• TABLE OF CONTENTS
HIDE
 Front Cover
 Title Page
 Table of Contents
 Introduction
 A brief description of mixture...
 Plotting considerations
 Program description
 Some popular examples of mixture...
 An agronomic experiment
 Conclusions
 Appendix 1. Program statements...
 Appendix 2. Program statements...
 Appendix 3. Program statements...
 Appendix 4. Program statements...
 Appendix 5. Program statements...
 Literature cited
 Back Cover
 Historic note






Group Title: Bulletin - Agricultural Experiment Stations, University of Florida ; 836
Title: Plotting three-dimensional response surfaces for three-component mixtures or two-factor systems
CITATION THUMBNAILS PAGE IMAGE ZOOMABLE
Full Citation
STANDARD VIEW MARC VIEW
Permanent Link: http://ufdc.ufl.edu/UF00027459/00001
 Material Information
Title: Plotting three-dimensional response surfaces for three-component mixtures or two-factor systems
Series Title: Bulletin Agricultural Experiment Stations, University of Florida
Physical Description: 31 p. : ill. ; 23 cm.
Language: English
Creator: Cornell, John A., 1941-
Publisher: Agricultural Experiment Stations, Institute of Food and Agricultural Sciences, University of Florida
Place of Publication: Gainesville Fla.
Publication Date: 1983
 Subjects
Subject: Mixtures -- Statistical methods   ( lcsh )
Mixtures -- Experiments   ( lcsh )
Response surfaces (Statistics)   ( lcsh )
Genre: government publication (state, provincial, terriorial, dependent)   ( marcgt )
bibliography   ( marcgt )
non-fiction   ( marcgt )
 Notes
Bibliography: Bibliography : p. 31.
Statement of Responsibility: John A. Cornell ... et al..
 Record Information
Bibliographic ID: UF00027459
Volume ID: VID00001
Source Institution: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
Resource Identifier: aleph - 000401519
oclc - 10680657
notis - ACE7367

Table of Contents
    Front Cover
        Front Cover
    Title Page
        Title Page
    Table of Contents
        Table of Contents
    Introduction
        Page 1
    A brief description of mixture experiments
        Page 2
        Page 3
        Page 4
    Plotting considerations
        Page 5
    Program description
        Page 6
        Page 7
    Some popular examples of mixture experiments
        Page 8
        Page 9
        Page 10
        Page 11
        Page 12
        Page 13
        Page 14
        Page 15
        Page 16
    An agronomic experiment
        Page 17
        Page 18
        Page 19
        Page 20
    Conclusions
        Page 21
        Page 22
        Page 23
        Page 24
    Appendix 1. Program statements for producing the surface contour plot in figure 3
        Page 25
    Appendix 2. Program statements for producing the 3D surface contour plot in figure 4
        Page 26
    Appendix 3. Program statements for producing the 3D plots in figures 5b, 5c, and 5d
        Page 27
        Page 28
    Appendix 4. Program statements for producing the contour plot in figure 10a
        Page 29
    Appendix 5. Program statements for producing the 3D surface plot in figure 10b
        Page 30
    Literature cited
        Page 31
    Back Cover
        Page 32
    Historic note
        Page 33
Full Text
)}


June 1983 Bulletin 836


Plotting Three-Dimensional Response Surfaces for
Three-Component Mixtures or Two-Factor Systems


John A. Cornell, John T. Shelton, Richard Lynch, and
Gregory F. Piepel


Agricultural Experiment Stations
Institute of Food and Agricultural Sciences
University of Florida, Gainesville
F. A. Wood, Dean for Research











Plotting Three-Dimensional Response Surfaces for
Three-Component Mixtures or Two-Factor Systems


John A. Cornell, John T. Shelton, Richard Lynch, and
Gregory F. Piepel


Department of Statistics, Institute of Food and
Agricultural Sciences, University of Florida





















AUTHORS
John A. Cornell is a professor in the Department of Statistics, Institute of Food
and Agricultural Sciences, University of Florida. John T. Shelton is a former
graduate student at the University of Florida, now an assistant professor in the
Department of Mathematics, University of North Carolina at Greensboro.
Richard Lynch and Gregory F. Piepel are graduate students in the Department of
Statistics, University of Florida.











CONTENTS

Introduction ................... ...... ...... ............... 1
A Brief Description of Mixture Experiments .......................... 2
Plotting Considerations ....................... ................. 5
Program Description ...................... ..................... 6
Some Popular Examples of Mixture Experiments ..................... 8
Example No. 1: Fruit Punch Experiment .......................... 9
Example No. 2: Fish Patty Experiment ........................... 12
Example No. 3: Rocket Propellant Experiment ..................... 13
An Agronomic Experiment .................. ..... ................. 17
Example No. 4: Peanut Yield Experiment ......................... 20
Conclusions .................................................. 21
Appendix 1 ..................... .......................... 25
Appendix 2 ......................... ...... .............. 26
Appendix 3 ...................................... ... 27
Appendix 4 ......................................... .... 29
Appendix 5 .................... ......................... 30
Literature Cited ................... .............. .... ........... 31









INTRODUCTION
Mixture experiments are performed in many areas of research where
basic ingredients are blended together (Cornell, 1981). Typical examples
in IFAS are research in food formulation and in blending chemicals to
form pesticides. Normally the experimenter or researcher seeks to deter-
mine how measured properties of the product formed from the blend
differ with changes made in the blends. A popular technique used for
studying changes in the measured response as a function of the ingre-
dients affecting the response is known as studying the response surface.
Estimated values of the response, obtained with some type of statistical
model or equation, are plotted against combinations of the settings of the
ingredients.
When a sufficient number of estimated response values are obtained,
the result is an estimated response surface. Three-dimensional repre-
sentations accentuate the surface shape characteristics by providing a
pictorial view of the changes in the response over the experimental
region. The shape of the surface can aid the experimenter in locating
maximum or minimum values of the response as well as in determining
what type of influence the ingredients have on the response both indi-
vidually and in combination with one another. The accentuation of the
surface shape is even more pronounced when the axes of the plane of the
experimental region that supports the surface is rotated, enabling the
three-dimensional surface to be viewed from different directions.
Of particular interest to us in this paper is the three-dimensional
plotting of a response surface positioned above the two-dimensional
triangle that defines the experimental region for a three-component
mixture experiment. (We shall also plot the response surface for an
agronomic experiment relating the yield of peanuts as a function of the
levels of two types of fertilizers where the experimental region of the
fertilizer levels is a square.) When the blends contain more than three
ingredients, the triangular region can accommodate values for only three
of the ingredients; therefore the values of the remaining ingredients are
held fixed. Ordinary contour plotting (Hare and Brown, 1977; Koons and
Heasley, 1981) is satisfactory for many types of mixture surfaces, but
when given the option of viewing a three-dimensional picture of the
shape of the surface, most experimenters would likely choose to exercise
this latter option to complement the surface contour plot.
SAS/GRAPH (SAS Institute, 1980) is a computer graphics system
that produces color plots, charts, and other displays on screens and
plotters. In SAS/GRAPH there is available a G3D procedure that plots
the values of the three variables, producing a three-dimensional surface.







