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A Retrospective View of Mixture ExperimentsJohn A. Cornell aa Department of Statistics , University of Florida , Gainesville, FloridaPublished online: 29 Aug 2011.

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A Retrospective View of MixtureExperiments

John A. Cornell

Department of Statistics,

University of Florida, Gainesville,

Florida

ABSTRACT When approached to present ‘‘A Retrospective View of Mixture

Experiments’’ at the 2010 Fall Technical Conference (FTC), I was happy to do

so. Why? Because I have been retired as a former member of the faculty of the

Department of Statistics for 7 years and I had the opportunity to speak in front

of many friends on a subject that has been part of my life for 45 years, since

1966. Following the invitation to speak, I looked up the meaning of retrospec-

tive in my copy of Webster’s Dictionary to find that retrospective means ‘‘to

reminisce or re-experience past events.’’ In other words, I was asked to look

back over the past 55 years or so of published research on mixture experi-

ments and share my thoughts with friends, many of whom were in the audi-

ence at the FTC and others who will read this article in Quality Engineering.

KEYWORDS blending ingredients, canonical polynomial, composition space or

experimental region, (q-1)-dimensional simplex, Henry Scheffe, John Gorman,

linear blending, mixture experiments, nonlinear blending, nonnegative propor-

tions, polyester yarn, fq, mg simplex-lattice, special-cubic model, upper and

lower bounds

INTRODUCTION

One spring afternoon in 1966 when I was assigned to the Applied Math

Group at Tennessee Eastman Company (TEC) in Kingsport, Tennessee,

my supervisor, Bob Brown, approached me and asked whether I had read

an article authored by Henry Scheffe in 1958 titled ‘‘Experiments with

Mixtures.’’ Mr. Brown was given a set of data, passed on to him from a

colleague working in the Polymer Division of TEC, and asked whether he

could determine whether elongation of polyester yarn, made by blending

three different polymers, could be optimized. When I said I had not read

the article, Mr. Brown gave me an assignment to read Scheffe’s article. He

wanted to know whether the elongation of spun yarn was a function of

the proportions of the three polymers that were mixed together and, if so,

did I know how to determine what the optimal blend was? Unfortunately,

I had not been taught during my graduate training how to find the optimal

blend, but the challenge somehow seemed exciting. Forty-five years later,

we are able to obtain a chronological listing of authors of selected statistical

literature on mixtures from 1953 to 2009 in Cornell (2011, pp. 14–15). A fully

Address correspondence to John A.Cornell, 2836 SW 92nd Terrace,Gainesville, FL 32608. E-mail:jcornell@stat.ufl.edu

Quality Engineering, 23:315–331, 2011Copyright # Taylor & Francis Group, LLCISSN: 0898-2112 print=1532-4222 onlineDOI: 10.1080/08982112.2011.602283

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comprehensive summary of statistical research

related to mixture experimental designs, models,

and data analysis topics from 1955 to 2004 can be

found in Piepel (2006).

Drs. Peter W. M. John and John W. Gorman copre-

sented the first invited talk on mixtures at the 1960

Gordon Research Conference on Statistics in Chemis-

try and Chemical Engineering, held at New Hamp-

ton, New Hampshire. The title of the talk was

‘‘Experiments with Mixtures’’ and highlighted the

designs and models introduced by Scheffe in 1958.

THE MIXTURE PROBLEM

The description of a mixture experiment is as

follows. Imagine the sweetening of a cup of coffee

by adding sugar or a sweetener or both sugar and

the sweetener to the coffee. Blends or mixtures of

coffee with sugar or with the sweetener or with both

are possible. Another example is the mixing of oil

with vinegar and some spices to create and flavor a

salad dressing. A third example might be the mixing

of 93-octane fuel in the family car that contains

87-octane fuel to improve driving performance; for

example, improving miles per gallon. In these exam-

ples, the adding and=or blending of ingredients in an

attempt to try to obtain a more desirable end product

is something all of us do in our everyday activities,

and these actions are known as mixture experiments;

see Figure 1.

In mixture experiments, the controllable variables

are nonnegative proportionate amounts of the

ingredients in a mixture in which the proportions

are by volume, by weight, or by mole fraction. In

a q component (or q ingredient) mixture in which

xi represents the proportion of the ith component

present in the mixture, the proportions are

nonnegative:

0 � xi � 1; i ¼ 1; 2; . . . ; q ½1�

and sum to unity or one,

Xqi¼1

xi ¼ 1: ½2�

The composition space, or experimental region, of

the q components, by virtue of restrictions [1] and

[2], takes the form of a (q� 1)-dimensional simplex.

When q¼ 2, the simplex is a straight line; when

q¼ 3, the simplex is an equilateral triangle; and

when q¼ 4, the simplex is a tetrahedron; see

Figure 2. When additional constraints are imposed

on the component proportions in the form of lower

Li and=or upper Ui bounds,

0 � Li � xi � Ui � 1; i ¼ 1; 2; . . . ; q ½3�

or as the linear multicomponent constraints,

Cj � A1jx1 þA2jx2 þ . . .þAqjxq � Dj ; j ¼ 1;2; . . . ;h;

½4�

where Aij, Cj, and Dj are scalar constants. These

additional constraints [3] and [4] can alter the shape

of the experimental region from that of a simplex

to one of an irregularly shaped convex polyhedron

inside the simplex. Thus, when we think about

experimental regions for mixture experiments, two

main shapes come to mind: a simplex-shaped region

FIGURE 1 Some common examples of mixing ingredients: (a)

sweetening coffee by mixing sugar and a sweetener; (b) mixing

oil and vinegar to flavor a salad; (c) blending 87-octane and

93-octane fuels.

