8
Response Surface Modeling with Constrained Systems

 “Every model is an approximation …. Don’t fall in love with a model.”

George E. P. Box, J. Stuart Hunter, and William G.  Hunter

Overview

In this part of the book, we have been discussing formulation experiments whose components are subject to single and multiple component constraints. After examining our formulation development strategy, we discussed screening experiments in the previous chapter. Now we discuss how to study further and optimize the critical components identified in the screening experiments. We focus on optimization of five-component systems. The methodology is appropriate for four- and six-component systems as well. In general, when six or more components are involved we suggest that screening experiments be run to reduce the number of components to three to five and then run optimization experiments on the smaller number of components. We discuss the designs and models used to plan and analyze optimization experiments including the use of computer algorithms in the design of the experiments. Several examples are used to illustrate the formulation situations and the proposed methodology.

CHAPTER CONTENTS

Overview

8.1 Design and Analysis Strategy for Response Surface Methodology

8.2 Plastic Part Optimization Study

8.3 Quadratic Blending Model Design Considerations

8.4 Example – Plastic Part Formulation

8.5 Example – Glass Formulation Optimization

8.6 Using the XVERT Algorithm to Create Designs for Quadratic Models

8.7 How to Use Computer-Aided Design of Experiments

8.8 Using JMP Custom Design

8.9 Blocking Formulation Experiments

8.10 Summary and Looking Forward

8.11 References

8.1 Design and Analysis Strategy for Response Surface Methodology

In Chapter 6 we discussed formulation experiments in which the components were subject to single and multiple component constraints. We focused on quadratic blending models for three- and four-component systems. We showed that effective and efficient quadratic model designs consisted of a subset of the vertices and centroids of the region. In Chapter 7 we presented approaches to screening constrained formulation systems. As noted there, our purpose for screening is usually to reduce the number of components down to a critical few and then to run optimization experiments with these components. In this chapter we extend this approach, focusing principally on response surface modeling of five-component systems.

Note that experimentation with six-component systems using quadratic models follows the same approach.  In general, when there are six or more components involved, formulation scientists may choose to run screening experiments, using the methods discussed in Chapter 7, before running the optimization experiments. Such optimization experiments typically involve five or fewer components. This strategy is particularly useful if one is not sure which components should be in the formulation and what component levels should be used. Also, quadratic model designs become quite large when there are six or more components, typically requiring 30 to 35 or even more blends.

To provide context for the type of problems we are addressing, we begin by discussing a five-component plastic formulation involving both single and multiple component constraints. This is followed by a discussion of the quadratic blending model design and analysis considerations used to develop formulation designs for optimization of constrained systems. Two approaches are introduced: one using the JMP Custom Design option, which is based on design optimality criteria, and the other using the XVERT algorithm discussed in Chapter 7.

8.2 Plastic Part Optimization Study

The objective of this plastic formulation study was to evaluate the effects of five ingredients on the hardness of the resulting plastic product (Snee 1979). The experimental region was defined by lower and upper constraints on the five components and two multiple component constraints as summarized below:

Ingredient	Ingredient Ranges
X1=Binder	0.50 – 0.70
X2=Cobinder	0.05 – 0.15
X3=Plasticizer	0.05 – 0.15
X4=Monomer A	0.10 – 0.25
X5=Monomer B	0.00 – 0.15

Here are the multiple component constraints:

•   Monomer:  0.18 < x₄ + x₅ < 0.26

•   Liquid:       0.00 < x₃ + x₄ + x₅ < 0.35

The blends in the design constructed from these constraints are shown in Table 8.1. The construction of this design will be discussed later in Section 8.4.

Table 8.1 – Plastic Part Formulation Experiment Design

Blend	Binder	Cobinder	Plasticizer	Monomer A	Monomer B	Hardness
1	0.700	0.050	0.050	0.197	0.000	162, 130
2	0.700	0.067	0.050	0.100	0.080	300, 230
3	0.617	0.150	0.050	0.180	0.000	60
4	0.617	0.050	0.150	0.180	0.000	7
5	0.517	0.150	0.150	0.100	0.080	4, 5
6	0.637	0.050	0.050	0.110	0.150	40
7	0.537	0.150	0.050	0.250	0.010	5
8	0.597	0.050	0.150	0.100	0.100	7
9	0.597	0.050	0.090	0.250	0.010	5, 4
10	0.500	0.147	0.090	0.110	0.150	6, 4
11	0.579	0.103	0.091	0.162	0.062	9
12	0.588	0.101	0.098	0.100	0.110	20
13	0.548	0.100	0.113	0.168	0.068	4
14	0.637	0.050	0.050	0.180	0.080	10
15	0.537	0.150	0.050	0.180	0.080	6
16	0.567	0.150	0.100	0.180	0.000	10
17	0.567	0.100	0.150	0.180	0.000	5
18	0.587	0.100	0.050	0.250	0.010	8
19	0.659	0.050	0.108	0.100	0.080	60
20	0.500	0.147	0.125	0.225	0.000	Too Soft
21	0.582	0.150	0.050	0.100	0.115	50

We now discuss the considerations that should be taken into account when constructing designs for five components intended to support a quadratic model. We then use this guidance in discussing the design of this experiment in Section 8.4.