Options are available to tilt and rotate the picture using different angles
in order to improve on the visibility of the surface. Response surface plots
for three separate mixture experiments as well as for a standard two-
factor agronomic peanut yield experiment are provided in this paper to
illustrate the usefulness of the three-dimensional plotting procedure.

A BRIEF DESCRIPTION OF MIXTURE EXPERIMENTS
Many products are formed by mixing together two or more ingre-
dients. Some examples are:
1. Cake formulations using baking powder, shortening, flour, sugar, and
water.
2. Sandwich fish patties made from blends of mullet, sheepshead, and
croaker (Cornell and Deng, 1982).
3. Tropical beverage blends consisting of juices from watermelon,
pineapple, and orange (Huor, et al. 1980).
4. Pesticide formulations made from blending the chemical constituents
Vendex, Omite, Kelthane, and Dibrom.
Properties of the various products such as the fluffiness or layer
appearance of the cake, the texture of the fish patties, the fruitiness flavor
of the beverage, and the toxicity of the pesticide depend on the relative
proportions of the ingredients that make up the blends. To study the
functional relationship between the property of a product and its com-
position, an experimenter varies the composition by changing the ingre-
dient proportions and observes the changes that occur in the property
being measured. Such a course of action is undertaken when seeking to
find the combination of the ingredients that is considered best in the sense
that it produces the optimum value for the property or response under
investigation (Cornell, 1981).
In mixture experiments, the ingredients (or components of the mix-
ture) represent proportionate amounts of the mixture and are measured
in units of a fractional amount of the total volume, total weight, or mole
fraction. The ingredient proportions are nonnegative and if expressed as
fractions of the mixture, they sum to unity. If the number of ingredients in
the system is denoted by q and if the proportion of the ith component in
the mixture is represented by xi, then
xi-0 i= 1,2,...,q (1)
and
q
X l+x2 -...-- + = 1. (2)
i=1
Of course the xi could represent nonnegative percentages of the total
mixture but when divided by 100%, they would again be fractions as in
Equation 2.







Since the proportions xi in Equation 2 are constrained variables, the
experimental region defined by the values of the xi is different from that in
standard factorial-type experiments. For example, in a 2q factorial ex-
periment, where the xi take the values 1, the experimental region is a
q-dimensional cuboid (square if q = 2, cube if q = 3). In a q-component
mixture experiment, the experimental region defined by the constraints
on the values of the xi in Equations 1 and 2 consists of all of the points of a
regular (q 1)-dimensional simplex. For q = 2 components, the region is
a straight line, while for three components, the region is an equilateral
triangle as shown in Figure la.
The coordinate system for the values of the xi's in a mixture experi-
ment is called a simplex coordinate system. With three components for
example, the coordinates can be plotted on triangular graph paper which
has lines parallel to the three sides of an equilateral triangle. The vertices
of the triangle represent the individual ingredients and are denoted by
xi = 1, xj = Xk = 0 for i, j, k = 1, 2, and 3 (Figure Ib). The interior points of
the triangle represent mixture blends in which all of the component
proportions are nonzero, that is, xl > 0, x2 > 0, and x3 > 0. The centroid
of the triangle corresponds to the mixture having equal proportions
(1/3,1/3,1/3) of each of the components.
Besides the difference in experimental regions, mixture experiments
differ from the ordinary regression problems in the form of the mathe-
matical or statistical model that is used to represent the functional rela-
tionship between the response measured and the ingredient proportions.
This is a consequence of the constraints (Equation 2) on the xi's. Some of


100% of Ingredient 1
= 1 (1,0,0)
1


Sx =1
S(1,0,0)
100% of Ingredient 3 100%:0%:0%
(0,0,1)
x3 = (f,4,o) / \,(,o0,)
x 12 50%:50%:0%/ 50%:0%:50%
(0,1,0)
100% of Ingredient 2 (1/3,1/3,1/3)
S 33%:33%:33% \3= 1
X2= If- ---- --- 3 ~
(0,1,0) (0,,) (0,0,1)
0%:100%:0% 0%:50%:50% 0%:0%:100%

Figure la. The three-component triangular experimental region. The coordi-
nates are denoted by (xl, x2, x3).
Figure lb. The triangular coordinates (x1, x2, x3) and the ingredient percentages
(% ingredient 1: % ingredient 2: % ingredient 3).







the more popular forms of mixture models, where 1 is the true response
and the xi's are ingredient proportions, are

Linear or first degree:
q
tl.= Pixi (3)
i=1
Quadratic or second degree:
q q
"q= 1 PiXi + ojixjxi (4)
i=1 i Cubic or third degree:
q q q
I= I Pixi +I I Pijxixj + I I Yixi x(Xi Xj)
=1 i q (5)
+ I, i3 jkXiXjXk
i
Special cubic:
q q q
1= 1 Px + PiXxx + i Ii pijkXiXjXk (6)
i = 1 i where the ~i, pi, Yii, and Pijk are regression coefficients that are to be
estimated. We notice that the mixture models of Equations 3-6 do not
contain a constant term, 3o, nor terms such as x?, x4, and xixi which are
present in the standard polynomial models.
In three components, the models of Equations 3-6 are