FIGURE 2 Experimental regions for q¼ 3 and q¼ 4 compo-

nents. For q¼ 3, the simplex is an equilateral triangle. For q¼ 4,

the simplex is a tetrahedron.

J. A. Cornell 316

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(generally the whole simplex) or a non-simplex-

shaped region inside the simplex. In both cases,

the dimensionality of the region is lower than q,

the number of components.

Mixture model forms most commonly used in

fitting mixture data, at least until 1965, were the

canonical polynomials introduced by Scheffe (1958,

1963). The first-degree or linear blending model is

yu ¼Xqi¼1

bixiuþeu; u¼ 1; 2; . . . ;N ½5�

where yu represents the uth observed value of the

response in N trials and the constant term b0 is not

included in the model due to restrictions [1] and [2].

The coefficient bi is the expected response to compo-

nent i (i.e., the response at xi¼ 1), and eu is the ran-

dom error in the uth observed response value. We

assume that the error eu is sampled from a distribution

with mean 0 and variance r2. In the case of testing the

magnitude of the estimate of one or more of the bi’s in

the model during analysis of the data, the errors in the

individual yu values are assumed to be sampled from

a normal N(0, r2) distribution. The second-degree or

binary nonlinear blending model is

yu ¼Xqi¼1

bixiu þXq�1

i¼1

Xqj¼2

bijxiuxju þ eu ½6�

where bij is a measure of the nonlinear (curvilinear)

blending of components i and j, i, j¼ 1, 2, . . ., q and

i< j. Higher-degree models contain nonlinear blend-

ing terms of the type bijkxixjxk and cijxixj(xi� xj), and

the canonical forms [5] and [6] will always contain a

lower number of terms than their standard poly-

nomial counterparts; see, for example, Cornell

(2002, chapter 2).

THE BEGINNING YEARS—1955

TO 1965

Many years ago it was rumored that Henry Scheffe

was a perfectionist when writing papers for publi-

cation. Rarely was he asked to revise a manuscript

following a review of the first draft. Figure 3 displays

the first page of the initial draft of the manuscript that

later became the seminal article entitled ‘‘Experi-

ments with Mixtures’’ that appeared in the Journal

of the Royal Statistical Society Series B. It was also

rumored that nearly 80% of the completed 34-page

manuscript was composed during a cross-country

flight from the East Coast of the United States to

Berkeley, California, in the spring of 1957.

This page, being the first page of the manuscript,

has two corrections on it. The first misprint is a cor-

rected spelling of the word resistance in line 3 and

the second misprint is the crossing out of gasolines,

and replaced with of different gasolines in line 5. In

the remaining 33 pages of the manuscript, only seven

additional corrections were discovered by Henry or

the copy editor. For many of us who never had the

pleasure of meeting Henry Scheffe, a couple of

pictures of Henry can be seen at http://www-history.

mcs.st-and.ac.uk/PictDisplay/Scheffe.html.

Three years prior to Scheffe’s (1958) seminal paper

on simplex-lattice designs and the associated canoni-

cal form of polynomial model, Claringbold (1955)

proposed a three-component simplex (equilateral

triangle) along with a second-degree model in two

independent variables with which to study the joint

action of three separate hormones estrone (x1), estra-

diol (x2), and estriol (x3) that were injected into

groups of mice. The design consisted of two 10-blend

sets of points administered at each of three different

amounts (low, middle, and high). One 10-blend set

is shown by the black dots positioned on the per-

imeter (vertices and edges) and center of the triangle

in Figure 4. The second 10-blend set is shown by the

boxes positioned at the vertices, middle of the edges,

inside the triangle, and at the center of the triangle.

The 20-blend design is illustrated in Figure 4.

Although many authors of articles on mixtures in

the 1960s and 1970s acknowledged Claringbold’s

(1953) work, most acknowledged Scheffe’s (1958)

paper to have a greater impact on published research

later than Claringbold’s due to the fact that refer-

ences to Scheffe’s paper was nearly triple the number

of references to Claringbold’s paper. Figure 5

displays the q¼ 3- and q¼ 4-component fq, mgsimplex-lattice designs for fitting the second- and

third-degree Scheffe models, respectively. In the

fq, mg notation, q is the number of components

and m is the degree of the model to be fitted. In

the fq, mg lattice, each component is assigned the

mþ 1 equally spaced values ranging from xi¼ 0,

1=m, 2=m, . . . , m=m¼ 1 and because of constraint

[2], with each blend or design point, the sum of the

xi’s must equal one or unity.

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Five years after his initial paper, Scheffe (1963)

introduced the simplex-centroid design and its asso-

ciated model. The simplex-centroid design consists

of 2q� 1 points: q permutations of (1, 0, 0, . . ., 0),

q(qþ 1)=2 permutations of (1=2, 1=2, 0, 0, . . . ,0),

q(q2 � 3qþ 2)=6 permutations of (1=3, 1=3, 1=3, 0,

0, . . . , 0), . . ., and one permutation of (1=q, 1=q, 1=

q, . . . , 1=q). The 2q� 1 points are the vertices, mid-

points of the edges, centroids of the faces, etc., of

the simplex. The corresponding model to be fitted

to data at the points of the simplex-centroid design is

y ¼Xqi¼1

bixi þXq�1

i¼1

Xqi<j

bijxixj

þXq�2

i¼1

Xq�1

i<j

Xqj<k

bijkxixjxk þ :::þ b123:::qx1x2x3:::xqþe

½7�

The number of terms in model [7] is 2q� 1 and there

exists a one-to-one correspondence between the num-

ber of points (or blends) in the design and the number

of terms in the model. Figure 6 shows the simplex-

centroid designs for q¼ 3 and q¼ 4.

