Corrigenda

Supersaturated designs and associated analysis methods have been proposed by several authors to identify active factors in situations in which only a very limited number of experimental runs is available. We use simulation to evaluate the abilities of the existing methods to achieve model identification–related objectives. The results motivate a new class of supersaturated designs, derived from simulation optimization, that maximize the probability that stepwise regression will identify the important main effects. Because the proposed designs depend on specific assumptions, we also investigate the sensitivity of the performances of the alternative supersaturated designs to these assumptions.     

An Efficient Adaptive Sequential Methodology for Expensive Response Surface Optimization

The preset response surface methodology (RSM) designs are commonly used in a wide range of process and design optimization applications. Although they offer ease of implementation and good performance, they are not sufficiently adaptive to reduce the required number of experiments and thus are not cost effective for applications with high cost of experimentation. We propose an efficient adaptive sequential methodology based on optimal design and experiments ranking for response surface optimization (O‐ASRSM) for industrial experiments with high experimentation cost, limited experimental resources, and requiring high design optimization performance. The proposed approach combines the concepts from optimal design of experiments, nonlinear optimization, and RSM. By using the information gained from the previous experiments, O‐ASRSM designs the subsequent experiment by simultaneously reducing the region of interest and by identifying factor combinations for new experiments. Given a given response target, O‐ASRSM identifies the input factor combination in less number of experiments than the classical single‐shot RSM designs. We conducted extensive simulated experiments involving quadratic and nonlinear response functions. The results show that the O‐ASRSM method outperforms the popular central composite design, the Box–Behnken design, and the optimal designs and is competitive with other sequential response surface methods in the literature. Furthermore, results indicate that O‐ASRSM's performance is robust with respect to the increasing number of factors. Copyright © 2012 John Wiley & Sons, Ltd.     

响应曲面建模的稳健M-回归方法

响应曲面方法是生产过程改进和优化的一种非常有效的方法。在传统的响应曲面模型的建立过程中,通常假定随机误差服从正态分布且相互独立具有相同的方差。但是实际生产中随机误差的方差并不是完全相同,观测值中会存在异常点,这就需要稳健的估计方法来抑制异常点对模型估计的影响。为了降低异常点对响应曲面模型最优值的影响,针对响应曲面方法中的中心复合设计,〖JP2〗充分考虑到不同实验设计位置上可能出现异常点的情况,对稳健M 回归方法：Huber 估计、Tukey 估计和Welsch 估计进行了理论比较研究。研究结果表明Welsch和Tukey 估计能有效改善异常点对响应曲面模型最优值的影响,消弱异常点对中心复合设计的干扰。通过一个来自化工方面的案例,计算了中心复合设计不同位置存在异常点与不存在异常点时,响应曲面模型的最优值,对比分析得出当异常点与响应均值的偏离程度较大时（10倍标准差）,稳健M 估计尤其是Welsch和Tukey 估计显著提高响应曲面建模的稳健性。     

Response surface optimization for a nonlinearly constrained irregular experimental design space

Finding optimum conditions for process factors in an engineering optimization problem with response surface functions requires structured data collection using experimental design. When the experimental design space is constrained owing to external factors, its design space may form an asymmetrical and irregular shape and thus standard experimental design methods become ineffective. Computer-generated optimal designs, such as D-optimal designs, provide alternatives. While several iterative exchange algorithms for D-optimal designs are available for a linearly constrained irregular design space, it has not been clearly understood how D-optimal design points need to be generated when the design space is nonlinearly constrained and how response surface models are optimized. This article proposes an algorithm for generating the D-optimal design points that satisfy both feasibility and optimality conditions by using piecewise linear functions on the design space. The D-optimality-based response surface design models are proposed and optimization procedures are then analysed.     