I = P1X1 + P2X2 + P3X3
T = PIXl + 2X2 + P3X3 + P12XlX2 + P13X1X3 + P23X2X3
71 = PiX1 + P2X2 + P3X3 + P12X1X2 + P13X1X3 + P23X2X3
+ Y12X1X2(X1 X2) + 13X1X3(X1 X3) + 723X2X3(X2 X3)
+ P123X1X2X3

Tr = P1X1 + P2X2 + 33X3 + 312X1X2 + P13X1X3 + P23X2X3 + P123X1X2X3
In an experimental program consisting of N trials, the observed value,
Y,, of the response at the uth trial, is assumed to vary about a mean of rq,
that is,
Yu= =r + Eu

where the experimental errors eu are assumed to be uncorrelated and
identically distributed with zero mean and common variance, a2, for all
u=1,2,...,N.
After the N observations have been collected, the unknown param-
eters or coefficients (Pi, Pi, etc.) in the model are estimated by the
method of least squares. The estimated coefficients are substituted into







the model and the resulting fitted model is then used as a deterministic
function to generate values of the response corresponding to numerous
combinations of the ingredient proportions throughout the mixture ex-
perimental region. For example, if in a three-component mixture setting
the estimates (bi and bij) of the coefficients in the quadratic model of
Equation 4 are calculated, having collected N- 6 observations, then the
second-order fitted model would be
S= bxxl + b2x2 + b3X3 + b12x1x2 + bl3X1X3 + b23X2x3
where Y (read "Y hat") represents the predicted or estimated value of r-
for given values of x1, x2, and x3. The fitted model is used for generating
values of the response in the contour plots as well as in the three-
dimensional plots that are presented in this paper.
Reiterating once again the strategy employed in this paper, we shall
limit the number of ingredients in our mixtures to three in studying
surface plots, keeping in mind that if q>3 the values of the remaining
q 3 ingredients are held fixed. For example, if we have a mixture system
consisting of q = 5 ingredients but we are interested only in studying
changes in the response brought about by changing the proportions xl, x2,
and x4, then the proportions x3 and x5 are held fixed (say for example,
x3 + x5 = 0.40). The proportions xl, x2, and x4 are limited in size to being
less than or equal to 1 0.40 = 0.60 and can be varied in such a way that
they must satisfy the equality x 2 + x4 = 1 (x3 + X5) = 0.60. Chang-
ing the scale of the proportions xi, i = 1,2, and 4 by writing x' = x1/0.60,
x = x2/0.60, and x4 = x4/0.60 forces the restriction on the x! to be
xi + x2 + x1 = 1.0 so that for convenience, we can then work with the xf,
i = 1,2, and 4 in place of the original xi.

PLOTTING CONSIDERATIONS
To use the PLOT procedure in SAS to plot contours of the surface or
to use the G3D procedure in SAS/GRAPH to plot a three-dimensional
surface over the three-component triangle, we first have to perform a
linear transformation from the simplex coordinate system to a two-
dimensional cartesian coordinate system. Several transformations are
available to do this (Cornell, 1981, Chapter 3). We choose the following
transformation
1 1
V (2 3x2 3x3), V2= (3 2). (7)

This transformation maps the triangular region in the coordinates (xl, x2,
x3) into another triangular region in the coordinates (V1, V2) so that every
point in and on the triangle can be expressed in units of V, and V2 (Fig-
ure 2). The centroid (xl, x2, 3) = (1/3, 1/3, 1/3) is expressable as
(V1, V2) = (0, 0).








x =


(0,0)2

3 = 1
x2 = x2 = 1 x =1

Figure 2. The mapping of the mixture component region into the two-
dimensional triangle of V1 and V2 by Equation 7.


PROGRAM DESCRIPTION
The contour option of the plot procedure in SAS produces contour
plots of the values of a response variable expressed as a function of the
levels of the two variables V1 and V2. You can specify the values of the
variables you want plotted in a PLOT statement; GCONTOUR automat-
ically scales the vertical and horizontal axes. Plotted in Figure 3 are the
values of Y from 4.60 to 7.10 using the second-degree polynomialt
l= 4.77x1 + 6.27x2 + 7.11x3 + 2.15xlx2 + 1.10x1x3 3.54x2x3 (8)
for values of V1 from 0.4 to 0.8 and values of V2 from 0.7 to 0.7 where
V, and V2 are functions of x2 andx3 (and realizing that x, = 1 x2 x3) as
shown in Equation 7. The extreme values 0.4 and 0.8 for V1 occur when
x2 + x3 = 1 and x2 + x3 = 0, respectively, as calculated using Equation 7,
while the extreme values 0.7 and 0.7 for V2 occur when x2 = 1 (and thus
Xl = x3 = 0) and when x3 = 1 (and thus xl = x2 = 0). A listing of the SAS
program statements used to generate the contour plot in Figure 3 is given
in Appendix 1.
To generate the plot in Figure 3, approximately 815 estimated re-
sponse values were calculated in SAS using a grid of values for xl, x2, and
x3 which were transformed to values of V, and V2. These response values,
when plotted on the triangle, completely fill the triangle, but only the
contour lines (or curves) that represent specific values of the estimated
response are plotted. To produce the contour plot in Figure 3, we
specified the estimated values to be Y = 4.9, 5.3, 5.8, 6.0, 6.3, 6.5, 6.8,
and 7.1.

tThe particular model of Equation 8 is taken from the fruit punch experiment that
is described in greater detail in Example No. 1 in the section on examples.









1.08 6--

0.952








0. 66-









\6.5
/1 / \o
1 : / "5.8 /










0 x2= A 3= 1



0. -06 -0.2 0.0 0.2 0.4 0.6 0.6
V2
Figure 3. Fruit punch experiment: contour plot for Equation 8.


The contour curves in Figure 3 show how the surface height decreases as
one approaches the x, = 1 vertex of the triangle and increases as one
approaches the x3 = 1 vertex of the triangle.
The G3D procedure in SAS/GRAPH plots the values of three vari-
ables, and V2, producing a three-dimensional plot. Plotted in










Figure 4 are the values of 2 from 4.60 to 7.10 using Equation 8 corre-
sponding to the same values of V, and V2 that produced the contour plot
in Figure 3. A listing of the program statements used to generate the
three-dimensional plot in Figure 4 is given in Appendix 2.
-0,6- 2


-0.3 -0.6 -O.'i -0.2 0.0 0.2 0./1 0.6 0.6


Figure 3. Fruit punch experiment: contour plot for Equation 8.


