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.     

Three‐level and mixed‐level orthogonal arrays for lean designs

Orthogonal arrays (OA's) are widely used in design of experiments. Each OA has a specific number of rows that is fixed by the number of factors in the OA and the number of levels in each factor. In a practical application of an industrial experiment, however, because of various operational constraints it could happen that the number of runs of the experiment cannot be set exactly equal to the number of rows of an OA. In this case, a lean design can be used. A lean design is obtained by removing some specific rows and columns from the extended design matrix formed from an OA, so that the resulting sub‐matrix still allows efficient estimation of the effects of some of the factors. Tables for 2‐level lean designs are already available in the literature. In this paper, the authors will investigate 3‐level lean designs and mixed‐level lean designs, and construct tables for such designs for convenient use. Copyright © 2009 John Wiley & Sons, Ltd.     

On the Analysis of Balanced Two‐Level Factorial Whole‐Plot Saturated Split‐Plot Designs

This paper considers an experimentation strategy when resource constraints permit only a single design replicate per time interval and one or more design variables are hard to change. The experimental designs considered are two‐level full‐factorial or fractional‐factorial designs run as balanced split plots. These designs are common in practice and appropriate for fitting a main‐effects‐plus‐interactions model, while minimizing the number of times the whole‐plot treatment combination is changed. Depending on the postulated model, single replicates of these designs can result in the inability to estimate error at the whole‐plot level, suggesting that formal statistical hypothesis testing on the whole‐plot effects is not possible. We refer to these designs as balanced two‐level whole‐plot saturated split‐plot designs. In this paper, we show that, for these designs, it is appropriate to use ordinary least squares to analyze the subplot factor effects at the ‘intermittent’ stage of the experiments (i.e., after a single design replicate is run); however, formal inference on the whole‐plot effects may or may not be possible at this point. We exploit the sensitivity of ordinary least squares in detecting whole‐plot effects in a split‐plot design and propose a data‐based strategy for determining whether to run an additional replicate following the intermittent analysis or whether to simply reduce the model at the whole‐plot level to facilitate testing. The performance of the proposed strategy is assessed using Monte Carlo simulation. The method is then illustrated using wind tunnel test data obtained from a NASCAR Winston Cup Chevrolet Monte Carlo stock car. Copyright © 2012 John Wiley & Sons, Ltd.     

Low‐cost response surface methods from simulation optimization

We propose ‘low‐cost response surface methods’ (LCRSMs) that typically require half the experimental runs of standard response surface methods based on central composite and Box Behnken designs, but yield comparable or lower modeling errors under realistic assumptions. In addition, the LCRSMs have substantially lower modeling errors and greater expected savings compared with alternatives with comparable numbers of runs, including small composite designs and computer‐generated designs based on popular criteria such as D‐optimality. The LCRSM procedures appear to be the first experimental design methods derived as the solution to a simulation optimization problem. Together with modern computers, simulation optimization offers unprecedented opportunities for applying clear, realistic multicriterion objectives and assumptions to produce useful experimental design methods. We compare the proposed LCRSMs with alternatives based on six criteria. We conclude that the proposed methods offer attractive alternatives when the experimenter is considering dropping factors to use standard response surface methods or would like to perform relatively few runs and stop with a second‐order model. Copyright © 2002 John Wiley & Sons, Ltd.     

Fraction of Design Space Plots for Split‐plot Designs

In industrial experiments, restrictions on the execution of the experimental runs or the existence of one or more hard‐to‐change factors often leads to split‐plot experiments, where there are two types of experimental units and two independent randomizations. The resulting compound symmetric error structure, as well as the settings of whole‐plot and subplot factors, play important roles in the performance of split‐plot experiments. When the practitioner is interested in predicting the response, a response surface design for a second‐order model such as a central composite design (CCD) is often used. The prediction variance of second‐order designs under a split‐plot error structure is often of interest. In this paper, fraction of design space (FDS) plots are adapted to split‐plot designs. In addition to the global curve exploring the entire design space, sliced curves at various whole‐plot levels are presented to study prediction performance for subregions in the design space. The different sizes of the constrained subregions are accounted for by the proportional size of the sliced curves. The construction and use of the FDS plots are demonstrated through two examples of the restricted CCD in split‐plot schemes. We also consider the impact of the variance ratio on design performance. Copyright © 2006 John Wiley & Sons, Ltd.     

