ROBUST PARAMETER DESIGN METHODOLOGY FOR MIXTURE PROCESS

This book on experimental design is somewhat similar in flavor to the classic book by G.E

The purpose of this article is to provide a survey of the various stages in the development of response surface methodology (RSM). The coverage of these stages is organized in three parts that describe the evolution of RSM since its introduction in the early 1950s. Part I covers the period, 1951–1975, during which the so‐called classical RSM was developed. This includes a review of basic experimental designs for fitting linear response surface models, in addition to a description of methods for the determination of optimum operating conditions. Part II, which covers the period, 1976–1999, discusses more recent modeling techniques in RSM, in addition to a coverage of Taguchi's robust parameter design and its response surface alternative approach. Part III provides a coverage of further extensions and research directions in modern RSM. This includes discussions concerning response surface models with random effects, generalized linear models, and graphical techniques for comparing response surface designs. Copyright © 2010 John Wiley & Sons, Inc.This article is categorized under:Statistical Models > Linear ModelsStatistical Models > Nonlinear Models

Mixture experiments in a robust parameter design (RPD) setting are commonly encountered in industry. The objective of a mixture-process RPD problem is to find levels for the controllable variables that result in a product that has a desirable mean response and that has small variability in the response transmitted from the noise variables. We consider statistical designs for experiments involving mixture variables, continuous noise variables, and categorical noise variables. We show how to construct designs for the RPD problem in the case of a mixture-process experiment with both continuous and categorical noise variables that simultaneously optimize the scaled prediction variances for the mean model and the slope model.

Dual response surface methodology (DRSM) and nonparametric methodology (NPM) are main approaches used to achieve robust parameter design (RPD) of industrial processes and products. When the relationship between influential input factors and output quality characteristic of a process is very complex, both approaches have their limitations. For DRSM, it fails to fit the real response surfaces of process mean and variance by using the second order polynomial models. For NPM, it is hard to optimize parameters of fitting equation, and it needs more experiments as well. From a machine learning perspective, this paper generalizes RPD as a restricted active learning problem and proposes a new approach to achieve it. It fits process mean and variance responses by support vector machines (SVM), and then optimizes levels of design parameters by genetic algorithm. In order to reduce experiment times, the influence of priori knowledge on generalized error of fitting model is studied. Then a prior knowledge based experiment design is developed. Moreover, the approach selects the form of kernel function and optimizes parameters in SVM by comparing the upper bounds of generalized error of different SVM models without extra samples. The example given in the paper shows that, the generalized error and the experiment times of the approach decrease by no less than 45% and 39% respectively, compared with traditional approaches. All these results demonstrate the adaptability and superiority of the approach proposed in the paper.

1. In this study, neural network (NN) and response surface (RS) models were developed to investigate the response [average daily gain (ADG) and feed efficiency (FE)] of young broiler chickens to dietary protein and lysine. For this purpose, data on their responses to dietary protein and lysine were extracted from the literature and separate NN and RS models were constructed. 2. Comparison between the NN and RS models revealed higher accuracy of prediction with the NN models compared to the RS models. In terms of R 2 values, the NN models developed for both ADG (R 2 = 0.923) and FE (R 2 = 0.904) were far superior to the RS models (R 2 for ADG = 0.511; R 2 for FE = 0.67). This suggests that the NN models can serve as an alternative option to conventional regression approaches including use of RS models. 3. Optimisation of the NN models developed for response to protein and lysine showed that diets containing 220.7 (g/kg of diet) protein and 12.85 (g/kg of diet) lysine maximise ADG, whereas maximum FE is achieved with diets containing 241.3 and 13.12 (g/kg) protein and lysine, respectively. Based on the optimisation results, optimal dietary protein and lysine concentrations for maximum FE in broiler chickens during the starting period are higher than for ADG.

Purpose: For more than 30 years, robust parameter design (RPD), which attempts to minimize the process bias (i.e., deviation between the mean and the target) and its variability simultaneously, has received consistent attention from researchers in academia and industry. Based on Taguchi’s philosophy, a number of RPD methodologies have been developed to improve the quality of products and processes. The primary purpose of this paper is to review and discuss existing RPD methodologies in terms of the three sequential RPD procedures of experimental design, parameter estimation, and optimization.Methods: This literature study composes three review aspects including experimental design, estimation modeling, and optimization methods.Results: To analyze the benefits and weaknesses of conventional RPD methods and investigate the requirements of future research, we first analyze a variety of experimental formats associated with input control and noise factors, output responses and replication, and estimation approaches. Secondly, existing estimation methods are categorized according to their implementation of least-squares, maximum likelihood estimation, generalized linear models, Bayesian techniques, or the response surface methodology. Thirdly, optimization models for single and multiple responses problems are analyzed within their historical and functional framework.Conclusion: This study identifies the current RPD foundations and unresolved problems, including ample discussion of further directions of study.