The contour curves in Figure 3 show how the surface height decreases as
one approaches the x1 = 1 vertex of the triangle and increases as one
approaches the x3 = 1 vertex of the triangle.
The G3D procedure in SAS/GRAPH plots the values of three vari-
ables, Y, V1, and V-2, producing a three-dimensional plot. Plotted in
Figure 4 are the values of Y from 4.60 to 7.10 using Equation 8 corre-
sponding to the same values of V1 and V2 that produced the contour plot
in Figure 3. A listing of the program statements used to generate the
three-dimensional plot in Figure 4 is given in Appendix 2.
The plane containing the two-dimensional triangle in V, and V2 in
Figure 4 is automatically set at a tilt angle of 700 and a rotation angle of






















6.06





0.90

5.03 xl




V2


000.90 -0.23
0.33

V) -0.23
2 -0.80.80
-0.80

Figure 4. Fruit punch experiment: 3D plot for Equation 8.



700. These values for the tilt and rotation angles are referred to as
comprising the default option. Another option provided in the G3D
procedure is the availability to change the tilt angle and/or rotation angle.
Performing such a change might enable the surface shape characteristics
to be viewed more clearly. We shall exercise this option in the rocket
propellant example No. 3.


SOME POPULAR EXAMPLES OF MIXTURE EXPERIMENTS

The following examples have appeared in the literature on mixture
experiments and help to illustrate the usefulness of the G3D procedure in
complementing the surface contour plots.







EXAMPLE NO. 1: FRUIT PUNCH EXPERIMENT
(Huor, et al., 1980)
Watermelon (xl), pineapple (x2), and orange (x3) juice concentrates
were used as primary ingredients of a fruit punch. Ten blends of the
three-juice concentrates were evaluated for overall general acceptance
by a sensory panel. The ingredient proportions and the average accept-
ance values scored on a scale of 1 (extremely poorer than reference) to 9
(extremely better than reference) for three replications of each blend are
listed in Table 1.

Table 1. Fruit punch general acceptance ratings.

Watermelon Pineapple Orange General
Acceptance Average
%W (xi)t %P (x2) %O (x3) (YU)

100 1.0 0 0 0 0 4.3, 4.7, 4.8 4.60
65 0.5 35 0.5 0 0 6.3, 5.8, 6.1 6.07
30 0 70 1.0 0 0 6.5, 6.2, 6.3 6.33
30 0 35 0.5 35 0.5 6.2, 6.2, 6.1 6.17
30 0 0 0 70 1.0 6.9, 7.0, 7.4 7.10
65 0.5 0 0 35 0.5 6.1, 6.5, 5.9 6.17
54 0.34 23 0.33 23 0.33 6.0, 5.8, 6.4 6.07
80 0.72 10 0.14 10 0.14 5.4, 5.8, 6.6 5.93
40 0.14 40 0.57 20 0.29 5.7, 5.0, 5.6 5.43
40 0.14 20 0.29 40 0.57 5.2, 6.4, 6.4 6.00

t %W-30 %P %0
x- -=, X2 ,X --
70 70 70


The second-degree model fitted to the 30 acceptance values in Table 1
is

Y = 4.77x1 + 6.27x2 + 7.11x3 + 2.15x1x2
+ 1.10x1X3 3.54x2X3
The contour plot and the 3D plots of the estimated acceptance surface are
shown in Figures 5a, 5b, 5c, and 5d. The 3D plots in Figures 5b, 5c, and 5d
are of the same surface shown in Equation 9 but viewed from different
directions. The directions taken in Figures 5c and 5d were obtained by
rewriting the two-dimensional triangle with the transformation of Equa-
tion 7 using the coordinates

U, 1- (2 3x3 3x), U2 ( 3) (10)

for Figure 5c, and









0.914 -
4




oo-jA
Xl= 1






l / "'... "-- --- "
i 5.8




,. -/ v. 6.0.. /


\6.0 6.3
- .18 1 6

i /1 / \ \ 6..
q 5.8
| -- L ', ', i3
1 XI1
-0.6 1X2 X3=


-0.3 -0.6 -0..r -0. C.0 0.2 ua.


Figure 5a. Fruit punch experiment: contour plot for Equation 8.


-J


.1.33


S -i


S0 ..- 0. 80
-O ol~ 0.00


Figure 5b. Fruit punch experiment.
Y = 4.77x1 + 6.27x2 + 7.11x3 + 2.15xix2 + 1.10x1x3
3.54x2X3










7.1.1

7. L7


7-.- 090
-/


Figure 5c. Fruit punch experiment.
Y= 4.77x1 + 6.27x2 + 7.11x3 + 2.15x1x2 + 1.10xix3
3.54x2x3


5.02


0.90


Figure 5d. Fruit punch experiment.
Y = 4.77x + 6.27x2 + 7.11x3 + 2.15x1x2 + l.10x1x3
3.54x2x3







1 1
Wi (2 3x 3x2), W2 = (x ) (11)