8.3 Quadratic Blending Model Design Considerations

As discussed in Chapter 6, we typically assume a quadratic blending model to design formulation experiments. The quadratic blending model is of the following form:

$$E(y)={\displaystyle \sum _{i=1}^q{b}_i{x}_i+{\displaystyle \sum {\displaystyle \sum _{1\le i<j}^q{b}_{ij}{x}_i{x}_j}}}$$

Such models have a total of q + q(q-1) coefficients. The bare minimum design size is the number of terms in the model. However, a general guideline is to select a design size that is 5 to 10 points more than the number of terms in the model. This guideline is intended to provide sufficient degrees of freedom to estimate all coefficients, and also to evaluate the adequacy of the model through residual plots, lack-of-fit tests, and so on. If it is important to include some replicates in the design to estimate experimental variation (error), then “5+5” is a good strategy: 5 points more than the number of coefficients plus 5 of the points in the design duplicated. Five degrees of freedom provides an F test for lack of fit that has adequate power.

As noted in Chapter 6, a good design to estimate the coefficients in a quadratic model with constraints will be a subset of the vertices and centroids of the experimental region. This subset can be quite large (Snee 1975, 1981). The question is how to select a subset of the vertices. There are two useful ways to select a subset of the vertices, both using computer algorithms:

•   D-Optimal or I-Optimal design algorithms

•   XVERT algorithm

The optimal design algorithms are generally the preferred method. Such algorithms are part of statistical software such as JMP, which includes routines to analyze the data. The XVERT algorithm is easy to implement in a spreadsheet and produces very good designs when the experimenter does not have access to optimal design algorithms.

Here are the two types of design optimality criteria that we consider:

Criteria	Objective
D-Optimality	Minimizes the volume of the confidence region for the model coefficients
I-Optimality	Minimizes the average prediction variance over the experimental region

The D-Optimal algorithm tends to select points at the extremes of the formulation region. The I-Optimal algorithm generally includes points at the interior of the region, which may be important depending on the application involved. Of course, one can always add points to the design points that are selected by the D-Optimality algorithms (or the I-Optimality designs) that are considered important to test. The optimality criteria and associated algorithms are an aid for experiment design construction, not the dictator of the design.

Letting the D-Optimal algorithm select from the vertices and centroids of the region increases the chance of interior points being included in the design. Algorithms such as some of those used in JMP work only with bounds on the components and do not select from a set of candidate points such as the vertices and centroids of the region.

An effective strategy is to use more than one criterion to select designs and then compare the designs, basing the comparison on the context and subject matter of the particular formulation being studied. Montgomery (2013) and Goos and Jones (2011) provide more information about the use of optimality criteria in designing experiments.

8.4 Example – Plastic Part Formulation

The single and multiple component constraints described in Section 8.2 produced a region that had 38 vertices, 76 edge centroids, 13 constraint plane centroids, and 1 overall centroid for a candidate list of 128 points as shown below:

Type of Point	Candidate List	Final design
Vertices	38	10 + 5
Centroids
Edge	76	8
Constraint Plane	13	2
Overall	1	1
Total	128	21 + 5

A D-Optimal design algorithm was used to select the design. A 20-point design was selected, and then the D-Optimal algorithm was used to select five points to be duplicated. As discussed in Section 7.3, five degrees of freedom provides an F test for lack of fit that has adequate power. It is not surprising to see that the five points selected were vertices, since D-Optimality puts greater weight on the extremes of the region. As expected, the D-Optimal algorithm did not select the overall centroid. The overall centroid was added to the design because it was of interest to obtain information about the hardness of the plastic at the center of the experimental region. The final design consisted of 10 vertices plus 5 duplicated vertices, 8 edge centroids, 2 constraint plane centroids, and the overall centroid, resulting in the 21 blends plus five replicate blends shown in Table 8.1.

Figure 8.1 – Plastic Part Formulation – Plots of Ln (Hardness) versus Component Levels

The first step in the modeling process is to plot ln(hardness) versus each of the component levels, as shown in Figure 8.1. As noted earlier, these plots enable one to assess the variation in the raw data and identify any components with dominant effects. In Figure 8.1 we see the large effect of Binder.

Hardness was converted to the natural log scale to ensure that the prediction variance would not vary with the level of the measurement. Use of transformations to stabilize the variance was discussed in Chapter 4. One gets a clue to the need for the log transformation when it is observed that the range of the data is more than an order of magnitude (max/min = 300/4 = 75). As a general rule of thumb, transformations are typically needed if the response range is more than an order of magnitude--i.e., if max/min > 10. In Figure 8.1 we see that Binder is a dominant component showing a strong positive effect. The other components show negative effects.

Unfortunately, the hardness measurement for blend 20 could not be made because the plastic was too soft to obtain a measurement. The 15-coefficient model fit (Analyze ► Fit Model) the remaining 25 hardness measurements and produced an adjusted R² of 0.98. The regression analysis statistics associated with fitting this model are shown below.

While the inability to measure the hardness of Blend 20 theoretically negates the optimality properties of the D-Optimal design, it does have a benefit; it enables us to get a check on the prediction accuracy of the model. The quadratic model predicted a hardness of 2.5 units for blend 20, which is smaller than the smallest measured hardness value of 4.0 units. Thus, a potential problem was turned into an opportunity; in this case it added credibility to the prediction accuracy of the model.

We fit the model and found that it gave a good fit to the data (adjusted R² =0.98) and accurately predicted the hardness of one blend that was not included in the model. Next, we look at the model more closely to understand the formulation system. One way to do this is to look at the component effects plot, which is shown in Figure 8.2.

