An interval approach to robust design with parameter uncertainty

In robust design, it is common to estimate empirical models that relate an output response variable to controllable input variables and uncontrollable noise variables from experimental data. However, when determining the optimal input settings that minimise output variability, parameter uncertainties in noise factors and response models are typically neglected. This article presents an interval robust design approach that takes parameter uncertainties into account through the confidence regions for these unknown parameters. To avoid obtaining an overly conservative design, the worst and best cases of mean squared error are both adopted to build an optimisation approach. The midpoint and radius of the interval are used to measure the location and dispersion performances, respectively. Meanwhile, a data-driven method is applied to obtain the relative weights of the location and dispersion performances in the optimisation approach. A simulation example and a case study using automobile manufacturing data from the dimensional tolerance design process are used to demonstrate the effectiveness of the proposed approach. The proposed approach of considering both uncertainties is shown to perform better than other approaches.     

Life-cycle reliability-based robust design optimization for GP model with response uncertainty

Reliability-based robust design optimization (RBRDO) is a crucial tool for life-cycle quality improvement. Gaussian process (GP) model is an effective alternative modeling technique that is widely used in robust parameter design. However, there are few studies to deal with reliability-based design problems by using GP model. This article proposes a novel life-cycle RBRDO approach concerning response uncertainty under the framework of GP modeling technique. First, the hyperparameters of GP model are estimated by using the Gibbs sampling procedure. Second, the expected partial derivative expression is derived based on GP modeling technique. Moreover, a novel failure risk cost function is constructed to assess the life-cycle reliability. Then, the quality loss function and confidence interval are constructed by simulated outputs to evaluate the robustness of optimal settings and response uncertainty, respectively. Finally, an optimization model integrating failure risk cost function, quality loss function, and confidence interval analysis approach is constructed to find reasonable optimal input settings. Two case studies are given to illustrate the performance of the proposed approach. The results show that the proposed approach can make better trade-offs between the quality characteristics and reliability requirements by considering response uncertainty.     

An investigation of noise factor effects in parameter design

Parameter design aims to determine nominal values of a set of design parameters so as to minimize variability in one or more key performance measures, in the presence of uncertainties in the design parameters, whilst maintaining the required nominal (design point) performance and the overall design concept. A theoretical analysis of some aspects of parameter design and of some related approximate methods is carried out and the results studied with the aim of determining the effect of changes in the magnitudes of the uncertainties in the design parameters on the resulting values of nominal parameter values in the optimum design. It is shown that there is reason to expect that, allowing for the effects of constraints, a given parameter design would be robust against overall changes, by a constant factor, in the parameter uncertainties but not in the presence of changes in individual parameter uncertainties, or by numerous such changes when the change factor varies between individual parameters. Some numerical results are obtained for problems in beam design and are shown to support the above assertions. The results imply that nominal values in a previously determined robust design may continue to be valid when parameter tolerances change, provided such changes are by the same factor for all free design parameters, i.e. those not fixed on a constraint boundary. Copyright © 2002 John Wiley & Sons, Ltd.     

Economic tolerance design for folded normal data

An important issue for design engineers is how to assign tolerance limits economically. Most work related to tolerance design is for nominal-is-best (N-type) quality characteristics and restricted by a normality assumption. However, smaller-is-better (S-type) quality characteristics and larger-is-better (L-type) quality characteristics are common in real applications. The practical distributions for S-type data or L-type data are typically skewed, and the normality assumption is violated. Determining tolerance with non-normal data using methodologies based on a normality assumption is not appropriate. This study considers the case in which measurements are recorded without their algebraic signs. The folded normal distribution works well to fit these absolute data. Based on the statistical properties of the folded normal distribution, this study develops an economic model encompassing quality loss, manufacturing costs, and re-work costs to determine tolerances. By minimising total costs, a procedure based on the Newton-Raphson method is utilised to obtain the optimal solution. Finally, a welding machine experiment is carried out to demonstrate the applicability of the proposed model.     