for Figure 5d. The program statements used to produce Figures 5b, 5c,
and 5d are given in Appendix 3.
The vertical heights of the surface above the triangle in Figures 5b, 5c,
and 5d represent the acceptance scores for the juice combinations 100%
watermelon, 70% watermelon with 30% pineapple, and 70% water-
melon with 30% orange, respectively. These heights are reflected by the
estimates (bi) of the coefficients (3;i) in the model of Equation 9. Note
that the height of the triangular base is Y = 4.0, which can be easily
readjusted in the program, if desired. Since b3 > b2 > bl, the combination
of 30% orange juice with watermelon juice appears to be more accept-
able than the pineapple juice combination, which is more acceptable than
the 100% watermelon juice.
The quadratic shape (represented by the crossproduct coefficient
estimates) of the juice acceptance surface stands out when viewing the
surface from the different directions. Figure 5b illustrates the synergistic
(beneficial) blending effect of pineapple (x2) with watermelon (xl); that
is, the acceptance score increases with a decrease in the percentage of
watermelon and an increase in the percentage of pineapple; Figures 5c
and 5d respectively show the synergistic blending effect of orange (x3)
with watermelon (xl) and the antagonistic (nonbeneficial) blending effect
of blending pineapple (x2) with orange (x3).
An obvious drawback to the use of the 3D plots only lies in attempting
to determine the mixture blend or blends that produce a specific value for
the height of the estimated surface. This is where the contour plot (Figure
5a) proves to be more useful than the 3D plot. For example, to estimate
several juice blends that provide a fruit punch with an acceptance score of
approximately 6.0, one may select any of the blends (xl, x2, x3) that are
singled out by the curves for Y = 6.0 in Figure 5a. One such blend,
denoted by the open circle in Figure 5a, has as coordinate values
x, = 0.42, x2 = 0.43, and x3 = 0.15, representing approximately 59.4%
watermelon, 30% pineapple, and 10.6% orange. Using Equation 9, this
combination would produce a fruit punch with an estimated acceptance
rating of Y= 5.998.

EXAMPLE NO. 2: FISH PATTY EXPERIMENT
(Cornell and Deng, 1982)
Three fish species, mullet (xl), sheepshead (x2), and croaker (x3),
were blended together to form patties. Also studied were the effects of
cooking temperature, cooking time, and deep fat frying time. At the
following combination of cooking temperature = 375F, cooking
time = 25 minutes, and deep fat frying time = 25 seconds, the average







texture of the patties (in grams of force x 10-3) was modeled with the
special cubic equation (6), to give
Y= 1.84x, + 0.67x2 + 1.51x3 + O.14x1x2 1.Olx1x3 (12)
+ 0.27x2X3 + 8.68x1X2X3
The contour plot in grams of force and the 3D plots of the estimated
texture surface are represented in Figures 6a, 6b, 6c, and 6d. The plots in
Figures 6c and 6d are views of the surface taken in the directions specified
by the transformations of Equations 10 and 11, respectively.
The shape characteristics of the special cubic texture surface are
approximately interpreted by the contour plot of Figure 6a. Of the
individual fish types, sheepshead (x2) is the softest (having the lowest
texture value) and mullet (xi) is the firmest. It is not particularly easy to
see what effect (synergistic or antagonistic) on texture is caused by
blending pairs of fish types. However, we can look at the fitted model of
Equation 12 and notice that perhaps the 50%:50% blend of mullet and
croaker (through the value of b13 = -1.10) lowered the texture ratings
compared to the texture ratings of the pure mullet and pure croaker
patties. Also, blending all three fish types appeared to raise the texture
response, as evidenced by the sign of b123 (= +8.68) in the special cubic
term of Equation 12.
The 3D plots in Figures 6b, 6c, and 6d provide a clearer description of
the shape characteristics of the texture surface. All three figures clearly
show the lower texture of the sheepshead patties (x2), while Figures 6c
and 6d nicely illustrate the effect of lowering the texture of patties due to
blending mullet (xi) with croaker (x3). Furthermore, Figure 6d repre-
sents the special cubic nature of the surface reflected by the 8.68x1X2X3
term in Equation 12. This mound-like appearance is only mildly antici-
pated by looking at Figures 6b and 6c and is hardly noticeable when
studying the contour plot of Figure 6a. Figure 6d thus illustrates the
usefulness of viewing the shape of the surface from all three directions or
from all three sides of the mixture triangle.

EXAMPLE NO. 3: ROCKET PROPELLANT EXPERIMENT
(Kurotori, 1966)
In making a certain type of propellant, a mixture of binder (B),
oxidizer (0), and fuel (F) is needed in the proportions
B 0.20, O0 0.40, F- 0.20.
The area of interest, in the triangle is the triangular subregion shown in
Figure 7. The response of interest is the modulus of elasticity value of the
fuel produced from blends of B, O, and F, and the elasticity values are
listed in Table 2.








Mullet
X,


X2
Sheepshead


Figure 6a. Contour plot of fish patty texture surface (breaking
force measurement in grams of force).


I1 11 i" 11 1 0.90
0.94 .xl',

0.33

V2




Vi -0.23
x2= 1 -0.80
-0 80

ker


Figure 6b. Fish patty experiment.
Y= 1.84x1 + 0.67x2 + 1.51x3 + O.14xlx2 1.01X1X3
+ 0.27x2X3 + 8.68x1X2X3









YIRT



1.83






1.39





t-A
('I
0.311






0,0 I
C.


Figure 6c. Fish patty experiment.
Y= 1.84x, + 0.67x2 + 1.51x3 + 0.14x1x2 1.01xO
+ 0.27x2X3 + 8.68x1X2X3


YHART



1.83






1.393





00.90


, 50 ""

0.- x3 I






N = 14 + X -0.80





Figure 6d. Fish patty experiment.
S= 1.84x, + 0.67x2 + 1.51x3 + 0.14xlx2 1.01xIx3
+ 0.27x2x3 + 8.68x x2X3







Binder (B)


Oxidizer (0) Fuel (F)

Figure 7. Rocket propellant subregion of interest shown as a triangle in the
variables xl, x2, and x3.


Concentrating on the surface in the triangular subregion is facilitated
by transforming from the original components B, 0, and F to scaled
components xi. The scaled components are defined as
B 0.20 0 0.40 F- 0.20 (
X1 =----, x2 ----, X ---- (13)
0.20 0.20 0.20

where the value of 0.20 in the denominator is calculated as
0.20 = 1 (0.20+0.40+0.20). The special cubic model in the scaled
components system fitted to the modulus of elasticity measurements is
Y = 2350xi + 2450x2 + 2650x3 + 0X1X2 + 1000X1X3 (14)
+ 1600x2X3 + 6150X1X2X3

The contour plot and 3D plots of the estimated elasticity surface are
presented in Figures 8a, 8b, 8c, and 8d.
The contour plot of Figure 8a nicely illustrates the special cubic nature
of the estimated elasticity surface by indicating the surface is rising to







Table 2. Rocket propellant elasticity measurements.