Figure 8.2 is a plot of the fitted response surface along the Cox axes (Cox 1971) discussed in Chapter 7. In effect, each frame of the plot shows the effect of the component from the low level of the component to the high level of the component. In Chapter 7 we defined the component effect as a point with perhaps some confidence limits. In the component effects plot such as Figure 8.2 the component effect is seen as a curve. This is because we are now considering quadratic models with non-linear blending. Further discussion of component effects plots can be found in Snee (2011) and Snee and Piepel (2013).

Figure 8.2 shows us the following:

•   Binder has a large positive effect. This effect is also seen in the plot of ln (Hardness) versus Binder level.

•   Cobinder has essentially no effect.

•   Plasticizer, Monomer A, and Monomer B have large negative effects.

•   A small amount of curvature is present because of the small nonlinear blending of Monomer A and Monomer B. The significant effects are strongly linear.

Figure 8.2 – Plastic Part Formulation: Component Effects as Shown by the JMP Prediction Profiler

Another way to observe these effects is to compute the linear effects along the Cox axes (Cox 1971) and display the effects with their 95% confidence intervals as shown in Figure 8.3. This is accomplished by fitting a linear blending model to the data and computing the component linear blending effects as shown in Chapter 7. The equation will not be used for prediction, but rather as a way of measuring and graphically displaying overall effects of the components. This is essentially what we are doing when we use a linear blending model to analyze the results of a screening experiment.

The component effects are shown in Table 8.2. In Figure 8.3 the effects are shown with 95% confidence intervals. Confidence limits that include zero indicate that the component has no statistically significant effect. Recall from Chapter 7 that Cox axes move along the line from the constrained centroid toward the pure blend. (See Figure 7.1.)

Evidence that linear blending is dominant for this system is further seen by the adjusted R² = 0.96 for the 5-coefficient linear blending model versus the adjusted R² = 0.98 for the 15-coefficient quadratic model. Figure 8.3 confirms the conclusions regarding the effects stated previously and may be preferred as a more compact graph than Figure 8.2.

Table 8.2 – Plastic Part Formulation: Component Effects along Cox Axes

Component	Coefficient	t Ratio	Prob	Range	Reference Blend	Effect	LWR CL	UPPR CL
Binder	5.3078	23.16	0.000	0.20	0.5972	2.6351	2.3977	2.8725
Cobinder	-0.0243	-0.04	0.970	0.10	0.0976	-0.0027	-0.1496	0.1443
Plasticizer	-10.464	-15.58	0.000	0.10	0.0871	-1.1462	-1.2997	-0.9927
Monomer A	-11.324	-16.63	0.000	0.15	0.1578	-2.0169	-2.2700	-1.7639
Monomer B	-7.7656	-10.45	0.000	0.15	0.0604	-1.2397	-1.4872	-0.9922

Figure 8.3 – Plastic Part Formulation: Component Linear Blending Effects

8.5 Example – Glass Formulation Optimization 

Chick and Piepel (1984) presented a glass formulation study that illustrates many of the formulation principles discussed in this book and expands our thinking to consider the case of multiple responses. This study involved five components with the following ranges:

Component	Lower and Upper Bounds
Cao	0.36 – 0.51
B2O3	0 – 0.13
SiO2	0 – 0.08
Na2O	0.15 – 0.35
Water Calcine	0.10 – 0.30

Chick and Piepel (1984) reported that the experimental region defined by these lower and upper bounds had 28 vertices, 56 edge centroids, and 48 faces, which included constraint plane centroids and other face centroids. As shown in Table 8.3, the D-Optimal design algorithm selected a 20-blend design that consisted of 13 vertices and 7 edge centroids. The overall centroid was added to the design as well as two other internal compositions.

The overall centroid was repeated five times. Testing was also performed on three more vertices and two edge centroids as confirmation blends to verify the prediction accuracy of the model. This resulted in a total of 32 melts being formulated and tested (Table 8.3). Melts 1 to 27 were used to develop the quadratic blending model. Melts 28 to 32 were used to check the prediction accuracy of the model.

Three measurements were made on each melt:

•   Temperature (deg C) at 10 Pa viscosity

•   Spinel phase yield (wt%)

•   Soxhlet leaching weight loss (wt%)

When multiple responses are measured, it is a good practice to look at the correlation among the responses. Strong correlations add to your understanding of the mixture system and also simplify the interpretation of the results. We discuss multiple responses in more detail in Chapter 10. When the numerical values of the responses are evaluated, we see that weight loss has a wide range (1.57 to 54.12) covering more than an order of magnitude. It was decided to express weight loss on a natural log scale in the analysis.

In Figure 8.4 we see strong correlations between temperature and yield and also temperature and weight loss. The correlation between yield and weight loss is present but at a lesser degree. Here are the correlation coefficients:

	Temperature	Yield
Yield	0.79
Log Weight Loss	-0.70	-0.54

Figure 8.4 – Glass Formulation Optimization: Correlations between Temperature, Yield, and Weight Loss

