RSM Simplified

Optimizing Processes Using Response Surface Methods for Design of Experiments, Second Edition

Mark J. Anderson, Patrick J. Whitcomb

• Publisher: Productivity Press
• Year: 2016
• Print ISBN: 9781138463905
• eBook ISBN: 9781315351728
• Edition: 2
• Topic: Commerce
• Subtopic: Commerce Général

This book continues where DOE Simplified leaves off in Chapter 8 with an introduction to "Response Surface Methods [RSM] for Optimization." It presents this advanced tool for design of experiments (DOE) in a way that anyone with a minimum of technical training can understand and appreciate. Unlike any other book of its kind, RSM Simplified keeps formulas to a minimum—making liberal use of figures, charts, graphs and checklists. It also offers many relevant examples, amusing and fun do-it-yourself exercises.

Chapter 1

Introduction to the Beauty of Response Surface Methods

For a research worker, the unforgotten moments of his [or her] life are those rare ones, which come after years of plodding work, when the veil over nature’s secret seems suddenly to lift, and when what was dark and chaotic appears in a clear and beautiful light and pattern.

Gerty Cori

The first American woman to win a Nobel Prize in science

Before we jump down to ground-level details on response surface methods (RSM), let’s get a bird’s-eye view of the lay of the land. First of all, we will assume that you have an interest in this topic from a practical perspective, not academic. A second big assumption is that you’ve mastered the simpler tools of design of experiments (DOE). (Don’t worry, we will do some review in the next few chapters!)

RSM offer DOE tools that lead to peak process performance. RSM produce precise maps based on mathematical models. This methodology facilitates putting all your responses together via sophisticated optimization approaches, which ultimately lead to the discovery of sweet spots where you meet all specifications at minimal cost.

This answers the question: What’s in it for me? Now let’s see how RSM fits into the overall framework of DOE and learn some historical background.

Strategy of Experimentation: Role for RSM

The development of RSM began with the publication of a landmark article by Box and Wilson (1951) titled “On the Experimental Attainment of Optimum Conditions.” In a retrospective of events leading up to this paper, Box (2000) recalled observing process improvement teams in the United Kingdom at Imperial Chemical Industries in the late 1940s. He and Wilson realized that, as a practical matter, statistical plans for experimentation must be very flexible and allow for a series of iterations.

Box and other industrial statisticians, notably Hunter (1958–1959) continued to hone the strategy of experimentation to the point where it became standard practice in chemical and other process industries in the United Kingdom and elsewhere. In the United States, Du Pont took the lead in making effective use of the tools of DOE, including RSM. Via their Management and Technology Center (sadly, now defunct), they took an in-house workshop called “Strategy of Experimentation” public and, over the last quarter of the twentieth century, trained legions of engineers, scientists, and quality professionals in these statistical methods for experimentation.

This now-proven strategy of experimentation, illustrated in Figure 1.1, begins with standard two-level fractional factorial design, mathematically described as “2^k−p” (Box and Hunter, 1961) or newer test plans with minimum runs (noted below), which provide a screening tool. During this phase, experimenters seek to discover the vital few factors that create statistically significant effects of practical importance for the goal of process improvement. To save time at this early stage where a number (k) of unknown factors must be quickly screened, the strategy calls for use of relatively low-resolution (“res”) fractions (p).

A QUICK PRIMER ON NOTATION AND TERMINOLOGY FOR STANDARD SCREENING DESIGNS

Two-level DOEs work very well as screening tools. If performed properly, they can reveal the vital few factors that significantly affect your process. To save on costly runs, experimenters often perform only a fraction of all the possible combinations. There are many varieties of fractional two-level designs, such as Taguchi or Plackett–Burman, but we will restrict our discussion to the standard ones that statisticians symbolize as “2^k−p,” where k refers to the number of factors and p is the degree of fractionation. Regardless of how you do it, cutting out runs reduces the ability of the design to resolve all possible effects, specifically the higher-order interactions. Minimal-run designs, such as seven factors in eight runs (2⁷⁻⁴)—a 1/16th (2⁻⁴) fraction, can only estimate main effects. Statisticians label these low-quality designs as “resolution III” to indicate that main effects will be aliased with two-factor interactions (2FIs). Resolution III designs can produce significant improvements, but it’s like kicking your PC (or slapping your laptop) to make it work: you won’t discover what really caused the failure.

To help you grasp the concept of resolution, think of main effects as one factor and add this to the number of factors it will be aliased with. In resolution III, it’s a 1-to-2 relation, which adds to 3. Resolution IV indicates a 1-to-3 aliasing (1 + 3 = 4). A resolution V design aliases main effects only with four factors (1 + 4 = 5).

Because of their ability to more clearly reveal main effects, resolution IV designs work much better than resolution III for screening purposes, but they still offer a large savings in experimental runs. For example, let’s say that you want to screen 10 process factors (k = 10). A full two-level factorial requires 2¹⁰ (2^k) combinations, way too many (1024!) for a practical experiment. However, the catalog of standard two-level designs offers a 1/32nd fraction that’s resolution IV, which will produce fairly clear estimates of main effects. To most efficiently describe this option mathematically, convert the fraction to 2^p (p = 5) scientific notation: 2⁻⁵ (=1/2⁵ = 1/(2 × 2 × 2 × 2 × 2) = 1/32). This yields 2¹⁰⁻⁵ (2^k−p) and by simple arithmetic in the exponent (10 − 5): 2⁵ runs. Now, we do the final calculation: 2 × 2 × 2 × 2 × 2 equals 32 runs in the resolution IV fraction (vs. 1024 in the full factorial).

P.S. A more efficient type of fractional two-level factorial screening design has been developed (Anderson and Whitcomb, 2004). These designs are referred to as “minimum-run resolution IV” (MR4) because they require a minimal number of factor combinations (runs) to resolve main effects from 2FIs (resolution IV). They compare favorably to the classical alternatives on the basis of required experimental runs. For example, 10 factors can be screened in only 20 runs via the MR4 whereas the standard (2^k−p) resolution IV design, a 1/32nd (2⁻⁵) fraction, requires 32 runs.

Along these same lines are minimum-run resolution V (MR5) designs, for example, one that characterizes six factors in only 22 runs versus 32 runs required by the standard test plan.

Figure 1.1 Strategy of experimentationStrategy of experimentation.

After throwing the many trivial factors off to the side (preferably by holding them fixed or blocking them out), the experimental program should enter the characterization phase where interactions become evident. This requires higher-resolution, or possibly full, two-level factorial designs. By definition, traditional one-factor-at-a-time (OFAT) approaches will never uncover interactions of factors that of...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Table of Contents
• Preface
• Authors
• 1 Introduction to the Beauty of Response Surface Methods
• 2 Lessons to Learn from Happenstance Regression
• 3 Factorials to Set the Stage for More Glamorous RSM Designs
• 4 Central Composite Design: Stars Added—RSM Show Begins
• 5 Three-Level Designs
• 6 Finding Your Sweet Spot for Multiple Responses
• 7 Computer-Generated Optimal Designs
• 8 Everything You Should Know about CCDs (but Dare Not Ask!)
• 9 RSM for Six Sigma
• 10 Other Applications for RSM
• 11 Applying RSM to Mixtures
• 12 Practical Aspects for RSM Success
• Glossary
• References
• Index
• About the Software