Binder Oxidizer Fuel
Elasticity
B (x,) 0 (x2) F (X3) Yu

0.40 1.0 0.40 0 0.20 0 2350
0.20 0 0.60 1.0 0.20 0 2450
0.20 0 0.40 0 0.40 1.0 2650
0.30 0.5 0.50 0.50 0.20 0 2400
0.30 0.5 0.40 0 0.30 0.50 2750
0.20 0 0.50 0.50 0.30 0.50 2950
0.266 0.33 0.466 0.33 0.266 0.33 3000



form a mound or hill in the lower right-hand portion of the triangle. From
the contour plot one might guess that the highest elasticity values are
produced from blends somewhere in the neighborhood of x1 0.20,
X2 0.30, and x3 0.50. Translating these scaled component values into
values for the original components by reversing Equation 13, we would
guess the original components values to be approximately B 0.24,
0 0.46, and F 0.30.
The mound-like appearance of the elasticity surface is illustrated
nicely in the 3D plots of Figures 8b, 8c, and 8d. Furthermore, the
quadratic nature of the surface along the sides x, = 1 to x3 = 1, and x2 = 1
to x3 = 1 of the scaled components triangle is easily discovered by looking
at the 3D plots. The linear blending effect along the x, = 1 to x2 = 1 edge
of the triangle is apparent from the surface plots of Figures 8b and 8d.
Figure 9 presents another view of the elasticity surface of Figure 8d. In
Figure 9 the plane of support in W1 and W2 is tilted to 850 and rotated to
800, whereas in Figure 8d, the plane is tilted at 700 and rotated to 70. The
extra degrees of tilt and rotation in Figure 9 accentuate the mound-
shaped appearance of the special cubic surface. To alter the degree of
tilt and/or rotation, the plot statement on line 0057 in Appendix 3,
which now reads as PLOT W1*W2 = YHAT, is changed to PLOT
W1*W2 = YHAT/TILT = 85 ROTATE = 80;

AN AGRONOMIC EXPERIMENT
The following example illustrates the plotting technique for a two-
factor system where the factors are not mixture ingredients but rather are
quantitative amounts of two types of fertilizer applied to experimental
plots laid out in the field. When there are three or more factors (indepen-
dent variables) present in an experiment, surface plots of the estimated
response defined as a function of the levels of two factors are generated at
fixed combinations of the levels of the remaining factors.


17









THRT


3056.58






109. 3S


" 0.90




1.33


352. 19 xl2






/:l l
L'o Iii '


2 0'






Figure 8b. Rocket propellant experiment.
S= 2350x1 + 2450x2 + 2650x3 + 1000x x3 + 1600x2X3
+ 6150xlx2x3


X, =1


x2= 1 3 = 1



Figure 8a. Contour plot of the rocket propellant elasticity surface.







THRT


3056.35





2709.23


0.90


2352.12





2000.00 -
0.1


Figure 8c. Rocket propellant experiment.
Y= 2350x, + 2450x2 + 2650x3 + 1000xlx3 + 1600x2x3
+ 6150x1X2X3


Figure 8d. Rocket propellant experiment.
Y = 2350x1 + 2450x2 + 2650x3 + 1000x1x3 + 1600x2x3
+ 6150x1X2X3


THAT














3056.35










2704. 23










2352. 12




0.90

3 0.33
V2
-0.23
2000.00
0.23 -0. 80
W1 Xl =l

Figure 9. Rocket propellant experiment.
S= 2350x1 + 2450x2 + 2650x3 + 1000x1x3 + 1600x2x3 + 6150x1x2x3
Tilt = 85, Rotate = 80



EXAMPLE NO. 4: PEANUT YIELD EXPERIMENT

Two brands of fertilizer (Fi and F2) were applied to experimental plots
(size of plot equals 140 ft2) to assess their effects on the yield of peanuts
(lbs/plot). The levels of the amounts of each fertilizer applied were 10, 20,
and 40 (x 10 lbs/acre) and nine fertilizer-level combinations were set up







as defined by a 3 x 3 factorial arrangement. Each of the nine fertilizer-
level combinations was replicated three times, and each number in Table
3 represents the measure of peanut yield in pounds per plot.
The second-degree model fitted to the 27 yield values in Table 3 is
Y = 6.179 + 0.374F, + 0.264F2 0.0001F F2 (15)
0.0076F2 0.0060F2
and the value of the coefficient of determination (R2) for the fitted model
is R2 = 0.9130. The contour plot and a 3D plot of the estimated surface
are presented in Figures 10a and 10b, and the program statements used
for producing the contour and 3D plots are listed in Appendices 4 and 5,
respectively.

Table 3. Peanut yields in pounds per plot.

Fertilizer (Fi)
Fertilizer (F2) 10 20 40 (x 10 lbs/acre)

10 11.2, 10.9, 11.0 12.0, 12.4, 12.8 10.9, 9.8, 10.0
20 12.6, 13.2, 12.7 14.0, 12.9, 13.7 11.3, 11.7, 10.9
40 (x 10 lbs/acre) 10.8, 11.3, 10.9 11.4, 11.7, 12.1 10.2, 9.7, 9.9


In Appendix 4, the statement on line 0010 specifies the estimated Y
values to be 10, 11, 12, 13, 13.5, and 14. However, only the contours for
the values of 10, 11, 12, 13, and 13.5 for Y are plotted, implying the
maximum value of 1is less than 14.0. This is where the 3D plot in Figure
10b becomes useful, for the highest estimated yield value is recorded on
the vertical scale to be 13.63 (lbs/plot). This estimated maximum yield
value occurs at the center of the smallest circle in Figure 10a, and the
levels of fertilizers 1 and 2 that produce the maximum yield value are
approximately F1 = 24.46 and F2 = 21.80 (x 10 lbs/acre), respectively.
Moving away from this centroid value by increasing or decreasing the
amounts of either or both fertilizers would result in lower estimated
yields.