In addition to the simplex-centroid design, Scheffe

(1963) introduced the inclusion of process variables

in mixture experiments. Process variables are not

mixture components but instead are independent

FIGURE 3 (a) The first page of Scheffe’s handwritten manuscript; (b) Page 1 of initial draft of the manuscript titled ‘‘Experiments with

Mixtures.’’

FIGURE 4 Design points for Claringbold’s study of three

hormones. The design consists two 10-blend sets administered

at each of three amounts. One set is denoted by dots . and the

other set is denoted by boxes &. The three vertices (single

hormone) and the centroid (all three hormones) are in each set.

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variables such as temperature, time, spin speed, etc.,

that could have an effect on the response or affect

the blending properties of the mixture components.

When considering designs for including process vari-

ables in mixture experiments, the most popular strat-

egy is to cross the points of the mixture design with

the points of a factorial arrangement in the process

variables. Similarly, combined models containing

both the mixture components and the process vari-

ables consist of combining or crossing the terms in

the respective model forms. Gorman and Hinman

(1962) extended the work of Scheffe’s designs and

models by expressing the estimation equations for

the parameters in the full cubic or third-degree and

full quartic or fourth-degree models in terms of the

average response values at the points of the full

cubic fq, 3g and quartic fq, 4g simplex-lattice

designs in q components. The formulas for estimat-

ing the coefficients in the fitted Scheffe cubic and

quartic models can also be found in Appendix 2B

of Cornell (2002).

The second half of the 1960s found modelers

transforming the q mixture components into q� 1

mathematically independent variables. Two reasons

cited by those who chose to work with independent

variables are (1) familiarity in knowing how to set

up designs, such as factorial arrangements, and fit

standard polynomial models in the independent

variables; and (2) belief in knowing how to interpret

the unknown parameter estimates in the fitted

models up to the third degree, as well as familiarity

with design optimality criteria.

Figure 7 displays a 12-point design in three compo-

nents, or two independent variables, w1 and w2, sug-

gested by Draper and Lawrence (1965). The design

consists of three sets of points (one set being triangu-

lar, a second a set of four clear dots, and a third set of

four dark dots; see figure). Draper and Lawrence also

introduced designs that minimize the bias in the fitted

model, or designs that minimize the variance of

prediction, or designs that minimize the integrated

mean square error of the estimate of the response

over the simplex region. The differences among the

designs were the result of spreading the points away

from the center point. Minimizing the integrated

mean square error of the predicted response value

y(w1, w2) was initially suggested by Box and Draper

(1959) when constructing response surface designs

in general.

FIGURE 7 Draper and Lawrence’s (1965) 12-point design

consisting of three sets of points (one triangular and two squares)

plus two center-point replicates.

FIGURE 6 Simplex-centroid designs for q¼ 3 and q¼4 compo-

nents.

FIGURE 5 Some f3, mg and f4, mg simplex-lattice arrange-

ments for m¼ 2 and m¼3.

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THE SECOND WAVE OF PROBLEMSTO BE SOLVED, 1966–1978: ENTER

THE COMPUTER

When one or more of the q component propor-

tions are restricted by lower bounds 0�Li� xi, i¼ 1,

2, . . . , q, Kurotori (1966) suggested transforming the

restricted components to pseudocomponents, which

were later called L-pseudocomponents, using

x0i ¼

xi � Li

1 �Pqi¼1

Li

; i¼ 1; 2; . . . ; q ½8�

The quantity 1 �Pq

i¼1 Li must be greater than zero

(that is,Pq

i¼1 Li < 1) so that the simplex in the x0i lies

entirely inside the simplex in the x0i. The denomi-

nator 1 �Pq

i¼1 Li in [8] is the height of the simplex

in the L-pseudocomponents x0i compared to unity

for the simplex in the x0i. Once the L-pseudocompo-

nent simplex is set up inside the original simplex,

Scheffe’s simplex-lattice or simplex-centroid designs

are easily set up in the L-pseudocomponents and

the Scheffe-type models are easily fitted in either or

both the original or the L-pseudocomponents.

Let us illustrate the use of L-pseudocomponents in

three components having the lower bound restrictions

0:20 � x1; 0:10 � x2; 0:25 � x3 ½9�

whereP3

i¼1 Li ¼ 0.20þ 0.10þ 0.25¼ 0.55 and 1�L¼ 0.45. In Figure 8, the L-pseudocomponent simplex

inside the simplex in the original components is

shown. The orientations of the L-pseudocomponent

simplex and the original simplex are the same.

When the restrictions on the component propor-

tions are in the form of lower and upper bounds

on the component proportions, such as

0 � Li � xi � Ui � 1; i ¼ 1; 2; . . . ; q ½10�

McLean and Anderson (1966) developed a technique

whereby the vertices of the constrained region

(convex polyhedron) can be located. The vertices

and convex combinations of the vertices can be used

as design points from which to collect data for fitting

the Scheffe-type models. Unfortunately, once the

limits or bounds on the component proportions are

specified, the possibility of obtaining a uniform

distribution of design points over the constrained

region or factor space is doubtful. When this

nonuniformity of the distribution of design points

exists, that is, when there are clusters of points in

some areas of the factor space and only a few points

in other areas, it is unlikely that the fitted model will

adequately predict the response over the factor

space. This is because the variance of the estimate

of the response will be affected by the distribution

of design points; poor precision will result in

areas of sparse experimentation, though there will

be good precision in areas with clusters of points.

