Scott M. Kowalski

Response Surface Methodology: A Retrospective and Literature Survey

Response surface methodology (RSM) is a collection of statistical design and numerical optimization techniques used to optimize processes and product designs. The original work in this area dates from the 1950s and has been widely used, especially in the chemical and process industries. The last 15 years have seen the widespread application of RSM and many new developments. In this review paper we focus on RSM activities since 1989. We discuss current areas of research and mention some areas for future research.

Response surface designs within a split-plot structure

In many industrial experiments, time and/or cost constraints often force certain factors in a designed experiment to be much harder to change than others. An appropriate approach to such an experiment restricts the randomization, which leads to a split-plot structure. This paper first establishes how one can modify the common central-composite design to efficiently accommodate a split-plot structure. The proposed designs allow pure-error estimates of the two variance components. We next discuss the conditions on the design that make ordinary least squares and weighted least squares estimates equivalent. These conditions are easy to obtain in practice. An important consequence is that people can use standard experimental design software to analyze second-order response surfaces. We give examples of central composite and Box-Behnken designs that meet the conditions. We conclude with an example.

Split-Plot Designs and Estimation Methods for Mixture Experiments With Process Variables

In mixture experiments with process variables, the response depends not only on the proportions of the mixture components, but also on the effects of the process variables. In many such mixture-process variable experiments, constraints such as time or cost prohibit the selection of treatments completely at random. In these situations, restrictions on the randomization force the level combinations of one group of factors to be fixed and the combinations of the other group of factors are run. Then a new level of the first factor group is fixed and combinations of the other group of factors are run. Earlier work referred to this restriction on randomization as a split-plot approach where several factor-level combinations among one or more groups of process variables defined the whole-plot treatments while a group of mixture blends defined the subplot treatments. New split-plot designs are presented for mixture experiments with process variables while considering a new model form. Three methods of estimation are considered for the terms in the model.

Construction of Balanced Equivalent Estimation Second-Order Split-Plot Designs

Practical restrictions on randomization are commonplace in industrial experiments due to the presence of hard-to-change or costly-to-change factors. Using a split-plot design (SPD) structure reduces the number of times that these hard-to-change factors are reset during the experiment. A class of second-order response surface SPDs has been proposed in which the ordinary least squares estimates of the model are equivalent to the generalized least squares estimates. Equivalent estimation designs provide best linear unbiased estimates that are independent of the variance components and can be obtained with standard statistical software. Moreover, design selection is robust to model misspecification and does not require previous knowledge of the variance components. This article expands the conditions to obtain equivalent estimation designs and outlines two systematic design construction techniques for building balanced versions of the central composite design. In addition, it presents an approach to generating equivalent estimation D-optimal designs. By applying these design construction techniques, a catalog of designs is generated. These methods provide practitioners with the necessary tools to build equivalent estimation SPDs for a wide variety of applications.

24 run split-plot experiments for robust parameter design

Split-plot experiments where the whole plot treatments and the subplot treatments are made up of combinations of two-level factors are considered. Often in industry, due to time and/or cost constraints, the size of an experiment needs to be kept small. Sometimes these experiments involve noise factors and design factors and can be thought of as robust parameter designs. Using fractional factorials and confounding, a method for constructing sixteen run designs is presented. Semifolding is used to add eight more runs. The resulting 24 run design breaks some of the alias chains and provides some degrees of freedom for estimating a subplot error variance. Also, an alternative 24 run design based on the balanced incomplete block design is presented.

Tutorial: Industrial Split-plot Experiments

ABSTRACT Many industrial experiments involve two types of factors: those that are hard-to-change and those that are easy-to-change (ETC). Hard-to-change (HTC) factors have levels that are difficult and/or expensive to change. As a result, the experimenter would prefer to run the experiment in such a manner as to minimize the number of times that he/she must change the levels of these factors. Unfortunately, it is precisely the changing of these levels that provides the information about the effects of the HTC factors. Consequently, when we minimize the number of times we change the levels of these factors, we also minimize the relevant information about their effects. This paper summarizes the structure and the analysis of industrial split-plot experiments. The purpose of this article is to teach practitioners how to identify split-plot experimental conditions, how to run the experiment efficiently, and then how to analyze the results. The article illustrates both first-order and second-order experiments. The first four sections provide a basic background on experimental design and an introduction to first-order split-plot experiments. The remainder of this article contains more advanced topics dealing with second-order, split-plot experiments.

A new model and class of designs for mixture experiments with process variables

Experiments that involve the blending of several components are known as mixture experiments. In some mixture experiments, the response depends not only on the proportion of the mixture components, but also on the processing conditions, A new combined model is proposed which is based on Taylor series approximation and is intended to be a compromise between standard mixture models and standard response surface models. Cost and/or time constraints often limit the size of industrial experiments. With this in mind, we present a new class of designs that will accommodate the fitting of the new combined model.

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.

Exact Inference for Response Surface Designs Within a Split-Plot Structure

Many industrial experiments involve factors that are hard to change as well as factors that are easy to change, which leads naturally to split-plot experiments. Unfortunately, the literature for second-order response surface designs traditionally assumes a completely randomized design. By carefully choosing the design for response surface experiments, it is possible to guarantee that the ordinary least-squares estimates of the model are equivalent to the generalized least-squares estimates. This paper uses designs satifying this result to derive conditions for exact tests of almost all of the model parameters. It then sets up the Satterthwaites procedure for the remaining parameters. An example illustrates the methodology. Comparisons between the exact estimators, estimators based on replicated points, and estimators using restricted maximum likelihood estimators (REML) are done using the example.

A Modified Path of Steepest Ascent for Split-Plot Experiments

Response surface methodology is applied often in industrial experiments to find the optimal conditions for the design factors. Response surface methodology involves three main steps: screening potential factors, seeking a region around the global optimum using the path of steepest ascent, and estimating the true model for the response. Many of these experiments involve randomization restrictions and can be thought of as split-plot experiments. This paper investigates the path of steepest ascent within a split-plot structure. Three methods are proposed for calculating the coordinates along the path.