Table 8.3 – Glass Formulation Optimization: Blends in Study Design

Melt	Point Type	CaO	B2O3	SiO2	Na2O	Water Calcine	Temp	Yield	Weight Loss
1	Vertex	0.077	0.125	0.346	0.144	0.308	1120	14	2.62
2	Vertex	0.077	0	0.491	0.144	0.288	1250	13	1.62
3	Vertex	0	0	0.488	0.182	0.33	1290	19	1.61
4	Vertex	0.079	0	0.503	0.305	0.113	1060	0	5.16
5	Vertex	0	0.128	0.503	0.256	0.113	930	0	19.44
6	Overall Centroid	0.039	0.062	0.42	0.248	0.231	980	2	3.25
7	Overall Centroid	0.039	0.062	0.42	0.248	0.231	954	2	2.5
8	Overall Centroid	0.039	0.062	0.42	0.248	0.231	954	6	2.45
9	Overall Centroid	0.039	0.062	0.42	0.248	0.231	959	4	2.22
10	Overall Centroid	0.039	0.062	0.42	0.248	0.231	966	9	2.61
11	Vertex	0	0	0.345	0.325	0.33	1060	12	9.1
12	Vertex	0	0.128	0.414	0.345	0.113	803	0	54.12
13	Edge Centroid	0.079	0.054	0.409	0.345	0.113	930	0	6.71
14	Vertex	0.079	0.128	0.5	0.147	0.146	1020	0	2.42
15	Vertex	0.077	0	0.35	0.339	0.234	1034	8	13.47
16	Vertex	0	0	0.499	0.343	0.158	1050	0	9.3
17	Vertex	0.079	0.128	0.355	0.325	0.113	838	0	29.47
18	Vertex	0	0.125	0.402	0.143	0.33	1189	18	2.63
19	Vertex	0.039	0	0.503	0.345	0.113	1000	0	12.93
20	Edge Centroid	0	0.063	0.348	0.339	0.25	950	10	5.65
21	Edge Centroid	0	0.062	0.345	0.263	0.33	1112	16	3.74
22	Edge Centroid	0.077	0.053	0.397	0.143	0.33	1240	17	1.57
23	Edge Centroid	0.039	0.127	0.497	0.146	0.191	1055	0	3.07
24	Edge Centroid	0.038	0	0.345	0.287	0.33	1149	18	3.35
25	Edge Centroid	0	0.126	0.347	0.271	0.256	900	4	6.06
26	Internal	0.046	0.074	0.451	0.222	0.207	982	0.5	2
27	Internal	0.043	0.07	0.423	0.27	0.194	947	0.5	2.67
28	Vertex	0	0.038	0.487	0.143	0.332	1360	18	0.95
29	Vertex	0	0	0.421	0.338	0.241	1010	0	7.7
30	Edge Centroid	0.03	0.127	0.352	0.343	0.148	770	0	37.06
31	Edge Centroid	0.077	0	0.343	0.248	0.332	1210	13	2.42
32	Vertex	0.03	0.009	0.502	0.345	0.114	980	0	14.21

Table 8.4 summarizes the fit of the linear and quadratic models. The model lack of fit is statistically significant for the temperature and ln(Weight Loss) responses but not yield.

Table 8.4 – Glass Formulation Optimization: Fit of Linear and Quadratic Models

Model	No. Melts	No. Terms	Statistic	Temp	Yield	Weight Loss
Linear	27	5	Adj R2	82	79	66
			Lack of Fit:  F-Ratio	27.55	1.18	18.09
			Lack 0f Fit:  Prob	0.003	0.487	0.006
Quadratic	27	15	Adj R2	97	87	90
			Lack of Fit:  F-Ratio	5.03	0.61	6.19
			Lack of Fit:  Prob	0.064	0.742	0.048

An examination of the component effects plots in Figure 8.5 reveals that the curvature is not as large as we would expect from the large difference between the adjusted R² values of the linear and quadratic models (Table 8.4).

Figure 8.5 – Glass Formulation Optimization: Component Effects Plots

In Figure 8.5 we see that CaO has a minimal effect, Na2O and Water Calcine have the largest effects, and B2O3 and SiO2 have smaller effects. The directions (positive and negative) of the effects are different depending on the response evaluated.

The strong linear blending portion of the component effects is shown in Figure 8.5. This suggests that a good way to visualize the component effects is to display them along the Cox axes (Cox 1971) as shown in Table 8.5 and Figure 8.6. This figure also allows us to see the correlation between the Temperature, Yield, and Weight Loss effects that we observed in Figure 8.4.

Figure 8.6 – Glass Formulation Optimization: Component Effects along Cox Axes

Table 8.5 – Glass Formulation Optimization: Component Effects along Cox Axes

Response	Component	Coefficient	t-Ratio	Prob	Range	Reference  Blend	Effect	LWR CL	UPPR CL
Temp	CaO	77.7	0.24	0.813	0.079	0.038	6.4	-48.4	61.1
	B2O3	-1187.8	-5.99	0.000	0.128	0.063	-162.3	-217.9	-106.7
	SiO2	331.2	3.23	0.004	0.157	0.421	89.8	32.8	146.8
	Na2O	-846.5	-7.31	0.000	0.202	0.254	-229.3	-293.7	-165.0
	Water Calcine	660.5	6.44	0.000	0.219	0.224	186.4	126.9	245.8
Yield	CaO	-10.7	-0.54	0.595	0.079	0.038	-0.9	-4.2	2.5
	B2O3	-26.6	-2.20	0.039	0.128	0.063	-3.6	-7.0	-0.2
	SiO2	-17.4	-2.77	0.011	0.157	0.421	-4.7	-8.2	-1.2
	Na2O	-17.0	-2.40	0.025	0.202	0.254	-4.6	-8.5	-0.7
	Water Calcine	61.2	9.76	0.000	0.219	0.224	17.3	13.6	20.9
Ln WtLoss	CaO	-1.8711	-1.26	0.220	0.079	0.038	-0.1536	-0.4036	0.0963
	B2O3	1.7729	1.96	0.063	0.128	0.063	0.2422	-0.0117	0.4961
	SiO2	-0.9274	-1.98	0.060	0.157	0.421	-0.2514	-0.5117	0.0089
	Na2O	3.0736	5.82	0.000	0.202	0.254	0.8327	0.5390	1.1265
	Water Calcine	-1.9315	-4.12	0.000	0.219	0.224	-0.5450	-0.8164	-0.2737

The concluding step in the analysis is to use the quadratic blending model to predict Temperature, Yield, and ln(Weight Loss) response measurements for the confirmation tests, which are melts 28 to 32.  In Table 8.6 we see that the models predict very well and show no lack of fit when comparing the prediction variance to the root mean square error (RMSE) statistics that were obtained from the fits of the quadratic models. That is, we formally test the hypothesis that these sample standard deviations, one from the original model fit to 27 points and the other from the predictions of points 28 to 32, come from the same population. Chick and Piepel (1984) concluded, “The models appear to be quite accurate in predicting the ranking of the glasses with respect to a property, which lends credence to the use of the effects plots to predict trends”.