Adapting second‐order response surface designs to specific needs

Experimental design strategies most often involve an initial choice of a classic factorial or response surface design and adapt that design to meet restrictions or unique requirements of the system under study. One such experience is described here, in which the objective was to develop an efficient experimental design strategy that would facilitate building second‐order response models with excellent prediction capabilities. In development, careful consideration was paid to the desirable properties of response surface designs. Once developed, the proposed design was evaluated using Monte Carlo simulation to prove the concept, a pilot implementation of the design carried out to evaluate the accuracy of the response models, and a set of validation runs enacted to look for potential weaknesses in the approach. The purpose of the exercise was to develop a procedure to efficiently and effectively calibrate strain‐gauge balances to be used in wind tunnel testing. The current calibration testing procedure is based on a time‐intensive one‐factor‐at‐a‐time method. In this study, response surface methods were used to reduce the number of calibration runs required during the labor‐intensive heavy load calibration, to leverage the prediction capabilities of response surface designs, and to provide an estimate of uncertainty for the calibration models. Results of the three‐phased approach for design evaluation are presented. The new calibration process will require significantly fewer tests to achieve the same or improved levels of precision in balance calibration. Copyright © 2008 John Wiley & Sons, Ltd.     

Central Composite Experimental Designs for Multiple Responses With Different Models

ABSTRACT

Central composite designs (CCDs) are widely accepted and used experimental designs for fitting second-order polynomial models in response surface methods. However, these designs are based only on the number of explanatory variables being investigated. In a multiresponse problem where prior information is available in the form of a screening experiment or previous process knowledge, investigators often know which factors will be used in the estimation of each response. This work presents an alternative design based on CCDs that allows main effects to be aliased for factors that are not related to the same response. This results in fewer required runs than current designs, saving investigators both time and money, by taking this prior information into account. R-package “DoE.multi.response” is included as a supplement for constructing these designs.     

Variance plus bias optimal response surface designs with qualitative factors applied to stem choice modeling

This paper explores the issue of model misspecification, or bias, in the context of response surface design problems involving quantitative and qualitative factors. New designs are proposed specifically to address bias and compared with five types of alternatives ranging from types of composite to D‐optimal designs using four criteria including D‐efficiency and measured accuracy on test problems. Findings include that certain designs from the literature are expected to cause prediction errors that practitioners would likely find unacceptable. A case study relating to the selection of science, technology, engineering, or mathematics majors by college students confirms that the expected substantial improvements in prediction accuracy using the proposed designs can be realized in relevant situations. Copyright © 2011 John Wiley & Sons, Ltd.     

ASRSM: A Sequential Experimental Design for Response Surface Optimization

Most preset response surface methodology (RSM) designs offer ease of implementation and good performance over a wide range of process and design optimization applications. These designs often lack the ability to adapt the design on the basis of the characteristics of application and experimental space so as to reduce the number of experiments necessary. Hence, they are not cost‐effective for applications where the cost of experimentation is high or when the experimentation resources are limited. In this paper, we present an adaptive sequential response surface methodology (ASRSM) for industrial experiments with high experimentation cost, limited experimental resources, and high design optimization performance requirement. The proposed approach is a sequential adaptive experimentation approach that combines concepts from nonlinear optimization, design of experiments, and response surface optimization. The ASRSM uses the information gained from the previous experiments to design the subsequent experiment by simultaneously reducing the region of interest and identifying factor combinations for new experiments. Its major advantage is the experimentation efficiency such that for a given response target, it identifies the input factor combination (or containing region) in less number of experiments than the classical single‐shot RSM designs. Through extensive simulated experiments and real‐world case studies, we show that the proposed ASRSM method outperforms the popular central composite design method and compares favorably with optimal designs. Copyright © 2012 John Wiley & Sons, Ltd.     

Studies on the Effects of Estimator Selection in Robust Parameter Design under Asymmetric Conditions

The primary goal of robust parameter design (RPD) is to determine the optimum operating conditions that achieve process performance targets while minimizing variability in the results. To achieve this goal, typical approaches to RPD problems use ordinary least squares methods to obtain response functions for the mean and variance by assuming that the experimental data follow a normal distribution and are relatively free of contaminants or outliers. Consequently, the most common estimators used in the initial tier of estimation are the sample mean and sample variance, as they are very good estimators when these assumptions hold. However, it is often the case that such assumed conditions do not exist in practice; notably, that inherent asymmetry pervades system outputs. If unaccounted for, such conditions can affect results tremendously by causing the quality of the estimates obtained using the sample mean and standard deviation to deteriorate. Focusing on asymmetric conditions, this paper examines several highly efficient estimators as alternatives to the sample mean and standard deviation. We then incorporate these estimators into RPD modeling and optimization approaches to ascertain which estimators tend to yield better solutions when skewness exists. Monte Carlo simulation and numerical studies are used to substantiate and compare the performance of the proposed methods with the traditional approach. Copyright © 2012 John Wiley & Sons, Ltd.     