CONCLUSIONS
In mixture experiments, a researcher employs a statistical model to
make inferences concerning the effects of the different ingredients on the
response of interest. The model is used to generate a response surface.
The shape of the response surface informs us how the individual ingre-
dients as well as combinations of the ingredients affect the response.
The Chinese proverb, "One picture is worth more than ten thousand
words," certainly seems true when we view a three-dimensional graphical







FERTI

S10.0

S12.0 ,11.
37 '


3,4
/1 "2'.613.0 \




2FE1










FER1 a n FER
S/ N

/ \







-e es a ps E s e s v f
Si i
I / \
\ / /
\ / /






/ /









characteristics of response surfaces. With a 3D plot, changes in the height






of the surface are easily visualized, and this information is used to sug-
gest which of the blends of ingredients produce the best values of the
response.
In this work, 3D plots of response surfaces from three separate
mixture experiments are presented. Each surface is viewed from three
different directions. Viewing the surface from the different directions











I.. -? -
I -

K1+' \


12.35 -I


./ /\ \ "\ C",, \1 u

*^v /

\ /30

/ / FERi2

L40 20

30
FERT1 20,.._
10

Figure 10b. Peanut yield experiment: quadratic model of yield as a function of
FERT1 and FERT2.


enables one to observe the effects on the response caused by blending
pairs of ingredients as well as all three of the ingredients.
Programming instructions are provided for producing a contour plot
as well as the three-dimensional plots of a response surface. The contour
plot was produced by the PLOT procedure of SAS 79.6 while the three-
dimensional plots were produced by the G3D procedure of SAS/GRAPH
79.6.







APPENDIX 1: PROGRAM STATEMENTS FOR PRODUCING
THE SURFACE CONTOUR PLOT IN FIGURE 3.
0000 //JOBCARD
0001 /*PASSWORD
0002 /*ROUTE PRINT LOCAL
0003 // EXEC SAS, REGION= 600K,PLOT=
0004 DATA;
0005 DO V1= -.6 TO 1 BY .02;
0006 DO V2 = -.8 TO .9 BY .02;
0007 X1= (SQRT(6)*V1+ 1)/3;
0008 X2= (1 -X1 -SQRT(2)*V2)/2;
0009 X3=1-X1-X2;
0010 YHAT=0;
0011 IF (0 <= X1 <= 1) AND (0 <= X2 <= 1) AND (0 <= X3
<= 1)THEN
t0012 YHAT=4.77*X1 + 6.27*X2 + 7.11*X3 + 2.15*X1*X2
t0013 +1.1*X1*X3-3.54*X2*X3;
0014 OUTPUT;
0015 END;
0016 END;
0017 PROC GCONTOUR;
t0018 PLOT V1*V2=YHAT/LEVELS=4.9 5.3 5.8 6 6.3 6.5 6.8 7.1
0019 LLEVELS = 12 3 4 5 12 3;
t0020 TITLE FRUIT PUNCH EXPERIMENT;
t0021 TITLE CONTOUR PLOT FOR EQ. (8);
0022 // EXEC PLOT
0023/*


tTo produce a contour plot for a model that is different from the model listed on
lines 0012 and 0013, only the title statements on lines 0020 and 0021 and the
model on lines 0012 and 0013 are changed. The statements on lines 0020 and
0021 define the title that is printed at the top of the contour plot, while the
statements on lines 0012 and 0013 define the model equation for the response
surface that is to be plotted. The statement on line 0018 specifies the values of Y
for which contours are desired.







APPENDIX 2: PROGRAM STATEMENTS FOR PRODUCING
THE 3D SURFACE PLOT IN FIGURE 4.

0000 //JOBCARD
0001 /*PASSWORD
0002 /*ROUTE PRINT LOCAL
0003 // EXEC SAS, REGION= 600K,PLOT=
t0004 TITLE FRUIT PUNCH EXPERIMENT;
t0005 TITLE2 ;
t0006 TITLE 3D PLOT FOR EQ.(8);
0007 DATA ONE;
0008 DO V1= -.8 TO .9 BY .05;
0009 DO V2= -.8 TO .9 BY .05;
0010 X1= (SQRT(6)*V1 +1)/3;
0011 X2= (1- X1 SQRT(2)*V2)/2;
0012 X3=1-X1-X2;
t0013 YHAT=4.0;
0014 IF (0<=X1 AND X1<=1) AND (0<=X2 AND
0015 X2<= 1) AND (0< = X3 AND X3< = 1) THEN DO;
t0016 YHAT = 4.77*X1 + 6.27*X2 + 7.11*X3 + 2.15*X1*X2
t0017 + 1.10*X1*X3 3.54*X2*X3;
0018 END;
0019 OUTPUT;
0020 END;
0021 END;
0022 PROC G3D;
0023 PLOT V1*V2= YHAT;
0024 // EXEC PLOT
0025 /*












tChange only these statements as needed for plotting the 3D response surface for
a different model. The statements on lines 0004, 0005, and 0006 produce the title
printed at the top of the 3D plot. The statement on line 0013 sets the height of the
plane in the Vi and V2 system that supports the surface. The statements on lines
0016 and 0017 define the model equation for the response surface to be plotted.







APPENDIX 3: PROGRAM STATEMENTS FOR PRODUCING THE
3D SURFACE PLOTS IN FIGURES 5b, 5c, and 5d.
0000 //JOBCARD
0001 /*PASSWORD
0002 /*ROUTE PRINT LOCAL
0003 // EXEC SAS, REGION = 600K,PLOT =
t0004 TITLE FRUIT PUNCH EXPERIMENT;
t0005 TITLE2 ;
t0006 TITLE3 YHAT = 4.77X1 + 6.27X2 + 7.11X3 + 2.15X1X2 +
1.10X1X3 -3.54X2X3;
0007 DATA ONE;
0008 DO V1= -.8 TO .9 BY .05;
0009 DO V2= -.8 TO .9 BY .05;
0010 X1= (SQRT(6)*V1+ 1)/3;
0011 X2=(1 -X1 -SQRT(2)*V2)/2;
0012 X3=1-X1-X2;
t0013 YHAT=4.0;
0014 IF (0<=X1 AND X1<=1) AND (0<=X2 AND
X2< = 1) AND (0< =X3 AND X3< = 1)
0015 THEN DO;
t0016 YHAT = 4.77*X1 + 6.27*X2 + 7.11*X3 + 2.15*X1*X2
t0017 + 1.10*X1*X3- 3.54*X2*X3;
0018 END;
0019 OUTPUT;
0020 END;
0021 END;
0022 PROC G3D;
0023 PLOT V1*V2= YHAT;
0024 DATA TWO;
0025 DO U1= -.8 TO .9 BY .05;
0026 DO U2= -.8 TO .9 BY .05;
0027 X2= (SQRT(6)*U1 + 1)/3;
0028 X3 = 1 X2 SQRT(2)*U2)/2;
0029 X1=1-X2-X3;
t0030 YHAT = 4.0;
0031 IF (0<=X1 AND X1<=1) AND (0<=X2 AND
X2< = 1) AND (0< = X3 AND X3< = 1)
0032 THEN DO;
t0033 YHAT = 4.77*X1 + 6.27*X2 + 7.11*X3 + 2.15*X1*X2
t0034 + 1.10*X1*X3- 3.54*X2*X3;
0035 END;
0036 OUTPUT;