Design of control charts with m‐of‐m runs rules

This paper investigates economic–statistical properties of the X? charts supplemented with m‐of‐m runs rules. An out‐of‐control condition for the chart is either a point beyond a control limit or a run of m‐of‐m successive points beyond a warning limit. The sampling process is modeled by a Markov chain with 2m states. The steady‐state probability for each state and the average run length (ARL) from each state of the Markov chain are derived in explicit formulas. Then the stationary average run length (SALR) is derived so as to develop an economic–statistical model. Using this model, the design parameters are optimized by minimizing the cost function with constraints on the average time to signal (ATS). The X? chart supplemented with m‐of‐m runs rules is compared with the Shewhart X? chart in terms of the SARL and the cost function. Sensitivity of the design parameters with respect to the cost function is also analyzed. General guidelines for implementing the X? chart with m‐of‐m runs rules are presented from those observations. It should be emphasized that supplementing run rules may provide feasible and efficient solutions even if the sample size is limited, while the Shewhart X? chart may not. Copyright © 2009 John Wiley & Sons, Ltd.     

Analysis and efficient 2k − 1 designs for experiments in blocks of size two

Two‐level factorial designs in blocks of size two are useful in a variety of experimental settings, including microarray experiments. Replication is typically used to allow estimation of the relevant effects, but when the number of factors is large this common practice can result in designs with a prohibitively large number of runs. One alternative is to use a design with fewer runs that allows estimation of both main effects and two‐factor interactions. Such designs are available in full factorial experiments, though they may still require a great many runs. In this article, we develop fractional factorial design in blocks of size two when the number of factors is less than nine, using just half of the runs needed for the designs given by Kerr (J Qual. Tech. 2006; 38 :309–318). Two approaches, the orthogonal array approach and the generator approach, are utilized to construct our designs. Analysis of the resulting experimental data from the suggested design is also given. Copyright © 2011 John Wiley & Sons, Ltd.     

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.     

Considerations for D‐Optimal Sequential Design

Classical D‐optimal design is used to create experimental designs for situations in which an underlying system model is known or assumed known. The D‐optimal strategy can also be used to add additional experimental runs to an existing design. This paper demonstrates a study of variable choices related to sequential D‐optimal design and how those choices influence the D‐efficiency of the resulting complete design. The variables studied are total sample size, initial experimental design size, step size, whether or not to include center points in the initial design, and complexity of initial model assumption. The results indicate that increasing total sample size improves the D‐efficiency of the design, less effort should be placed in the initial design, especially when the true underlying system model isn't known, and it is better to start off with assuming a simpler model form, rather than a complex model, assuming that the experimenter can reach the true model form during the sequential experiments. Copyright © 2013 John Wiley & Sons, Ltd.     

Robust split‐plot designs

In many experimental situations, practitioners are confronted with costly, time consuming, or hard‐to‐change (HTC) factors. These practical or economic restrictions on randomization can be accommodated with a split‐plot design structure that minimizes the manipulation of the HTC factors. Selecting a good design is a challenging task and requires knowledge of the opportunities and restrictions imposed by the experimental apparatus and an evaluation of statistical performance among competing designs. Building on the well‐established evaluation criteria for the completely randomized context, we emphasize the unique qualitative and quantitative evaluation criteria for split‐plot designs. An example from hypersonic propulsion research is used to demonstrate the consideration of multiple design evaluation criteria. Published in 2007 by 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.     

Semifold plans for mixed‐level designs

This article presents a more efficient method for sequential augmentation of mixed‐level designs. The proposed approach reduces the optimal foldover plan of a mixed‐level design to a semifold plan by selecting half of the treatment combinations of the foldover fraction using exhaustive search and the criterion of general balance metric. The resulting design is a more economic run size augmented fraction that possesses good balance and orthogonality properties for main effects and two‐factor interactions. Three efficient arrays consisting of 20, 24 and 30 runs were selected for the analysis. Efficient arrays composed of a higher number of runs can be semifolded in a similar manner. Copyright © 2011 John Wiley & Sons, Ltd.     