Cost‐constrained G‐efficient Response Surface Designs for Cuboidal Regions

In many industrial experiments there are restrictions on the resource (or cost) required for performing the runs in a response surface design. This will require practitioners to choose some subset of the candidate set of experimental runs. The appropriate selection of design points under resource constraints is an important aspect of multi‐factor experimentation. A well‐planned experiment should consist of factor‐level combinations selected such that the resulting design will have desirable statistical properties but the resource constraints should not be violated or the experimental cost should be minimized. The resulting designs are referred to as cost‐efficient designs. We use a genetic algorithm for constructing cost‐constrained G‐efficient second‐order response surface designs over cuboidal regions when an experimental cost at a certain factor level is high and a resource constraint exists. Consideration of practical resource (or cost) restrictions and different cost structures will provide valuable information for planning effective and economical experiments when optimizing statistical design properties. Copyright © 2005 John Wiley & Sons, Ltd.     

Partial Duplication of Response Surface Designs

Continuing the author's earlier work [6] a method is described which requires that certain experimental runs of a central composite, second-order, response surface design be repeated, thereby providing a more general estimate of the experimental error, at the same time providing more reliable estimates of the effects.

The partial duplication of the factorial portion as well as the partial duplication of the star portion has been considered and described. The response surface designs with the star portion duplicated seem to have more potential than the designs with their factorial portions duplicated or partially duplicated.     

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.     

Partition experimental designs for sequential processes: Part II—second‐order models

Second‐order experimental designs are employed when an experimenter wishes to fit a second‐order model to account for response curvature over the region of interest. Partition designs are utilized when the output quality or performance characteristics of a product depend not only on the effect of the factors in the current process, but the effects of factors from preceding processes. Standard experimental design methods are often difficult to apply to several sequential processes. We present an approach to building second‐order response models for sequential processes with several design factors and multiple responses. The proposed design expands current experimental designs to incorporate two processes into one partitioned design. Potential advantages include a reduction in the time required to execute the experiment, a decrease in the number of experimental runs, and improved understanding of the process variables and their influence on the responses. Copyright © 2002 John Wiley & Sons, Ltd.     

Small,balanced, efficient,optimal, and near rotatable response surface designs for factorial experiments asymmetrical in some quantitative,qualitative factors

We present a structured method of constructing cost-efficient response surface designs (RSDs) as compared to the replicated central composite designs (RCCDs), that are useful for modelling and optimization of the experiments asymmetric in some qualitative, quantitative factors with at least two unrestricted quantitative factors while the remaining take two or three levels. We demonstrate the method by designing various experimental situations in 3 to 6 factors, and the analysis competency of the RSDs by analysing a published data set from an optimization experiment. The structural and prediction properties make our RSDs a good alternative to the known RSDs.     

Batch Sample Design from Databases for Logistic Regression

The prevalence of large observational databases offers potential for identifying predictive relationships among variables of interest, although observational data are generally far less informative and less reliable than experimental data. We consider the problem of selecting a subset of records from a large observational database, for the purpose of designing a small but powerful experiment involving the selected records. It is assumed that the database contains the predictor variables but is missing the response variable, and that the purpose is to fit a logistic regression model after the response is obtained via the experiment. Active learning methods, which treat a similar problem, usually select records sequentially and focus on the single objective of classification accuracy. In contrast, many emerging applications require batch sample designs and have a variety of objectives that may include classification accuracy or accuracy of the estimated parameters, the latter being more in line with the optimal design of experiments (DOE) paradigm. The aim of this paper is to explore batch sampling from databases from a DOE perspective, particularly regarding the configuration, performance, and robustness of the designs that result from the different criteria. Through extensive simulation, we show that DOE‐based batch sampling methods can substantially outperform random sampling and the entropy method that is popular in active learning. We also provide insight and guidelines for selecting appropriate design criteria and modeling assumptions. Copyright © 2016 John Wiley & Sons, Ltd.     