Chick and Piepel (1984) also used a specialized optimization algorithm to identify optimal glass compositions. We will discuss mathematical optimization of models in Chapter 10.

Table 8.6 – Glass Formulation Optimization: Quadratic Model Prediction Accuracy

Melt	Temp	Pred Temp	Resid Temp	Yield	Pred Yield	Resid Yield	Log Wt Loss	Pred Log WtLoss	Resid Log WtLoss
28	1360	1351.0	9.0	18	21.4	-3.4	-0.0223	0.0753	-0.0976
29	1010	976.7	33.3	0	3.4	-3.4	0.8865	0.8768	0.0097
30	770	795.7	-25.7	0	1.0	-1.0	1.5689	1.4562	0.1127
31	1210	1188.0	22.0	13	18.4	-5.4	0.3838	0.7490	-0.3652
32	980	1016.9	-36.9	0	0.3	-0.3	1.1526	1.0171	0.1354
Pred Resid Std Dev			27.19			3.27			0.1866
RMSE			21.29			2.56			0.1295
F-Ratio			1.63			1.63			2.08
Prob			0.226			0.226			0.139

8.6 Using the XVERT Algorithm to Create Designs for Quadratic Models

The XVERT algorithm can be used in the construction of quadratic model designs; it produces a design that is a combination of vertices, constraint plane centroids, and the overall centroid of the region. The XVERT algorithm is used to calculate the vertices and develop a subset of the vertices if needed. After the vertices have been selected, the vertices are used to calculate the center point of each of the constraint planes and the overall centroid of the region.

Conceptually, such a design is similar to the face-centered-cube design (FCCD) for independent variables (Montgomery 2013). The FCCD consists of the corners of the region (vertices), the centers of the faces of the region (constraint plane centroids), and the center point (overall centroid). In the case of three factors, the FCCD would consist of the eight corners of the three-dimensional cube, the center points of each of the six two-dimensional faces, and the overall center point for a total of 8+6+1=15 points.

The design construction process proceeds as follows:

1.   Select a subset of the vertices using XVERT. After creating candidate groups, select the first point in each candidate subgroup to be in the design. As we will see below, in the case of five components a useful design will contain approximately 16 vertices.

2.   Calculate the constraint plane centroids from the vertices that were selected in step 1. This will produce a maximum of 2q points. Each component will produce a maximum of two points: one point each for the lower and upper bounds of the component.

3.   Add the overall centroid of the region.

The resulting design for five components will contain a maximum of 16+10+1=27 points. The quadratic model for five components has 15 coefficients, leaving 12 residual degrees of freedom, which estimate the experimental error when there is no model lack of fit. Replication is a separate consideration. As noted previously, at least five blends should be duplicated when the goal is to obtain an estimate of experimental error from replicates.

We will illustrate this design construction process using the five-component example from Snee and Marquardt (1974), which has the following component ranges:

Component	Lower and Upper Bounds	Range
X1	0.00 - 0.10	0.10
X2	0.00 – 0.10	0.10
X3	0.05 – 0.15	0.10
X4	0.20 – 0.40	0.20
X5	0.40 – 0.60	0.20

The XVERT design was developed from the first four columns of a 16-run Plackett-Burman design (Plackett and Burman 1946), which is identical to a 2⁴ factorial design. The result is ten core points and six candidate subgroups, as shown in Table 8.7. Each candidate subgroup produces three points for an additional 6x3=18 vertices. These 18 vertices plus the ten core points produce a total of 28 vertices.

 

Table 8.7 – Five-Component Quadratic Model Design Example: Vertices of Region

Group	Vertex	X1	X2	X3	X4	X5
Core Points	1	0.10	0.10	0.05	0.20	0.55
Core Points	2	0.10	0.00	0.15	0.20	0.55
Core Points	3	0.00	0.10	0.15	0.20	0.55
Core Points	4	0.10	0.10	0.15	0.20	0.45
Core Points	5	0.00	0.00	0.05	0.40	0.55
Core Points	6	0.10	0.00	0.05	0.40	0.45
Core Points	7	0.00	0.10	0.05	0.40	0.45
Core Points	8	0.00	0.00	0.15	0.40	0.45
Core Points	9	0.00	0.00	0.05	0.35	0.60
Core Points	10	0.10	0.10	0.15	0.25	0.40
Candidate Subgroup 1	11	0.10	0.00	0.05	0.25	0.60
Candidate Subgroup 1	12	0.10	0.00	0.10	0.20	0.60
Candidate Subgroup 1	13	0.10	0.05	0.05	0.20	0.60
Candidate Subgroup 2	14	0.00	0.10	0.05	0.25	0.60
Candidate Subgroup 2	15	0.00	0.10	0.10	0.20	0.60
Candidate Subgroup 2	16	0.05	0.10	0.05	0.20	0.60
Candidate Subgroup 3	17	0.00	0.00	0.15	0.25	0.60
Candidate Subgroup 3	18	0.00	0.05	0.15	0.20	0.60
Candidate Subgroup 3	19	0.05	0.00	0.15	0.20	0.60
Candidate Subgroup 4	20	0.10	0.10	0.05	0.35	0.40
Candidate Subgroup 4	21	0.10	0.05	0.05	0.40	0.40
Candidate Subgroup 4	22	0.05	0.10	0.05	0.40	0.40
Candidate Subgroup 5	23	0.10	0.00	0.15	0.35	0.40
Candidate Subgroup 5	24	0.10	0.00	0.10	0.40	0.40
Candidate Subgroup 5	25	0.05	0.00	0.15	0.40	0.40
Candidate Subgroup 6	26	0.00	0.10	0.15	0.35	0.40
Candidate Subgroup 6	27	0.00	0.10	0.10	0.40	0.40
Candidate Subgroup 6	28	0.00	0.05	0.15	0.40	0.40