An example provided by McLean and Anderson

(1966) is the development of flares using the chemi-

cal constituents magnesium (x1), sodium nitrate (x2),

strontium nitrate (x3), and binder (x4) with the

following constraints:

0:40 � x1 � 0:60; 0:10 � x2 � 0:50;

0:10 � x3 � 0:50; 0:03 � x4 � 0:08½11�

The constrained region defined by [11] has 8 extreme

vertices, 12 edges connecting the vertices, and 6

two-dimensional faces as shown in Figure 9. McLean

and Anderson (1966) chose the 8 vertices, the cen-

troids of the 6 faces, and the overall centroid of the

region as 15 design points in which to collect data

for fitting the 10-term Scheffe quadratic model

y ¼ b1x1 þ b2x2 þ b3x3 þ b4x4 þ b12x1x2 þ b13x1x3

þ b14x1x4 þ b23x2x3 þ b24x2x4 þ b34x3x4 þ e

½12�

Although there are shortcomings to the extreme

vertices design when constraints of the type in [11]

exist, in particular 0.03� x4� 0.08, in 1966 computer

software for generating optimal designs as we knowFIGURE 8 The L-pseudocomponent (Xi

0) simplex inside the

original component simplex (Xi) i¼ 1,2,3.

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it today did not exist. Therefore, McLean and

Anderson’s paper was an important contribution that

soon became the impetus or stimulus for other

researchers to develop algorithms for generating

optimal designs. Gorman (1966) discussed ways to

modify some of the constraints in an effort to remove

clusters of points.

Thompson and Myers (1968) further expanded the

design possibilities by defining an ellipsoidal region

of interest centered about a point of maximum

interest in the simplex. The point of maximum

interest might be the mixture that the current product

is made of or just a convenient starting point for the

experimentation as well as a base point from which

to construct a design. The ellipsoidal region of

interest is contained entirely within the simplex,

and the shape of the region is determined by the

experimenter. See discussion of Cornell and Good

(1970).

Thompson and Myers (1968) showed how poly-

nomial models of any degree can be used to esti-

mate the response over the region of interest by

first transforming from the set of q mixture compo-

nents to a set of q� 1 linearly independent vari-

ables. The transformation to independent variables

(Draper and Lawrence [1965a, 1965b]) enables the

use of standard methods of design construction

(they suggest using rotatable designs) as well as

facilitates the use of the criterion the average mean

square error of the estimate of the response for

determining optimal design configurations. An illus-

tration of a region of interest for three components,

centered at the point of main interest x0, is shown

in Figure 10.

A generalization of Scheffe’s (1958) simplex-

lattices and canonical form of polynomials was con-

sidered by Lambrakis (1968, 1969) where he defined

the following: let the p mixture components be

defined as ‘‘major’’ components or M-components

and the proportion of the ith (1� i� p) M-compo-

nent present in the mixture is denoted by ci, so that

ci> 0; i¼ 1; 2; . . . ; p;Xpi¼1

ci ¼ 1: ½13�

Each M-component is a mixture of ni� 2 ‘‘minor’’ or

m-components. Let us denote by Xij the proportion

of the jth minor component in the ith major

component as

0 � Xij � ci; i ¼ 1; 2; . . . ; p; j ¼ 1; 2; . . . ;ni; ½14�

andPni

j¼1 Xij ¼ ci in the simplex corresponding to

the ith M-component. The proportion of the jth

m-component of the ith M-component is then

represented by Xij¼ cixij, so that

Xpi¼1

Xni

j¼1

Xij ¼ 1: ½15�

Because each M-component is a mixture of ni

m-components, each major component can be

represented geometrically by a regular (ni� 1)-

dimensional simplex. If we recall Scheffe’s (1958)

fq, mg lattice notation, and realizing that a

Scheffe-type polynomial may be fitted to the points

of the fni, mig lattice associated with the ith M-

component, then Lambrakis’s (1968, 1969) generali-

zation of the simplex-lattice design is the result of

FIGURE 10 Ellipsoidal regions for three components. The

inner (solid) ellipse corresponds to the unit spherical region in

W1 and W2 and the larger (dashed) ellipse corresponds to the

largest spherical region, centered at x0, that will fit inside the

simplex.FIGURE 9 McLean and Anderson’s (1966) 15-point constrained

region design for fitting the quadratic model in four components.

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combining the p fni, mig simplex lattices. These

combinations are called multiple lattices and consist

of

Ppi¼1 ¼

ni þmi � 1mi

� �

points or blends.

To illustrate the configuration of the double-lattice

for p¼ 2 M-components where each M-component

contains two M-components, n1¼n2¼ 2, suppose

that it is desired to fit a third-order polynomial to

approximate the surface over the lattice correspond-

ing to M-component 1 and to fit a second-order poly-

nomial to the surface over the lattice corresponding

to M-component 2. Let us set c1¼ c2¼ 12, which

forces each M-component to make up 0.50 or 50%

of the mixture. Denoting the two m-components in

M-component 1 by x1 and x2 and denoting the two

m-components in M-component 2 by x3 and x4,

respectively, the double-lattice configuration is the

result of multiplying or crossing the 4 points of a

f2, 3g simplex-lattice and the 3 points of a f2, 2gsimplex-lattice, producing the 12 points of the fn1,

n2; m1, m2g¼f2, 2; 3, 2g double-lattice shown in

Figure 11.

