cellular manufacturing systems, which involves the formation of part families and machine cells, is difficult. This is due to the fact that the machines need to be relocated as per the requirements if adaptive designs are used. This paper presents a new approach (robust design) for forming part families and machine cells, which can handle all the changes in demands and product mixes without any relocations. This method suggests fixed machine cells for the dynamic nature of the production environment by considering a multi-period forecast of product mix and demand. A genetic algorithm based solution procedure is adopted to solve the problem. The results thus obtained were compared with the adaptive design proposed by Wicks and Reasor (1999 Wicks, EM and Reasor, RJ. 1999. Designing cellular manufacturing systems with dynamic part populations. IIE Trans., 31: 11–20. [Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]). It is found that the robust design performs better than the adaptive design for the problems attempted.     

Robust design employing a genetic algorithm

The minimization of variability in a key design feature or performance measure, in the presence of variability in the realized values of design parameters, is discussed and an analytic solution for quadratic performance measures is provided. Solutions are based on the determination of optimum nominal (or design point) values for the design parameters, subject to constraints in the form of a given nominal performance at the design point and limits on the nominal values of the design parameters, which preserve the design concept. The more general, numerical, problem solution is addressed and a previously described deterministic procedure which generated multiple local optima is improved by the replacement of a simplex search method with a sophisticated genetic algorithm which, with suitable parameter values and choice of Lagrange multiplier, converges only to the required global minimum within the specified design parameter limits. Further improvements are discussed. Copyright © 2000 John Wiley & Sons, Ltd.     

Reliability-based optimal design using sample average approximations

An algorithm for reliability-based optimal design is developed using sampling techniques for estimating the failure probability. The algorithm applies a new method for sensitivity calculations of the failure probability. Initially, the estimates of the failure probability are coarse. As the algorithm progresses towards an optimal design, the number of sample points is increased in an adaptive way leading to better estimates of the failure probability. The algorithm is proven to converge to an optimal design. The applicability of the algorithm is shown in an example from the area of highway bridge design.     

Robust design using Bayesian Monte Carlo

In this paper, we propose an efficient strategy for robust design based on Bayesian Monte Carlo simulation. Robust design is formulated as a multiobjective problem to allow explicit trade‐off between the mean performance and variability. The proposed method is applied to a compressor blade design in the presence of manufacturing uncertainty. Process capability data are utilized in conjunction with a parametric geometry model for manufacturing uncertainty quantification. High‐fidelity computational fluid dynamics simulations are used to evaluate the aerodynamic performance of the compressor blade. A probabilistic analysis for estimating the effect of manufacturing variations on the aerodynamic performance of the blade is performed and a case for the application of robust design is established. The proposed approach is applied to robust design of compressor blades and a selected design from the final Pareto set is compared with an optimal design obtained by minimizing the nominal performance. The selected robust blade has substantial improvement in robustness against manufacturing variations in comparison with the deterministic optimal blade. Significant savings in computational effort using the proposed method are also illustrated. Copyright © 2007 John Wiley & Sons, Ltd.     

Robust design of an interior hard trim to improve occupant safety in a vehicle crash

Head injury ranks among the top contributors in automobile accidents. Consequently, although styling is treated important, safety of occupants in a crash receives preemptive priority in the design of automotive interior components. Additionally, the Federal Motor Vehicle Safety Standard (FMVSS) 201 has laid down certain requirements to be fulfilled by automobile manufacturers for producing a safe vehicle. One of the requirements stipulate dummy equivalent of the Head Injury Criteria, i.e. HIC(d) value for the interior components of a vehicle to be below 1000 under certain stated conditions. In this paper, we provide a robust design approach to achieve the requirements for one such interior component, viz. an interior hard trim that covers the pillar closest to the driver's head on the left-hand side of the vehicle.     

A method for the preliminary design of gears using a reduced number of American Gear Manufacturers Association (AGMA) correction factors

Typically, the preliminary design of mechanical components such as gears is carried out using standardized design processes such as those developed by the American Gear Manufacturers Association (AGMA). These design standards include a large number of ‘correction factors’ to account for various uncertainties. As the knowledge about these uncertainties increases, it becomes possible to include them systematically in the design procedure, thereby reducing the number of empirical correction factors. Robust design provides a way to design in the presence of various uncertainties. In this article, a design method is proposed to eliminate empirical correction factors and is demonstrated by eliminating two correction factors from the AGMA design standards for a spur gear, namely, the factor of safety in contact and the reliability factor by the formal introduction of uncertainty in the magnitude of load and material properties. The proposed method is illustrated with the design of an automotive gear with desired reliability, cost and robustness.