The vertices in the quadratic model design are the ten core points and the first point in each of the six candidate subgroups—namely, points 1 to 10, 11, 14, 17, 20, 23, and 26.

Next, the constraint plane centroids (centers of the faces of the experimental region) are calculated from the 16 vertices. Adding the overall average of the 16 vertices completes the design. The design is tabulated in Table 8.8 along with the points used to calculate the centroids.

Table 8.8 – Five-Component Quadratic Model Design Example: Quadratic Model

Group	Vertex	X1	X2	X3	X4	X5
Core Points	1	0.1	0.1	0.05	0.2	0.55
Core Points	2	0.1	0	0.15	0.2	0.55
Core Points	3	0	0.1	0.15	0.2	0.55
Core Points	4	0.1	0.1	0.15	0.2	0.45
Core Points	5	0	0	0.05	0.4	0.55
Core Points	6	0.1	0	0.05	0.4	0.45
Core Points	7	0	0.1	0.05	0.4	0.45
Core Points	8	0	0	0.15	0.4	0.45
Core Points	9	0	0	0.05	0.35	0.6
Core Points	10	0.1	0.1	0.15	0.25	0.4
Candidate Subgroup 1	11	0.1	0	0.05	0.25	0.6
Candidate Subgroup 2	14	0	0.1	0.05	0.25	0.6
Candidate Subgroup 3	17	0	0	0.15	0.25	0.6
Candidate Subgroup 4	20	0.1	0.1	0.05	0.35	0.4
Candidate Subgroup 5	23	0.1	0	0.15	0.35	0.4
Candidate Subgroup 6	26	0	0.1	0.15	0.35	0.4
CP Centroid X1 = 0.00	3,5,7,8,9,14,17,26	0	0.05	0.1	0.325	0.525
CP Centroid X1 = 0.10	1,2,4,6,10,11,20,23,	0.1	0.05	0.1	0.275	0.475
CP Centroid X2 = 0.00	2,5,6,8,9,11,17,23,	0.05	0	0.1	0.325	0.525
CP Centroid X2 = 0.10	1,3,4,7,10,14,20,26,	0.05	0.1	0.1	0.275	0.475
CP Centroid X3 = 0.05	1,5,6,7,9,11,14,20,	0.05	0.05	0.05	0.325	0.525
CP Centroid X3 = 0.15	2,3,4,8,10,17,23,26	0.05	0.05	0.15	0.275	0.475
CP Centroid X4 = 0.20	1,2,3,4,	0.075	0.075	0.125	0.2	0.525
CP Centroid X4 = 0.40	4,6,7,8,	0.025	0.025	0.075	0.4	0.475
CP Centroid X5 = 0.40	10, 20,23,26	0.075	0.075	0.125	0.325	0.4
CP Centroid X5 = 0.60	9,11,14,17,	0.025	0.025	0.075	0.275	0.6
Overall Centroid	1-10,11,14,17,20,23,26	0.05	0.05	0.1	0.3	0.5

In the end, we have generated a very good design using only a spreadsheet and the XVERT algorithm. Of course, regression analysis software that can fit mixture models is required to analyze the data.

8.7 How to Use Computer-Aided Design of Experiments 

As we have seen in Chapters 6, 7, and 8, components subject to single and multiple component constraints produce experimental regions that are irregular in shape. The most effective approach is to use computer algorithms to aid in constructing the design. The operative word here is aid in the design process; the algorithms do not dictate the design.

We have found the following procedure to be useful in these situations. Note that the optimal design algorithms are particularly useful for steps 3 and 4, and to some degree 5. They do not help with steps 1 and 2.

1.   Select the region of interest given the objectives of the study and the associated subject matter knowledge of the formulation system being studied.

2.   Incorporate the principles of good experimental design, such as those shown in Table 8.9 below.

3.   Decide how the design points will be selected.

4.   Select the points in the region defined by the component constraints, or

5.   Subset the vertices and centroids of the region defined by the component constraints.

6.   Select a set of blends to form a good design for the formulation experiment. Consider making several alternative designs by using different design scenarios depending on the practical considerations that are associated with the study. For example one can vary the levels of one or more of the components in the formulation and/or the number of blends in the design.

7.   Evaluate the resulting designs graphically and analytically in order to decide which design to use.

Table 8.9, adapted from the work of Box and Draper (1975), summarizes some characteristics of good experimental designs. Many of these characteristics define how the design will be constructed and are not amenable to codifying in a computer algorithm. This emphasizes the point of computer-aided design, not computer-generated design.