The 12-term double-Scheffe model is constructed

by crossing the terms in the two single-lattice mod-

els. Let the cubic model in the m-components x1

and x2 be of the form g1¼ a1x1þ a2x2þ a12x1x2þc12x1x2(x1� x2) and the quadratic model in the

minor components x3 and x4 be of the form g2¼b1x3þ b2x4þ b12x3x4 where g1 and g2 represent the

expected values of y1 and y2 over the M-components

1 and 2, respectively. The corresponding 12-term

double-Scheffe model becomes

g12 ¼ g1 � g2 ¼ c13x1x3 þ c14x1x4 þ c23x2x3 þ c24x2x4

þ c123x1x2x3 þ c124x1x2x4 þ d1234x1x2x3x4

þ c134x1x3x4 þ c234x2x3x4 þ c123x1x2x3ðx1 � x2Þþ c124x1x2x4ðx1 � x2Þ þ d1234x1x2x3x4ðx1 � x2Þ

:

½16�

A slightly different approach to categorizing the com-

ponents and blending the categories was considered

by Cornell and Good (1970). The q mixture com-

ponents are assumed to belong to k rather than p

(k� 2) distinct categories where a category is a group

of components considered to be similar; for example,

a category of acid constituents, a category of bases,

etc. (Let us use the notation that Cornell and Good

used by switching the total number of M-components

from p to k and by letting k be the number of cate-

gories and q be the total number of components in

the k categories.)

The number of categories of mixture components

is general (k<1), and each category is represented

in every mixture by one or more of its member com-

ponents; that is, if ni� 2 components belong to the

ith category, thenPni

j¼1 xij ¼ ci,Pk

i¼1 ci ¼ 1, andPki¼1 ni ¼ q:

Similar to the approach used by Thomson and

Myers (1968), Cornell and Good (1970) assumed that

the experimenter’s interest is concentrated in a

region of interest centered at a point of main interest

denoted by x0. The region of interest in the space of

the mixture components is defined analytically to be

ellipsoidal in shape and of the form

Xki¼1

xi � x0i

hi

� �2

� 1 ½17�

where the x0i and hi (1� i� k) are chosen by the

experimenter. The x0i denotes the center of the

interval of interest for the ith component and hi is

a constant that allows for the spread of the symmetric

interval of interest for the ith component. This defi-

nition of a region of interest isolates the attention

to a specific area of interest and therefore enables

one to ignore other areas of the simplex factor space

FIGURE 11 A 12-point double-lattice configulation. The

blending of M-component 1 is vertical and the blending of

M-component 2 is horizontal.

J. A. Cornell 322

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that are not of interest. The factor space is a (q� k)-

dimensional convex polytope.

A transformation of the q linearly dependent mix-

ture components is made to q� k mathematically

independent variables, which are denoted by w1,

w2, . . .,wq-k. In addition to removing the physical

units associated with the mixture components, the

transformation simplifies the overall problem in that

the region of interest becomes a unit sphere in the

metric of the wi centered at wi¼ 0 (1� i� q� k),

which is much easier to work with than an ellipsoidal

region is, particularly when constructing designs.

To facilitate the understanding of the method-

ology, an example is presented involving the pro-

duction of polyethylene teraphthalate (a thin plastic

film for coating) from mixtures of two acids with

two glycols. The relation between the mixture

components and the independent design variables

is discussed and formulas are derived for the

relationship. Although the two- and three-category

situations were illustrated in detail by Cornell and

Good (1970), the methods are completely general

for larger numbers of categories. See Figure 12

where two acids x1 and x2 are blended with two

glycols x3 and x4.

The next 10 to 15 years brought authors from the

industrial sector. From DuPont in particular, Drs.

Ronald Snee and Don Marquardt were quick to

introduce the use of computers for designing and

modeling mixture data. The following articles were

a must-read for researchers that posed questions

such as:

. How do I know that I have a mixture experiment

rather than the usual independent variable

problem?

. If I have several (q� 3) mixture components, how

do I choose between using a simplex-lattice design

or a simplex-centroid design? If there are restric-

tions such as lower and=or upper bounds on the

components proportions, how do I proceed?

. In addition to mixture components, are there also

process variables that might affect the response or

affect the blending properties of the mixture

components?

Answers to these and other questions are readily

available from Snee (1971, 1973) and Marquart and

Snee (1974). In each of the two papers, Snee (1971,

1973) presented a lucid discussion of several fitted

model forms as well as ways of analyzing mixture

data. These papers were a must-read for the new-to-

mixtures reader because they appeared in print prior

to the review article by Cornell (1973). The paper by

Marquardt and Snee (1974) titled ‘‘Test Statistics for

Mixture Models’’ was the winner of The Jack Youden

Prize for the most outstanding expository paper to

appear in Technometrics in 1974. This paper dis-

cusses the correct test statistics for testing hypotheses

about the parameters (coefficients) in the Scheffe-

type mixture models. As an example, suppose for

q¼ 3 we wish to fit the second-degree model of Eq. [6],

y ¼ b1x1 þ b2x2 þ b3x3 þ b12x1x2

þ b13x1x3 þ b23x2x3 þ e

and test the null hypothesis,

H0: The response does not depend on the mixture

components.

or,

H0: b1 ¼ b2 ¼ b3 ¼ b0 and b12 ¼ b13 ¼ b23 ¼ 0

½18�

against the alternative hypothesis,

HA: The response does depend on the mixture

components.

FIGURE 12 Estimated response contours plotted over the

largest spherical region. Two acids x1 and x2 are blended with

two glycols x3 and x4 as categorized components.

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or,

HA: One or more of the ¼ in 18½ � is 6¼ ½19�

Then the test statistic is

Fcal ¼SSRegression ðp� 1Þ=

SSError ðN � pÞ=½20�

where p¼ 6 is the number of terms in the model and

N¼ the total number of observations. If in [20] the

value of FCal is <F(p-1,N-p,a) from the F-tables, then

do not reject H0 in [18], in which case the response

surface above the triangle looks approximately like

that shown in Figure 13.