SummaryRobust parameter design is an effective methodology for reducing variance and improving the quality of a product and a process. Recent work has mainly concentrated on two‐level robust parameter designs. We consider general robust parameter designs with factors having two or more or mixed levels these levels being either qualitative or quantitative. We propose a methodology and develop a generalised minimum aberration optimality criterion for selecting optimal robust parameter designs. A catalogue of 18‐run optimal designs is constructed and tabulated.

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.

In recent years, the demand for plastic products has increased, and along with the deepening of academics, mass production, weight reduction, and high precision are progressing. In the fields of design development and production technology, there are many issues related to quality assurance such as molding defects and product strength. In particular, in the resin molding process, there is a high degree of freedom in product shape and mold structure, and it is an important issue to create quality functions that apply analysis of complex multidimensional information. In this study, the important factors of the resin molding process related to the optimization of resin strength are extracted by applying the multivariate analysis method and robust parameter design. As a result of verification of the proposed method, it is clarified that uniformization of the resin filling density in the mold is extremely important for stabilizing the resin strength.

Response surface methods have been used for a variety of applications in aerospace engineering, particularly in multidisciplinary design optimization. We investigate the use of kriging models as alternatives to traditional second-order polynomial response surfaces for constructing global approximations for use in a real aerospace engineering application, namely, the design of an aerospike nozzle. Our objective is to examine the difeculties in building and using kriging models to create accurate global approximations to facilitate multidisciplinary design optimization. Error analysis of the response surface and kriging models is performed, along with a graphical comparison of the approximations. Four optimization problems are also formulated and solved using both sets of approximation models to gain insight into their use for multidisciplinary design optimization. We end that the kriging models, which use only a constant “global” model and a Gaussian correlation function, yield global approximations that are slightly more accurate than the response surface models.

ABSTRACTThe response surface-based robust parameter design, with its extensive use of optimization techniques and statistical tools, is known as an effective engineering design methodology for improving production processes, when input variables are quantitative on a continuous scale. In many engineering settings, however, there are situations where both qualitative and quantitative variables are considered. In such situations, traditional response surface designs may not be effective. To rectify this problem, this article lays out a foundation by embedding those input variables into a factorial design with pseudo-centre points. A 0–1 mixed-integer nonlinear programming model is then developed and the solutions found using three optimization tools, namely the outer approximation method, the branch-and-bound technique and the hybrid branch-and-cut algorithm, are compared with traditional counterparts. The numerical example shows that the proposed models result in better robust parameter design solutions than the traditional models.

A new procedure for the optimization of aerodynamic properties of trains is presented, where simple response surface (RS) models are used as a basis for optimization instead of a large number of evaluations of the Navier—Stokes solver. The suggested optimization strategy is demonstrated in two flow optimization cases: optimization of the train's front for the crosswind stability and optimization of vortex generators (VGs) for the purpose of drag reduction. Besides finding the global minimum for each aerodynamic objective, a strategy for finding a set of optimal solutions is demonstrated. This is based on the use of genetic algorithms on RS models. The resulting Pareto-optimal solutions are used to explore the extreme designs and find trade-offs between design objectives. For the optimization of VGs, three different RS models are used: polynomial functions, radial basis neural networks (RBNN), and RBNN-enhanced polynomial RSs. The three approaches produce different results, and the combination of RBNN and polynomial functions in the last approach is found to be the best as it enables the construction of high-order polynomial functions, and the model's fit with the data is the best.

As products/processes often possess several quality characteristics,Robust Parameter Design (RPD) problems are likely to be dealt with by considering multiple responses. Several researches have proposed solutions to this problem by using loss function. A weighted loss function approach to make a good trade-off between bias and variation components of the objective function built on these researches is suggested in this paper. In addition, an algorithm of the weight parameter is also proposed, which can be used when the prior information is vague. An example from the literature is given to illustrate the proposed method, and a numerical comparative analysis of the results from different articles is also provided.

Design of systems based on computer simulations is prevalent. An important idea to improve design quality, called robust parameter design (RPD), is to optimize control factors based on the expectation of a loss function so that the design is robust to noise factor variations. When computer simulations are time consuming, optimizing the simulator based on a Gaussian process (GP) emulator for the response is a computationally efficient approach. For this purpose, acquisition functions (AFs) are used to sequentially determine the next design point so that the GP emulator can more accurately locate the optimal setting. Despite this, few articles consider AFs for positive definite quadratic forms such as the expected quadratic loss (EQL) function, which is the standard expected loss function for RPD with nominally-the-best responses. This paper proposes new AFs for optimizing the EQL, analyzes their convergence, and develops quick and accurate methods based on the characteristic function of the EQL to compute them. We apply the AFs to RPD problems with internal noise factors based on a GP model and an initial design tailored for such problems. Numerical results indicate that all four AFs considered have similar performance, and they outperform an optimization approach based on modeling the quadratic loss as a GP and maximin Latin hypercube designs. MATLAB codes for reproducing reported results are given in the online supplement.

7 Robust design: Experiments for improving quality