Table 8.9 – Characteristics of a “Good” Experimental Design

•   Offers clear objectives and criteria for success

•   Generates useful information throughout the region of interest

•   Offers good prediction accuracy

•   Detects lack of fit

•   Permits blocking

•   Allows sequential buildup of design order--e.g., screening to response surface

•   Provides an internal estimate of experimental variation

•   Requires a minimum number of runs

•   Provides data that allows visual analysis

•   Requires a practical number of variable levels

•   Plans for confirmation experiments to check model prediction accuracy

JMP has a very useful design construction option called Custom Design that is accessed by DOE ► Custom Design. Examples of the use of this option are offered in Sections 8.8 and 8.9 below. The Custom Design option can be used in a variety of situations. Custom Design is particularly useful in cases in which the classic experimental designs are not effective.  Such situations often result when there are constraints on the variables in the design. Component constraints are certainly a frequently encountered situation in formulation experiments.

Since the generated designs are for the most part one-of-a-kind, it is important to have some criteria to evaluate their quality. JMP provides an option to compute Design Diagnostics, which includes D-Efficiency, G-Efficiency, A-Efficiency, and Average Variance of Prediction. G-Efficiency is useful as it is a lower bound on D-Efficiency (Atwood 1969). It is also useful because it enables us to compute the maximum prediction variance of the model:

Maximum Prediction Variance = p/ (n*G-Efficiency)

Where p = number of terms in the model and n = number of blends in the design.

We routinely use these criteria as follows:

Criteria	Desired Value
G- Efficiency	Greater than 50%
Average Prediction Variance	Lower is Better
Maximum Prediction Variance	Less than 1.0

The recommendation that G-Efficiency should be greater than 50% is due to Wheeler (1972). A maximum prediction variance =1.0 implies that the variance of the model prediction is no larger than the variance of the measurements that are used to create the model. Lucas (1976) showed that the routinely used response surface designs generally have maximum prediction variances less than 1.0. We have used these criteria for many years (Snee 1975, 1985) and found the associated designs to work well in practice.

8.8 Using JMP Custom Design   

At this point, we have developed strategies for formulation development, and we have understood the effect of mixture constraints on the experimental region, formulation blending models, and the associated graphical displays. So, we are in a position to make effective use of the Custom Design option in JMP.

Here are the basic steps:

1.   Define the responses.

2.   Define the factors (components in the case of formulations) and factor levels.

3.   Define factor constraints.

4.   Specify the model.

5.   Generate the design: Specify blocking, replication, and number of runs.

6.   Click Make Design to generate the design.

7.   Evaluate the design diagnostics.

Shown below are the results of these steps in the case of the Plastic Part Formulation example introduced in Section 8.2. In this example, we have five components, two multiple component constraints, and a desired design size of 26 blends that are to include replicate blends and the overall centroid blend. In the Design Diagnostics screen shown below we find that this design has a G-Efficiency of 80.4%, which indicates that the maximum prediction variance of the design is 0.72. The average prediction variance for this design is 0.42.  As specified by the guidelines defined in Section 8.7, this is considered a good design; perhaps it is even a very good design.

It is noted that this design contains some blends that have levels involving as many as six decimal places. These levels can be rounded to fewer decimals (e.g., three or four decimals) if that will be helpful in carrying out the design. Such rounding will have no practical effect on the statistical properties of the design.

We also note that because of the manner in which the designs are computed by JMP another run of Custom Design with the same input specifications will produce a slightly different design. The good news is that the statistical properties of the two designs will be essentially the same.

8.9 Blocking Formulation Experiments 

In Chapter 2 we introduced the subject of blocking as a mechanism for removing the effects of important extraneous variables that may be present. Some examples include raw material lots, teams, operators, and day of the week. The variation induced by these so-called noise or environmental variables can be accounted for by blocking. In essence, we introduce a blocking variable, perhaps equal to 1 for day one and 2 for day two, and incorporate that in the model. The effects of the blocked variables are still present but they are isolated in the statistical analysis so that the effects of the components and other variables are not affected.

Blocking of formulation experiments can be complicated but, thanks to the availability of the Custom Design function in JMP, blocking of formulation experiment designs can be fairly straightforward. We will illustrate the use of Custom Design in blocking experiments using the bread flour experiment discussed by Draper et al. (1993). In this case there are four ingredients being studied. The response of interest is the volume of the loaf. Draper et al. developed a four-component design in four blocks of nine blends each producing an experiment involving a total of 36 blends, as shown in Table 8.10.

Table 8.10 – Bread Flour Design for Four Components in Four Random Blocks (Draper et al. 1993)

Blend	Block	X1	X2	X3	X4		Blend	Block	X1	X2	X3	X4
1	1	0	0.25	0	0.75		19	2	0	0.75	0	0.25
2	1	0.25	0	0.75	0		20	2	0.25	0	0.75	0
3	1	0	0.75	0	0.25		21	2	0	0.25	0	0.75
4	1	0.75	0	0.25	0		22	2	0.75	0	0.25	0
5	1	0	0.75	0.25	0		23	2	0	0	0.25	0.75
6	1	0.25	0	0	0.75		24	2	0.25	0.75	0	0
7	1	0	0	0.75	0.25		25	2	0	0.25	0.75	0
8	1	0.75	0.25	0	0		26	2	0.75	0	0	0.25
9	1	0.25	0.25	0.25	0.25		27	2	0.25	0.25	0.25	0.25
10	3	0	0.25	0	0.75		28	4	0	0.75	0	0.25
11	3	0.25	0	0.75	0		29	4	0.25	0	0.75	0
12	3	0	0.75	0	0.25		30	4	0	0.25	0	0.75
13	3	0.75	0	0.25	0		31	4	0.75	0	0.25	0
14	3	0	0.75	0.25	0		32	4	0	0	0.25	0.75
15	3	0.25	0	0	0.75		33	4	0.25	0.75	0	0
16	3	0	0	0.75	0.25		34	4	0	0.25	0.75	0
17	3	0.75	0.25	0	0		35	4	0.75	0	0	0.25
18	3	0.25	0.25	0.25	0.25		36	4	0.25	0.25	0.25	0.25