All‐subsets regression under effect heredity restrictions for experimental designs with complex aliasing

We consider an all‐subsets regression method for models under effect heredity restrictions for experimental designs with complex aliasing, whose number of potential main effects and two‐factor interactions exceed the number of runs. In this paper, we present an algorithm that systematically attempts to fit all such models. We illustrate the algorithm with two published experiments. Copyright © 2009 John Wiley & Sons, Ltd.     

Classes of Split‐plot Response Surface Designs for Equivalent Estimation

When planning an experimental investigation, we are frequently faced with factors that are difficult or time consuming to manipulate, thereby making complete randomization impractical. A split‐plot structure differentiates between the experimental units associated with these hard‐to‐change factors and those that are relatively easy‐to‐change. Furthermore, it provides an efficient strategy that integrates the restrictions imposed by the experimental apparatus into the design structure. In this paper, several industrial and scientific examples are presented to highlight design considerations when a restriction on randomization is encountered. We propose classes of split‐plot response designs that provide an intuitive and natural extension from the completely randomized context. For these designs, the ordinary least‐squares estimates of the model are equivalent to the generalized least‐squares estimates. This property provides best linear unbiased estimators and simplifies model estimation. The design conditions that provide equivalent estimation are presented and lead to design construction strategies to transform completely randomized Box–Behnken, equiradial and small composite designs into a split‐plot structure. Published in 2006 by John Wiley & Sons, Ltd.     

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.     

Cost‐penalized estimation and prediction evaluation for split‐plot designs

Comparisons between different designs have traditionally focused on balancing the quality of estimation or prediction relative to the overall size of the design. For split‐plot designs with two levels of randomization, the total number of observations may not accurately summarize the true cost of the experiment, because different costs are likely associated with setting up the whole and subplot levels. In this paper, we present several flexible measures for design assessment based on D‐, G‐ and V‐optimality criteria that take into account potentially different cost structures for the split‐plot designs. The new measures are illustrated with two examples: a 2³ factorial experiment for first‐order models, where all possible designs are considered, and selective designs for a three‐factor second‐order model. Copyright © 2006 John Wiley & Sons, Ltd.     

Economic‐statistical design of 2‐of‐2 and 2‐of‐3 runs rule scheme

In this paper, we investigate the economic‐statistical design method for the 2‐of‐2 runs rule and the 2‐of‐3 runs rule. The Markov chain approach is used to obtain the average run length and the process cycle time. In addition, a simplified algorithm is presented to search the optimal setting of the design parameters. A numerical example and sensitivity analysis are also provided to compare the performances of the runs rules. The results show that the use of runs rule scheme can reduce operating cost comparing with the Shewhart control chart while maintaining a good statistical performance. Copyright © 2008 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.     

Correlation‐Based r‐plot for Evaluating Supersaturated Designs

Orthogonality or near‐orthogonality is an important property in the design of experiments. Supersaturated designs are natural when we wish to investigate the main effects for a large number of factors but are restricted to a small number of runs. These supersaturated designs, by definition, cannot satisfy pairwise orthogonality of all the factor columns in the design matrix. Hence, we need a means to evaluate the degree of near‐orthogonality of different alternative supersaturated designs. It is usual to use numerical measures that condense the rich information from many pairwise column measures to assess the degree of orthogonality of given supersaturated designs, but we propose using graphical methods to better understand patterns between sets of columns and evaluate the degree of near‐orthogonality to compare and select between alternative supersaturated designs. The methods are illustrated with a number of diverse examples to illustrate the information that can be extracted from the summary. Copyright © 2013 John Wiley & Sons, Ltd.     

Using a genetic algorithm to generate D‐optimal designs for mixture‐process variable experiments

This article presents and develops a genetic algorithm (GA) to generate D‐efficient designs for mixture‐process variable experiments. It is assumed the levels of a process variable are controlled during the process. The GA approach searches design points from a set of possible points over a continuous region and works without having a finite user‐defined candidate set. We compare the performance of designs generated by the GA with designs generated by two exchange algorithms (DETMAX and k‐exchange) in terms of D‐efficiencies and fraction of design space (FDS) plots which are used to evaluate a design's prediction variance properties. To illustrate the methodology, examples involving three and four mixture components and one process variable are proposed for creating the optimal designs. The results show that GA designs have superior prediction variance properties in comparison with the DETMAX and k‐exchange algorithm designs when the design space is the simplex or is a highly‐constrained subspace of the simplex.