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209 Pages·1963·6.62 MB·English

by  Romney

Formulation Simplified

Formulation Simplified
Finding the Sweet Spot through Design and
Analysis of Experiments with Mixtures
Mark J. Anderson
Patrick J. Whitcomb
Martin A. Bezener
A PRODUCTIVITY PRESS BOOK
Routledge
Taylor & Francis Group
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© 2018 by Mark J. Anderson, Patrick J. Whitcomb, Martin A. Bezener
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Contents
Preface ........................................................................................................ix
Acknowledgments ....................................................................................xi
Authors .....................................................................................................xiii
Introduction .............................................................................................xv
1 Getting Your Toe into Mixtures ...................................................1
All That Glitters Is Not Gold.......................................................................3
Generating a Beautiful Response Surface—Like a String of Rubies
on a Gold Strand! ........................................................................................7
Details on Modeling the Performance of a Two-Component Mixture ........11
Practice Problems ......................................................................................14
Problem 1.1 ............................................................................................15
Problem 1.2 ............................................................................................17
Appendix 1A: Cubic Equations for Mixture Modeling (and Beyond) .....21
Appendix 1B: Statistical Details on How to Order Up Just the Right
Model .........................................................................................................25
Testing for Lack of Fit ...........................................................................26
Model Summary Statistics .....................................................................29
2 Triangulating Your Region of Formulation ...............................31
The Simplex Centroid Design ...................................................................33
The Black and Blue Moon Beer Cocktail ................................................35
Diving Under the Response Surface to Detail the Underlying
Predictive Model .......................................................................................41
Taking Cost into Account .........................................................................44
Do Not Put a Square Peg into a Triangular Hole ....................................46
v
vi ◾ Contents
Practice Problems ......................................................................................48
Problem 2.1 ............................................................................................48
Problem 2.2 ...........................................................................................49
Appendix 2A: The Special Cubic (and Advice on Interpreting
Coefficients) ...............................................................................................50
3 Simplex Lattice Designs to Any Degree You Like .....................55
Working with Four Components in Tetrahedral Space ...........................55
Building a Simplex Lattice Design ............................................................57
Augmented Simplex Lattice: When in Doubt, Build Them Stout ............60
Using Augmented Simplex Lattice Mixture Design to Optimize
an Extra-Virgin Olive Oil ..........................................................................62
Practice Problems ......................................................................................70
Problem 3.1 ............................................................................................71
Problem 3.2 ...........................................................................................71
4 Mixture Constraints That Keep Recipes Reasonable .................73
Setting Minimum Constraints ...................................................................73
Expanding the Constrained Space via Mathematical Coding..................75
Why It Was Worth Reading This Chapter and What’s in It for You
as a Formulator .........................................................................................79
Practice Problem .......................................................................................80
Problem 4.1 ............................................................................................80
Appendix 4A: Upper (“U”) Pseudo Coding to Invert Mixture Space ......82
5 Optimal Design to Customize Your Experiment .......................87
Extreme Vertices Design: Shampoo Experiment .....................................89
Optimal Designs Customize Your Experiment as You See Fit for
Any Feasible Region ..................................................................................93
Take This Tableting Case-Study Now and Call the Statistician
in the Morning ..........................................................................................94
Optimal Design Simplified .......................................................................95
Take-Home Advice on Deploying Optimal Design .................................97
Practice Problem .......................................................................................98
Problem 5.1 ............................................................................................98
Appendix 5A: An Algorithm for Finding Vertices ....................................99
6 Getting Crafty with Multicomponent Constraints ...................101
How Multicomponent Constraints Differ from Simple Ones ................101
MCCs Made as Easy as Making a Pound Cake ......................................104
Contents ◾ vii
Ratio Constraints .....................................................................................107
Appendix 6A: Combining Components .................................................108
7 Multiple Response Optimization Hits the Spot .......................113
Desirability Simplified .............................................................................114
Framing the Sweet Spot and Narrowing It Down to Achieve
Quality by Design (QbD) ........................................................................119
Practice Problem .....................................................................................124
Problem 7.1 ..........................................................................................124
8 Screening for Vital Components ..............................................127
High-Octane Simplex Screening Designs ..............................................128
Measuring the Effect of a Component ...................................................133
Extreme Vertices Design for Non-Simplex Screening ............................134
Appendix 8A: Trace Plots—Cox versus Piepel Direction ......................138
9 Working Amounts, Categorical and Process Factors
into the Mix .............................................................................143
Mixture-Amount Experiments—Not Just the Composition but
How Much of It .......................................................................................143
Contending with Categorical Variables ..................................................149
Practice Problem .....................................................................................152
Problem 9.1 ..........................................................................................152
Appendix 9A: Alternatives for Modeling Results from
Combined Designs ..................................................................................152
10 Blocking and Splitting Designs for Ease of Experimentation .....155
Blocking to Remove Known Sources of Variation .................................155
Split Plots to Handle Hard-to-Change Factors or Components
in Combined Designs .............................................................................160
A Case Where the Process, Rather Than the Mixture, Is Hard
to Change ................................................................................................164
11 Practical Magic for Making the Most of a Mixture ..................165
The Failure of Fillers and Perils of Parts ................................................165
Strategy for Formulation Experimentation .............................................171
References ......................................................................................175
About the Software ........................................................................179
Index ..............................................................................................181

Preface
All that is gold does not glitter; Not all those who wander are lost.
—J. R. R. Tolkien (The Fellowship of the Ring)
This book rounds out our series of “Simplified” books (Anderson and
Whitcomb, 2015, 2016) into a trilogy on the design of experiments (DOE).
It may not achieve the stature of Tolkien’s towering trio—The Lord of the
Rings, but the detailing of mixture design completes our quest to provide
the statistical tools needed by modern-day industrial experimenters. The
beneficiaries of this third “Simplified” book will be formulators of alloys,
beverages, chemicals, cosmetics, construction materials (such as concrete),
food, flavors, pharmaceuticals, paints, plastics, pulp, paper rubber, textiles,
and so forth, that is, any product made from stuff.
Formulation Simplified is derived from a popular workshop on mixture
design that my coauthor, Pat, developed over twenty years ago. He’s worked
unstintingly to continuously incorporate new statistical methods that prove
to be of practical use. More recently, statistician, Martin Bezener, joined our
team at Stat-Ease and took to mixtures like a barista to coffee. However, it’s
one thing to be trained intensively by expert instructors like Pat or Martin, but
another thing to learn on your own from a book. That’s where I come in by
making these powerful statistical tools of experimental design and analysis as
unintimidating as possible in a self-study, written format. Luckily, I can rely on
Pat and Martin to bolster any inadequate mathematical details, thus helping us
maintain statistical rigor throughout. If we feel that this may create too much
information (TMI) for some readers, the in-depth explanations go into sidebars
or appendices that can be glossed over (at least on the first go-through!).
What differentiates Formulation Simplified from the standard statisti-
cal texts on mixture design by Cornell (2002) and Smith (2005) is that we
make things relatively easy and fun to read. To convey my experience that
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