0037 END;
0038 END;
0039 PROC G3D;
0040 PLOT U1*U2=YHAT;
0041 DATA THREE;
0042 DO W1= -.8 TO .9 BY .05;
0043 DO W2= -.8 TO .9 BY .05;
0044 X3 = (SQRT(6)*W1+ 1)/3;
0045 X1 = (1- X3 SQRT(2)*W2)/2;
0046 X2=1-X1-X3;
t0047 YHAT=4.0;
0048 IF (0<=X1 AND X1<=1) AND (0<=X2 AND
X2< = 1) AND (0< = X3 AND X3< = 1)
0049 THEN DO;
t0050 YHAT = 4.77*X1 + 6.27*X2 + 7.11*X3+ 2.15*X1*X2
t0051 +1.10*X1*X3- 3.54*X2*X3;
0052 END;
0053 OUTPUT;
0054 END;
0055 END;
0056 PROC G3D;
0057 PLOT W1*W2= YHAT;
0058 // EXEC PLOT
0059 /*















tChange these statements as required to plot the three views of the 3D surface for
a different model. The statements on lines 0004, 0005, and 0006 define the title of
the surface to be printed at the top of the three plots. The statements on lines
0013, 0030, and 0047 set the height of the planes in the V1 and V2 system, the U1
and U2 system, and the W1 and W2 system, respectively, that support the
response surfaces to be plotted. The statements on lines 0016-0017, 0033-0034,
and 0050-0051 define the model equation for the response surface to be plotted.







APPENDIX 4: PROGRAM STATEMENTS FOR PRODUCING
THE CONTOUR PLOT IN FIGURE 10a.
0000 //JOBCARD
0001 /*PASSWORD
0002 /*ROUTE PRINT LOCAL
0003 // EXEC SAS, REGION= 600K,PLOT=
0004 DATA;
t0005 DO FERT1 = 10 TO 40;
t0006 DO FERT2= 10 TO 40;
t0007 Y = 6.179+ .374*FERT1+ .264*FERT2- .0001*FERT1*FERT2
t0008 .0076*FERT1*FERT1- .0060*FERT2*FERT2;
0009 OUTPUT;END;END;
t0010 PROC GCONTOUR;PLOT FERT1*FERT2
=Y/LEVELS=10 11 12 13 13.5 14
0011 LLEVELS= 12 5 4 5 1;
t0012 TITLE PEANUT YIELD EXPERIMENT;
t0013 TITLE QUADRATIC MODEL OF YIELD;
t0014 TITLE AS A FUNCTION OF FERT1 AND FERT2;
0015 // EXEC PLOT
0016 //*




















tChange only these statements as needed for plotting contours for a different
surface. The statements on lines 0005 and 0006 specify the levels of the two
independent variables; the statements on lines 0007 and 0008 define the model
equation for the surface, and line 0010 specifies the values of the surface for
which contours are desired. The statements on lines 0012, 0013, and 0014
produce the title printed at the top of the page.







APPENDIX 5: PROGRAM STATEMENTS FOR PRODUCING
THE 3D SURFACE PLOT IN FIGURE 10b.
0000 //JOBCARD
0001 /*PASSWORD
0002 /*ROUTE PRINT LOCAL
0003 // EXEC SAS,REGION = 600K,PLOT=
0004 DATA;
t0005 DO FERT1= 10 TO 40;
t0006 DO FERT2= 10 TO 40;
t0007 Y= 6.179+ .374*FERT1 + .264*FERT2- .0001*FERT1*FERT2
t0008 .0076*FERT1*FERT1- .0060*FERT2*FERT2;
0009 OUTPUT;END;END;
0010 PROC G3D;PLOT FERT1*FERT2= Y;
t0011 TITLE PEANUT YIELD EXPERIMENT;
t0012 TITLE QUADRATIC MODEL OF YIELD;
t0013 TITLE AS A FUNCTION OF FERT1 AND FERT2;
0014 // EXEC PLOT
0015 /*


























tSee comments in Appendix 4.










LITERATURE CITED


1. Cornell, J. A. 1981. Experiments with mixtures: Designs, models, and the
analysis of mixture data. John Wiley & Sons, Inc., New York.
2. Cornell, J. A., and J. C. Deng. 1982. Combining process variables and
ingredient components in mixing experiments. J. of Food Sci. 47:836-843.
3. Hare, L. B., and P. L. Brown. 1977. Plotting response surface contours for
three-component mixtures. J. of Quality Technology 9:193-196.
4. Huor, S. S., E. M. Ahmed, P. V. Rao, and J. A. Cornell. 1980. Formulation
and sensory evaluation of a fruit punch containing watermelon juice. J. of
Food Sci. 45:809-813.
5. Koons, G. F., and R. H. Heasley. 1981. Response surface contour plots for
mixture problems. J. of Quality Technology 13:207-212.
6. Kurotori, I. S. 1966. Experiments with mixtures of components having lower
bounds. J. of Indus. Quality Control 22:592-596.
7. SAS Institute. 1980. SAS/GRAPH User's Guide. Raleigh, N. C., SAS
Institute.










































This public document was promulgated at an annual cost of $1880
or a cost of $1.25 a copy to provide information on the statistical
technique of plotting three-dimensional response surfaces.


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ISSN 0096-607X









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