Each of the components varied from 0 to 75% of the blend. The design consisted of eight binary blends representing 25-75 and 75-25 mixtures in each block. The overall centroid blend (0.25, 0.25, 0.25, 0.25) is also contained in each block--namely blends 9, 18, 27, and 36.

The design created for the bread flour experiment using JMP Custom Design is shown below. The steps in design creation are the same as those used previously in Section 8.7 for the Plastic Part Formulation Study except there are no linear constraints associated with this design.

Note that when there is no clearly optimal design, Custom Design will not print efficiencies, as it has no base of reference. This may occur, as in this case, when blocking formulation designs.

We note that this design is very similar to the design proposed by Draper et al. (see Table 8.10) but there are some differences:

•   The overall centroid, after rounding, is contained in blocks 2, 3, and 4 but not in block 1.

•   Some of the binary blends are a 50-50 composition (blends 1, 6 to 8, 10, 27, 29, 31, and 36) rather than the 75-25 and 25-75 composition blends in the Draper et al. design.

We do not see this as a problem as the Average Prediction Variance of the Draper and JMP designs are essentially equal, being 0.440 and 0.452, respectively.

8.10 Summary and Looking Forward 

In this chapter we have shown how to design and analyze optimization experiments for formulations. We have focused on optimization of five-component systems. The methodology is appropriate for four- and six-component systems as well. In general, when six or more components are involved we suggest screening experiments be run first to reduce the number of components to 3 to 5, and then run optimization experiments on the smaller number of components.

This strategy takes advantage of what we have learned from the screening designs to construct more effective optimization experiments. The overall amount of experimentation can be reduced, thereby speeding up the formulation development process. In essence, we are able to use hindsight to our advantage. Commercial software such as JMP is available to do the necessary design, analysis, and graphics.

In Chapter 9 we expand the formulation development methodology further to include experimenting with process variables as well as formulation components. Again, we see the strategy of speeding up the process of formulation development, including both formulation and process variables. Recently published strategies (Snee et al. 2016) that can reduce the total experimentation required by 50% or more are included in Chapter 9.

8.11 References 

Atwood, C. L. (1969) “Optimal and Efficient Designs of Experiments”, The Annals of Mathematical Statistics, 40 (5), 1570-1602.

Box, G. E. P and N. R. Draper. (1975) “Robust Designs.” Biometrika, Vol. 62 (2), 347-352.

Box, G. E. P, J. S. Hunter and W. G. Hunter. (2005) Statistics for Experimenters: Design, Innovation, and Discovery. 2nd Ed., Wiley-Interscience, Hoboken, NJ.

Chick, L. A and G. F. Piepel. (1984) “Statistically Designed Optimization of a Glass Composition.” Journal of the American Ceramic Society, 67 (11), November 1984, 763-768.

Cox, D. R. (1971) “A note on polynomial response functions for mixtures.” Biometrika, 58 (1), 155-159.

Draper, N. R., P. Prescott, S. M. Lewis, A. M. Dean, P. W. M. John and M. G. Tuck. (1993) “Mixture Designs for Four Components in Orthogonal Blocks.” Technometrics, 35 (3), 268-276.

Goos, P. and B. Jones. (2011) Optimal Design of Experiments: a Case Study Approach. John Wiley & Sons, Hoboken, NJ.

Lucas, J. M. (1976) “Which response surface design is best: A performance comparison of several types of quadratic response surface designs in symmetric regions.” Technometrics, 18 (4), Nov., 1976, pp. 411-417.

Montgomery, D. C. (2012) Design and Analysis of Experiments, 8^th Edition, John Wiley & Sons, Hoboken, NJ.

Plackett, R. L. and J.P. Burman. (1946) "The Design of Optimum Multifactorial Experiments." Biometrika 33 (4), 305-25.

Snee, R. D. and D. W. Marquardt. (1974) “Extreme vertices designs for linear mixture models.” Technometrics, 16 (3), 399-408.

Snee, R. D. (1975) “Experimental designs for quadratic models in constrained mixture spaces.” Technometrics, 17 (2), 149-159.

Snee, R. D. (1979) “Experimental designs for mixture systems with multicomponent constraints.” Communications in Statistics, Theory and Methods, 8 (4), 303-326.

Snee, R. D. (1981) “Developing blending models for gasoline and other mixtures.” Technometrics, 23 (2), 119-130.

Snee, R. D. (1985) “Computer-aided Design of Experiments – Some Practical Experiences.” Journal of Quality Technology, 17 (4), 222-236.

Snee, R. D. (2011) “Understanding Formulation Systems – A Six Sigma Approach.” Quality Engineering, 23 (3), July-September, 278-286.

Snee, R. D., R. W. Hoerl and G. Bucci. (2016) “A Statistical Engineering Strategy for Mixture Problems with Process Variables.” Quality Engineering, 28 (3), 263-279.