Economic tolerance design for quality

This paper provides a few general mathematical models for determining product tolerances which minimize the combined manufacturing costs and quality loss. The models contain quality cost with a quadratic loss function and represent manufacturing costs with geometrical decay functions. The models are also formulated with multiple variables which represent the set of characteristics in a part. Applications of these models include minimizing the total cost with effective tolerance allocation in product design.     

Sequential optimization of parameter and tolerance design for multiple dynamic quality characteristics

System design, parameter design and tolerance design are the three stages of design process as presented by G. Taguchi. Systems design identifies the basic elements of the design to provide new or improved products to customers. Parameter design determines the optimal parameter settings, which will minimize variation from the target performance of the product. Tolerance design finally identifies the components of the design, which are sensitive in terms of affecting the quality of the product, and establishes tolerance limits that will give the required level of variation in the design. Most studies have focused primarily on optimizing the parameter design or tolerance design for multiple static quality characteristics. In this paper, a mathematical formula corresponding to the model is derived from Taguchi's quadratic quality loss function to minimize the expected total cost for the parameter design of multiple dynamic quality characteristics. When the optimal parameter design is not sufficient to reduce the output variation, the first-order Taylor series expansion is then used to analyse the variations of noise factors for optimizing the tolerance design. It concludes with an example demonstrating this approach.     

Economic considerations on parameter design

A simultaneous consideration of process mean and variance in product design stages has been considered one of the most significant of Taguchi's contributions. Among his quality improvement methods, parameter design has drawn a great deal of particular attention from researchers. The ultimate objective of Taguchi's parameter design is to find control factor settings to achieve an on‐target process mean with a minimum variance. There is no doubt regarding the virtue of the minimum variance. However, considering a variety of economic aspects related to product specifications as well as a quality loss, the on‐target process mean may not necessarily be economical. This paper investigates the parameter design problem from an economic point of view and proposes an alternative procedure to achieve the most economical process mean as well as the minimum variance by taking product specifications and an asymmetric quality loss into consideration. It is shown through an illustrative example that a significant cost saving can be accrued from the proposed procedure. Copyright © 2000 John Wiley & Sons, Ltd.     

Optimization of structures under material parameter uncertainty using evidence theory

An evidence-based approach is developed for optimization of structural components under material parameter uncertainty. The approach is applied to evidence-based design optimization (EBDO) of externally stiffened circular tubes under axial impact load using an isotropic–elastic–plastic plasticity model to simulate dynamic material behaviour. Uncertainty modelling considers the changes in material parameters that are caused by variability in material properties as well as incertitude and errors in experimental data and procedure to determine the material parameters. Spatial variation of material parameters across the structural component is modelled using a field joint belief structure and propagated for the calculation of evidence-based objective function and design constraints. Surrogate models are used in both uncertainty propagation and solution of the optimization problem. The methodology and the solution to the EBDO example problem are presented and discussed.     

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.     

An efficient metamodeling approach for uncertainty quantification of complex systems with arbitrary parameter probability distributions

This paper proposes an efficient metamodeling approach for uncertainty quantification of complex system based on Gaussian process model (GPM). The proposed GPM‐based method is able to efficiently and accurately calculate the mean and variance of model outputs with uncertain parameters specified by arbitrary probability distributions. Because of the use of GPM, the closed form expressions of mean and variance can be derived by decomposing high‐dimensional integrals into one‐dimensional integrals. This paper details on how to efficiently compute the one‐dimensional integrals. When the parameters are either uniformly or normally distributed, the one‐dimensional integrals can be analytically evaluated, while when parameters do not follow normal or uniform distributions, this paper adopts the effective Gaussian quadrature technique for the fast computation of the one‐dimensional integrals. As a result, the developed GPM method is able to calculate mean and variance of model outputs in an efficient manner independent of parameter distributions. The proposed GPM method is applied to a collection of examples. And its accuracy and efficiency is compared with Monte Carlo simulation, which is used as benchmark solution. Results show that the proposed GPM method is feasible and reliable for efficient uncertainty quantification of complex systems in terms of the computational accuracy and efficiency. Copyright © 2016 John Wiley & Sons, Ltd.     