On the other hand, if the value of FCal in [20] is

greater than or equal to the value of F(5,N-6,a), then

the shape of the surface above the triangle is not a

level plane as shown in Figure 13 but is a plane as

shown in Figure 14a or there is curvature in the

surface above the triangle as shown in 14b.

ADDITIONAL TOOLS USED FORSOLVING MORE DIFFICULTPROBLEMS, 1979–1989

Snee (1979a) suggested steps for generating the

coordinates of the extreme vertices of a highly

constrained region defined by the placing of con-

straints on the component proportions of the form

in [4]. Cornell and Khuri (1979) provided a transform-

ation for obtaining contours of constant prediction

variance on concentric triangles for ternary mixture

systems as shown in Figure 15.

Cox (1971) was the first to suggest measuring

component effects along rays inside the simplex by

fitting the standard polynomial model subject to

constraints placed on the coefficient estimates.

Piepel (1982) suggested a different direction by uti-

lizing the L-pseudocomponent simplex in defining

his direction to measure both partial and total effects

of the components; see Figures 16a and 16b.

Snee and Rayner (1982) assessed the accuracy of

mixture model regression calculations, and Gorman

and Cornell (1982) proposed a technique for reduc-

ing the form of the combined model containing both

mixture components and process variables.

Cornell and Gorman (1984) introduced fractional

design plans for including process variables in

mixture experiments as shown in Figures 17a

and 17b.

FIGURE 14 (a) Planar surface above the three-component triangle. (b) Curvature of the surface above the triangle along the edge x1–x2.

FIGURE 13 Surface defined by the hypothesis H0: b1¼ b2¼b3¼ b0 and b12¼b13¼ b23¼0.

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Gorman and Cornell (1985) fit equations to freez-

ing point data exhibiting eutectics for binary and

ternary mixture systems; see Figures 18a–18c.

Crosier (1986) outlined the geometry of con-

strained mixture experiments by providing a for-

mula for calculating the number of d-dimensional

boundaries (d¼ 0, 1, 2, . . ., q� 2) of a region

defined by the set of consistent constraints,

x1 þ x2 þ . . .þ xq ¼ 1; 0 � Li � xi � Ui � 1;

is

FIGURE 15 Obtaining constant prediction variance on concentric triangles for ternary mixture systems. Steps in the transformation of

circular variance contours to triangular-shaped contours. Four stages of the transformation are shown in Figures 8.8–8.11 of Chapter 8 in

Cornell (2002, p. 469–475).

FIGURE 16 Comparing the directions taken by Cox and by Piepel for measuring the effect of component 1: (a) Cox’s (1971) direction

and (b) Piepel’s (1982) direction.

FIGURE 17 Same half fraction of cooking conditions versus mixed half fraction of cooking conditions at the seven blends of a

simplex-centroid design consisting of three types of fish. See Cornell (2002, Fig. 7.9, p. 394).

325 A Retrospective View of Mixture Experiments

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Nd ¼ Cðq; q � d � 1Þ

þXq�d�1

r¼1

LðrÞCðq � r; q � r � d � 1Þ

�Xq

r¼dþ1

LðrÞ þ EðrÞ½ �Cðr ; r � d � 1Þ

½21�

where for r¼ 1, 2, . . ., q and d¼ 1, 2, . . ., q� 2,

RL ¼ 1 �Xqi¼1

Li and Rp ¼ Min RL;RUð Þ;

where L(r) is the number of combinations of

component ranges that sum to a number that is

lower than Rp; E(r) is the number of combinations

of component ranges that sum to Rp; and G(r) is

the number of combinations of component ranges

that sum to a number that is higher than Rp.

For d¼ 0 (the vertices), Eq. [21] simplifies to

N0 ¼ q þXqr¼1

LðrÞðq � 2rÞ � EðrÞðr � 1Þ½ � ½22�

where C(q,r)¼ q!=[r!(q� r)!]. Also, C(q, r)¼ L(r)þE(r)þG(r); see Cornell (2002, pp. 156–157).

Piepel and Cornell (1985) proposed models for

mixture experiments when the response also

depends on the total amount and, two years later,

Piepel and Cornell (1987) suggested designs for mix-

ture amount experiments; see Figure 19.

In believing that many mixture surfaces, when

viewed over the entire three-component triangle,

look and are actually more complicated in shape

than can be modeled with a first- or second-degree

Scheffe-type model, Cornell (1986) compared two

10-point designs for studying three-component

mixture systems that are capable of supporting a

cubic model or allow fitting a special quartic model;

see Figures 20a and 20b.

Sahrmann et al. (1987) searched for the optimum

Harvey Wallbanger recipe via mixture experiment

techniques. This tongue-in-cheek real-life experiment

was performed with 3 M design class employees by

showing them how to score flavor and how to use a

balanced incomplete block design to generate the

data followed by the fitting of a Scheffe-type quadratic

model to approximate the shape of the flavor surface

above the constrained region in Figure 21a. Figure 21b

is a copy of a typical flavor rating score sheet.

FIGURE 18 Three-dimensional plots of the freezing temperature surface for the biphenyl-bibenzyl-naphthalene system. Part (c) is part

(a) tilted to show the ternary eutectic T¼max(TI, TII, TIII) at the bottom of three intersecting surfaces TI, TII and TIII.

FIGURE 20 Three-component 10-point designs: (a) a f3, 3gsimplex-lattice; (b) a simplex-centroid design augmented with

three interior points.

FIGURE 19 DN-optimal designs for two levels of amount.

Designs 1 and 2 are equivalent for a fixed number of points.

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Cornell (1988) analyzed data from a real-life

mixture experiment containing process variables

using a split-plot approach.