Construction of mixed-level supersaturated designs and comparison of their performance: Application to a gas chromatographic method

A supersaturated design is a design for which there are fewer runs than effects to be estimated. Although two-level supersaturated designs are becoming increasingly popular, mixed-level designs are scarcely used. Mixed-level designs are needed when the response is based on a polynomial response surface model or in situations where factors are nominal variables (with more than two modalities). The aim of this study is to explore the construction of mixed-level supersaturated designs and to evaluate their performance from the analysis of peppermint oil using a gas chromatographic method as application. This experimental setup requires the study of seven factors at two levels and five factors at three levels. Different building methods are tested from asymmetric or symmetric supersaturated designs. The mixed-level supersaturated designs obtained are compared from the point of view of a priori criteria with the aim of evaluating which criteria are better suited to judge the quality and fitness for purpose of these experimental designs. Finally, the results of the supersaturated designs are compared to the complete classical design.     

A comparison of designs for one‐step screening and response surface estimation

The sequential design approach to response surface exploration is often viewed as advantageous as it provides the opportunity to learn from each successive experiment with the ultimate goal of determining optimum operating conditions for the system or process under study. Recent literature has explored factor screening and response surface optimization using only one three‐level design to handle situations where conducting multiple experiments is prohibitive. The most straightforward and accessible analysis strategy for such designs is to first perform a main‐effects only analysis to screen important factors before projecting the design onto these factors to conduct response surface exploration. This article proposes the use of optimal designs with minimal aliasing (MA designs) and demonstrates that they are more effective at screening important factors than the existing designs recommended for single‐design response surface exploration. For comparison purposes, we construct 27‐run MA designs with up to 13 factors and demonstrate their utility using established design criterion and a simulation study. Copyright 2011 © John Wiley & Sons, Ltd.     

Nonhierarchical multi‐model fusion using spatial random processes

New model fusion techniques based on spatial‐random‐process modeling are developed in this work for combining multi‐fidelity data from simulations and experiments. Existing works in multi‐fidelity modeling generally assume a hierarchical structure in which the levels of fidelity of the simulation models can be clearly ranked. In contrast, we consider the nonhierarchical situation in which one wishes to incorporate multiple models whose levels of fidelity are unknown or cannot be differentiated (e.g., if the fidelity of the models changes over the input domain). We propose three new nonhierarchical multi‐model fusion approaches with different assumptions or structures regarding the relationships between the simulation models and physical observations. One approach models the true response as a weighted sum of the multiple simulation models and a single discrepancy function. The other two approaches model the true response as the sum of one simulation model and a corresponding discrepancy function, and differ in their assumptions regarding the statistical behavior of the discrepancy functions, such as independence with the true response or a common spatial correlation function. The proposed approaches are compared via numerical examples and a real engineering application. Furthermore, the effectiveness and relative merits of the different approaches are discussed. Copyright © 2015 John Wiley & Sons, Ltd.     

No‐confounding designs with 20 runs—Alternatives to resolution IV screening designs

When experimental resources are significantly constrained, resolution V fractional factorial designs are often prohibitively large for experiments with 6 or more factors. Resolution IV designs may also be cost prohibitive, as additional experimentation may be required to de‐alias active 2‐factor interactions (2FI). This paper introduces 20‐run no‐confounding screening designs for 6 to 12 factors as alternatives to resolution IV designs. No‐confounding designs have orthogonal main effects, and since no 2FI is completely confounded with another main effects or 2FI, the experimental results can be analyzed without follow‐on experimentation. The paper concludes with the results of a Monte Carlo simulation used to assess the model‐fitting accuracy of the recommended designs.     

The Impact of Response Measurement Error on the Analysis of Designed Experiments

This article considers the analysis of designed experiments when there is measurement error in the true response or so‐called response measurement error. We consider both additive and multiplicative response measurement errors. Through a simulation study, we investigate the impact of ignoring the response measurement error in the analysis, that is, by using a standard analysis based on t‐tests. In addition, we examine the role of repeat measurements in improving the quality of estimation and prediction in the presence of response measurement error. We also study a Bayesian approach that accounts for the response measurement error directly through the specification of the model, and allows including additional information about variability in the analysis. We consider the impact on power, prediction, and optimization. Copyright © 2015 John Wiley & Sons, Ltd.