Economic parameter design for ultra-fast laser micro-drilling process

The basic requirement in this type of micro-drilling process is to achieve high product quality with the minimum machining cost, which can be realised through parameter design. In this paper, we propose a new economic parameter design under the framework of Bayesian modelling and optimisation. First of all, the Bayesian seemingly unrelated regression (SUR) models are utilised to develop the relationship models between input factors and output responses in the laser micro-drilling process. After that, simulated response values which reflect the real laser micro-drilling process are obtained by using the Gibbs sampling procedure. Moreover, a novel rejection cost function and a quality loss function are constructed based on the simulated responses. Finally, an optimisation scheme integrating the rejection cost (i.e. rework cost and scrap cost) function and the quality loss function is implemented by using multi-objective genetic algorithm to find feasible economic parameter settings for laser micro-drilling process.     

A multi-objective optimization model for hub network design under uncertainty: An inexact rough-interval fuzzy approach

This article proposes a multi-objective mixed-integer model to optimize the location of hubs within a hub network design problem under uncertainty. The considered objectives include minimizing the maximum accumulated travel time, minimizing the total costs including transportation, fuel consumption and greenhouse emissions costs, and finally maximizing the minimum service reliability. In the proposed model, it is assumed that for connecting two nodes, there are several types of arc in which their capacity, transportation mode, travel time, and transportation and construction costs are different. Moreover, in this model, determining the capacity of the hubs is part of the decision-making procedure and balancing requirements are imposed on the network. To solve the model, a hybrid solution approach is utilized based on inexact programming, interval-valued fuzzy programming and rough interval programming. Furthermore, a hybrid multi-objective metaheuristic algorithm, namely multi-objective invasive weed optimization (MOIWO), is developed for the given problem. Finally, various computational experiments are carried out to assess the proposed model and solution approaches.     

An inductive design exploration method for hierarchical systems design under uncertainty

Model structure uncertainty, originating from assumptions and idealizations in modelling processes, is a form of uncertainty that is often hard to quantify. In this article, the authors propose and demonstrate a method, the inductive design exploration method (IDEM), which facilitates robust design in the presence of model structure uncertainty. The approach in this method is achieving robustness by compromising between the degree of system performance and the degree of reliability based on structure uncertainty associated with system models (i.e. models for performances and constraints). The main strategies in the IDEM include: (i) identifying feasible ranged sets of design space instead of single (or optimized) design solution, considering all types of quantifiable uncertainties and (ii) systematically compromising target achievement with provision for potential uncertainty. The IDEM is successfully demonstrated in a clay-filled polyethylene cantilever beam design example, which is a simple but representative example of integrated materials and product design problems.     

A new interval optimization method considering tolerance design

This study considers the design variable uncertainty in the actual manufacturing process for a product or structure and proposes a new interval optimization method based on tolerance design, which can provide not only an optimal design but also the allowable maximal manufacturing errors that the design can bear. The design variables' manufacturing errors are depicted using the interval method, and an interval optimization model for the structure is constructed. A dimensionless design tolerance index is defined to describe the overall uncertainty of all design variables, and by combining the nominal objective function, a deterministic two-objective optimization model is built. The possibility degree of interval is used to represent the reliability of the constraints under uncertainty, through which the model is transformed to a deterministic optimization problem. Three numerical examples are investigated to verify the effectiveness of the present method.     