THE LAST TWO DECADES, 1990–2009,FOCUSING ON APPLIED RESEARCH

In the five sections presented thus far, we intro-

duced the mixture problem and focused mainly on

the development of designs along with Scheffe-type

models. Time and space limitations have not allowed

us to cover and discuss the many different model

forms that have been suggested during the past 55

years. For the readers of this article who are inter-

ested in learning which empirical model forms have

proved or proven to work best, we recommend

Cornell (2002, chapters 6–8).

Additional topics of interest that have appeared

during the last two decades are listed below with full

reference information provided in the References

section.

1. Smith and Cornell (1993): ‘‘Biplot Displays for

Looking at Multiple Response Data in Mixture

Experiments’’ as shown in Figure 22.

2. Vining et al. (1993): ‘‘A Graphical Approach for

Evaluating Mixture Designs.’’

3. Montgomery and Voth (1994): ‘‘Multi-Collinearity

and Leverage in Mixture Experiments.’’

4. Cornell (1995): ‘‘Fitting Models to Data from

Mixture Experiments Containing Other Factors.’’

5. Heinsman and Montgomery (1995): ‘‘Optimiza-

tion of a Household Product Formulation Using

a Mixture Experiment.’’

6. Bowles and Montgomery (1997): ‘‘How to

Formulate the Ultimate Margarita: A Tutorial on

Experiments with Mixtures.’’

7. Cornell and Ramsey (1998): ‘‘A Generalized

Mixture Model for Categorized-Components

Problems with an Application to a Photoresist-

Coating Experiment.’’

8. Khuri et al. (1999): ‘‘Using Quantile Plots of the

Prediction Variance for Comparing Designs for

a Constrained Mixture Region: An Application

Involving a Fertilizer Experiment.’’

9. Anderson and Whitcomb (2002): ‘‘Designing

Experiments That Combine Mixture Compo-

nents with Process Factors.’’

10. Draper and Pukelsheim (2002): ‘‘Generalized

Ridge Analysis under Linear Restrictions with

FIGURE 21 (a) The seven Harvey Wallbanger blends and their locations in the triangle relative to the ratio variables constraint region.

(b) Official Judge ‘‘Harvey Wallbanger’’ flavor evaluation scoring sheet.

FIGURE 22 Biplot illustrating the effects of glass fibers, resin,

and microspheres on two strength responses, two modulus

responses, and warp of a plastic compound. The projection arrow

from the resin component onto the wrap vector illustrates the

positive effect of resin on the wrap.

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Particular Applications to Mixture Experiment

Problems.’’

11. Kowalski et al. (2002): ‘‘Split-Plot Designs and

Estimation Methods for Mixture Experiments

with Process Variables.’’

12. Prescott et al. (2002): ‘‘Mixture Experiments:

Ill-Conditioning And Quadratic Model

Specification.’’

13. Cornell and Gorman (2003): ‘‘Two New Mixture

Models: Living with Collinearity but Removing

Its Influence.’’

14. Goldfarb et al. (2003): ‘‘Mixture-Process Variable

Experiments with Noise Variables.’’

15. Goldfarb et al. (2004): ‘‘Evaluating Mixture-

Process Designs with Control and Noise Variables.’’

16. Goos and Donev (2006): ‘‘The D-Optimal

Design of Blocked Experiments with Mixture

Components.’’

The following textbooks and a booklet from the

American Society for Quality have provided infor-

mation on experiments with mixtures since 1981:

Cornell (1981, 1983, 1990a, 1990b, 2002, 2011) and

Smith (2005). Currently a primer is in press with a

publication date scheduled for summer 2011.

TOPICS FOR FUTURE RESEARCH

We come to the point in time now where it seems

only fitting to ask, ‘‘Have most of the questions

surrounding designs and models for setting up, for car-

rying out, and used to analyze data from experiments

with mixtures raised during the past half century been

answered? Or are there still questions that are unan-

swered? The retrospective journey taken in this article

was mainly ‘‘What’s been done?’’ Certain unanswered

questions still remain that create a void in my mixtures

toolbox, and some of them are listed here:

1. Are there ways to measure the effects of the

components other than using response trace plots

when the experimental region is a constrained

non-simplex-shaped region inside the original

simplex? See Figure 23.

2. Are there model-free techniques for fractionating

designs particularly with mixture experiments

containing other variables? See Figure 24.

3. Today’s software has allowed standard forms of

polynomial models to be fitted to mixture data

and to create excellent and sophisticated plots.

Very little has been done to explore surface

shapes using logistic regression and loglinear

models. Why is this?

4. Has today’s software been updated to include the

modified L-pseudocomponents or the centered

and scaled intercept model of Cornell and

Gorman (2003)?

REMEMBERING DR. JOHN W.GORMAN; A FRIEND TO EVERYONEWHO WORKED WITH MIXTURES

Dr. John W. Gorman was born in 1925 near Sioux

Falls, South Dakota. As a youth he joined the

Friendly Indians, a forerunner of the Boy Scouts.

Growing up on a farm, John felt that farm life had

a way of building up strength, energy, and an interest

in life and learning.

At the age of 17, John joined the Army Specialized

Training Program (ASTP), which offered enlistees an

opportunity to attend a college of their choice once

they completed basic training. Later the ASTP was

called the GI Bill. By April of 1944, John, then 18,

was stationed in England, where he volunteered for

the U.S. Army Rangers. He landed in Normandy less

than a week after D-Day and for the next 18 months

served as an infantry scout in the 2nd Ranger

Battalion, later to be recognized as one of the most

decorated units of the war. John served with distinc-

tion during WWII, earning the Combat Infantry

Badge, Purple Heart, and Bronze Star Medal while

with the 2nd Ranger Battalion in the European

Theater.