新安江模型参数有效优化及不确定性评估

应用新安江模型进行水文模拟时,由于模型本身的不足及参数多、信息量少等原因,会出现率定的最优参数组不唯一、不稳定等问题。考虑到以往的参数优选,都只得出一个参数组,不能反映出其不确定性状况。提出应用基于马尔可夫链蒙特卡罗(MCMC)理论的SCEM-UA算法,通过双牌流域以1 h为时段间隔的36场典型洪水数据对新安江模型参数进行优选和不确定性评估。结果表明,该算法能很好地推出新安江模型参数的后验概率分布;率定和检验结果分析也表明,应用SCEM-UA算法对新安江模型进行优选和不确定评估是有效和可行的。     

Optimization of engineering tolerance design using revised loss functions

Engineering tolerance design plays an important role in modern manufacturing. Both symmetric and asymmetric tolerances are common in many manufacturing processes. Recently, various revised loss functions have been proposed for overcoming the drawbacks of Taguchi's loss function. In this article, Kapur's economic tolerance design model is modified and the economic specification limits for both symmetric and asymmetric losses are established. Three different loss functions are compared in the optimal symmetric and asymmetric tolerance design: a revised Taguchi quadratic loss function, an inverted normal loss function and a revised inverted normal loss function. The relationships among the three loss functions and process capability indices are established. A numerical example is given to compare the economic specification limits established by using the three loss functions. The results suggest that the revised inverted normal loss function be used in determining economic specification limits.     

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.     

An Integrative Loss Function Approach to Multi‐Response Optimization

Loss function approach is effective for multi‐response optimization. However, previous loss function approaches ignore the dispersion performance of squared error loss and model uncertainty. In this paper, a weighted loss function is proposed to simultaneously consider the location and dispersion performances of squared error loss to optimize correlated multiple responses with model uncertainty. We propose an approach to minimize the weighted loss function under the constraint that the confidence intervals of future predictions for the multiple responses should be contained in specification limits of the responses. An example is illustrated to verify the effectiveness of the proposed method. The results show that the proposed method can achieve reliable optimal operating condition under model uncertainty. Copyright © 2013 John Wiley & Sons, Ltd.     

Bivariate tolerance design for lock wheels by considering quality loss

In the design of tolerance allocation the cost–tolerance function is usually employed to represent the objective function which is to be minimized. The traditional cost–tolerance functions in the literature are concerned with only one characteristic. In this paper we obtain a bivariate cost–tolerance function to describe the relationship between the cost and tolerances of two characteristics (i.e. the thickness and inner diameter) of a lock wheel. Then the bivariate loss function is combined with the bivariate cost–tolerance function to determine the optimal tolerances for the thickness and inner diameter of a lock wheel such that the user's potential loss/cost may be evaluated. When the quality loss is considered, the tolerances of both characteristics become tighter. By including the effect of product degradation, the present work of expected bivariate quality loss is then introduced as a quality performance measure. By assuming linear drifts on both the thickness and inner diameter of the lock wheels, the model with the present worth of quality loss leads to tighter tolerances of both characteristics. In addition, a longer planning horizon (or a longer useful life of the product) leads to tighter tolerances and a larger user's discount rate results in looser tolerances for both characteristics. Copyright © 2000 John Wiley & Sons, Ltd.     

Development of a sequential optimization procedure for robust design and tolerance design within a bi-objective paradigm

Many practitioners and researchers have implemented robust design and tolerance design as quality improvement and process optimization tools for more than two decades. Robust design is an enhanced process/product design methodology for determining the best settings of control factors while minimizing process bias and variability. Tolerance design is aimed at determining the best tolerance limits for minimizing the total cost incurred by both the customer and manufacturer by balancing quality loss due to variations in product performance and the cost of controlling these variations. Although robust design and tolerance design have received much attention from researchers and practitioners, there is ample room for improvement. First, most researchers consider robust design and tolerance design as separate research fields. Second, most research work is based on a single quality characteristic. The primary goal of this paper is to integrate a sequential robust design–tolerance design optimization procedure within a bi-objective paradigm, which, the authors believe, is the first attempt in the robust design and tolerance design literature. Models are proposed and numerical examples along with sensitivity analysis are performed for verification purposes.