Following the war, John enrolled at the University

of Minnesota and earned a B.S. in chemistry and a

Ph.D. in chemical engineering. After graduation, he

became a process engineer at the Knolls Atomic

Power Laboratory in Schenectady, New York. A

major portion of his professional life, however, was

spent with Amoco Oil Company as a research associ-

ate specializing in product and process R&D. Like

Jack Youden, John taught himself statistical design

and analysis of experiments. John was especially

fond of mixture experiments.

My initial correspondence with John was in 1972,

when he served as an associate editor for Techno-

metrics. An eventual paper, titled ‘‘Experiments with

J. A. Cornell 328

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Mixtures: A Review,’’ was handled by John. I first

met John in person at the 1975 Gordon Research

Conference and, as luck would have it, we were

assigned as roommates. It was during that week

that I was able to learn a year’s worth of mixtures

mainly through questions I asked John as well as

from the answers he gave me. He had a keen sense

and feel for mixture designs and was always willing

to share his wealth of knowledge on the fitting

and testing of mixture models. Including process

variables in mixture experiments was an area that

he and I shared a common interest in, and Cuthbert

Daniel encouraged us to work together, which

we did.

As I look back at the many highlights that John

and I experienced during a 40-year period of doing

research on mixtures, some of my fondest memories

were working with John while putting together

Cornell and Gorman (1978, 1984, 2003) and Gorman

and Cornell (1982, 1985).

John’s many contributions to the American

Society for Quality Control (ASQC), now (ASQ),

and the American Statistical Association (ASA)

earned him the honorary rank of Fellow of ASQ

and ASA. He was a member of the WWII Rangers

Association, a member of the Unity Men’s Club,

the Skylight Club, and the American Legion. Those

who had the good fortune of knowing John have

undoubtedly missed his compassionate love of

family, gardening, and friends; his thirst for solving

research problems; and, most of all, his warm and

characteristic smile.

SUMMARY

Fifty-eight years have passed since the first men-

tion of a mixture experiment appeared in the statistics

literature. Very few authors list Quenouille’s book,

which first appeared in 1953, in their references,

however. I believe the reason is because they either

are not aware of this book or, like Scheffe (1961),

they believe that Quenouille’s approach to modeling

ingredient blending is different from Scheffe’s and

they prefer Scheffe’s. And why not? Piepel (2006)

compiled a bibliography of mixture experiment pub-

lications that numbers over 700 entries, with roughly

half of them appearing in non-statistics literature.

Piepel’s (2006) chapter lists 360 references and, yes,

he does include Quenouille’s book.

This retrospective view of the past 57 years has

been slightly more extensive than I had planned

on reporting when I initially agreed to take this pro-

ject on. Beginning with the section titled The Mixture

Problem we began with a few examples of everyday

activities such as sweetening coffee or tea, mixing oil

and vinegar to create an Italian dressing for a tossed

salad, and blending higher 93-octane fuel with

87-octane fuel presently in the family car to improve

miles per gallon or driving performance. These

activities fall into the class of mixture experiments.

The next ten sections cover The Beginning

Years—1955 to 1965; The Second Wave of Problems

to Be Solved, 1966–1978: Enter the Computer;

Additional Tools Used for Solving More Difficult

Problems, 1979–1989; The Last Two Decades,

1990–2006; and Topics for Future Research, respect-

ively. In the next to last section, a tribute is paid to

Dr. John W. Gorman, a friend to everyone who

worked with mixtures. John, age 83, passed away

on June 4, 2009, at his home in Plymouth, Minnesota.

He was a quiet gentleman with a warm and charis-

matic smile who loved and inspired others to work

on mixture problems. He will be missed.

Finally, it is time to thank those individuals who

helped me put together this piece of history. I want

to thank Professors Geoff Vining and Douglas C. Mon-

tgomery for serving as discussants at the Friday morn-

ing Invited Session on October 8, 2010, sponsored by

the CPI Division at 54th Fall Technical Conference,

held in Birmingham, Alabama. It is a nice feeling to

know that the topic of mixture experiments is in the

good hands of these two pros as well as others cur-

rently working with mixtures. One of those is the Edi-

tor of Quality Engineering, Dr. Connie Borror, whose

name appears with others in the list of references.

Another special thanks goes out to Ms. Elizabeth Leis,

Production Editor, who graciously pushed me to fin-

ish this article by the publication month, and last but

certainly not least is Dr. Busaba Laungrungrong,

who checked my spelling and helped me put this arti-

cle in proper order.

ABOUT THE AUTHOR

John A. Cornell is a Emeritus Professor of Statistics

who formerly spent 36 years with the Agricultural

Experiment Station at the University of Florida,

Gainesville. The author of more than 150 technical

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articles, and three books titled Experiments with Mix-

tures: Designs, Models, and the Analysis of Mixture

Data, A Primer on Experiments with Mixtures, and

Response Surfaces: Designs and Analyses (co-

authored with A. I. Khuri), he is a past Editor of

the Journal of Quality Technology. A past recipient

of the W. J. Youden Prize, the Shewell Prize, the

Brumbaugh Award and The Shewhart Medal from

the American Society for Quality (ASQ), he is a

Fellow of the American Statistical Association and

the ASQ, and a past elected member of the Inter-

national Statistical Institure. Dr. Cornell received the

B.S.E. (1962) and M.Stat. (1965) degrees from the

University of Florida, Gainesville, FL, and the Ph.D.

(1968) in statistics from Virginia Polytechnic Institute

and State University, Blacksburg, Va.

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331 A Retrospective View of Mixture Experiments

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