Development of a Rubber-Based Product Using a Mixture ... · Progress in Rubber, Plastics and Recycling Technology, Vol. 29, No. 3, 2013 123 Development of a Rubber-Based Product

123Progress in Rubber, Plastics and Recycling Technology, Vol. 29, No. 3, 2013

Development of a Rubber-Based Product Using a Mixture Experiment: A Challenging Case Study

*Corresponding author. Email address: [email protected]

©Smithers Rapra Technology, 2013


Yahya Kaya,1* Greg F. Piepel,2 and Erdal Caniyilmaz3

1Department of Mechanical Engineering, Erciyes University, 38039 Kayseri, Turkey2Pacific Northwest National Laboratory, Richland, WA 99354, USA3Department of Industrial Engineering, Erciyes University, 38039 Kayseri, Turkey

Received: 13 June 2012, Accepted: 20 September 2012

SUMMARY

Many products used in daily life are made by blending two or more components. The properties of such products typically depend on the relative proportions of the components. Experimental design, modeling, and data analysis methods for mixture experiments provide for efficiently determining the component proportions that will yield a product with desired properties. This article presents a case study of the work performed to develop a new rubber formulation for an o-ring (a circular gasket) with requirements specified on 10 product properties. Each step of the study is discussed, including: 1) identifying the objective of the study and requirements for properties of the o-ring, 2) selecting the components to vary and specifying the component constraints, 3) constructing a mixture experiment design, 4) measuring the responses and assessing the data, 5) developing property-composition models, 6) selecting the new product formulation, and 7) confirming the selected formulation in manufacturing. The case study includes some challenging and new aspects, which are discussed in the article.

Keywords: Rubber formulation, Product development, Experimental design, Mixture experiment, Mixture models, Optimization

iNTRoducTioN

To survive in today’s competitive world, companies must develop and manufacture, in a timely fashion, products that meet the ever-changing requirements and expectations of customers. Many products are formed by blending two or more ingredients (components), such as rubber-based

124 Progress in Rubber, Plastics and Recycling Technology, Vol. 29, No. 3, 2013

Yahya Kaya, Greg F. Piepel, and Erdal Caniyilmaz

products, plastics, alloys, concrete, and various food products. Companies must efficiently develop product formulations (proportions of components) that meet customer requirements on several properties of the product. When the product properties also depend on the amount of the product mixture (e.g., the amount of fertilizer applied), or on the settings of one or more process variables during manufacturing (e.g., temperature), companies must also efficiently determine the best values of such variables.

Statistical methods for the design and analysis of mixture experiments, combined with knowledge and experience of subject-matter experts, provide for efficiently developing product formulations and, if applicable, settings of other non-mixture variables. The knowledge and experience of subject-matter experts is needed to specify the components, as well as the constraints on their proportions that define the composition region to be explored experimentally. Also, experts must specify the product properties and the desired values or ranges for the properties. Mixture experiment methods for experimental design, property modeling, optimization, and confirmation, combined with the expert’s inputs, provide for efficiently developing product formulations and the settings of other variables to meet the requirements on product properties.

This article presents a case study in which mixture experiment methods were applied to develop a rubber-compound formulation for o-rings. The article describes each step of the study and what was performed to complete that step. The steps included 1) identifying the objective of the study and requirements for properties of the new o-ring, 2) selecting the components to vary in the mixture experiment and specifying the constraints on the component proportions that identify the experimental region, 3) constructing the mixture experiment design, 4) measuring the 10 properties and assessing the data for the design points, 5) developing property-composition models, 6) selecting a product formulation considering requirements on the 10 properties, and 7) confirming the selected formulation meets all requirements. Subsequent sections of the article successively address these steps. First, however, the mixture experiment approach for formulation development is discussed and compared to non-mixture approaches.

BACKGROUND

Overview of the Mixture Experiment Approach

In a product made of a mixture of components, the components have linear and possibly nonlinear blending effects (which may be synergistic or antagonistic)



on the product properties. A mixture experiment provides a structured way to investigate these effects and to discover the functional relationship between components and properties. By using an appropriate mixture experiment design and modeling methods, the experimenter can select the component proportions that yield the product with desired properties.

Piepel [1] identifies eight mixture experiment approaches: component proportions, mathematically independent variables, slack variable, mixture amount, component amounts, mixture-process variable, mixture of mixtures, and multifactor mixture. Piepel and Cornell [2] discussed some of these approaches, gave illustrating examples, and made recommendations for selecting a suitable approach for designing a mixture experiment and analyzing the data. The component-proportion approach (which is the most commonly used mixture experiment approach) was used in the o-ring study discussed in this article. In the component-proportion approach, the properties of the product depend on the relative proportions of the components and not on the amount of the mixture or on process-variable settings.

In a mixture experiment with q components, we denote by xi the proportion of the i-th component in a mixture. The xi are subject to the constraints:

0 ≤ xi ≤ 1, i = 1, 2, ... , q (1)

x

ii=1

q

∑ = 1 (2)

where q is the number of mixture components. Because of (1) and (2), the component proportions are dependent variables and the mixture space is a (q-1)-dimensional simplex. Sometimes the component proportions are subject to lower (Li) and/or upper (Ui) bounds:

0 ≤ Li ≤ xi ≤ Ui ≤ 1, i = 1, 2, ... , q (3)

and possibly multi-component constraints of the form:

C

j≤ A

1kx

1+ A

2kx

2+ ...+ A

qkx

q≤D

j, j = 1, 2, ... , K (4)

Constraints of the form (3) can yield a simplex mixture space, but constraints of the form (3) and/or (4) generally yield a polyhedral mixture space. Subject-matter experts must utilize their knowledge and experience to select the components and their constraints that specify the product composition space to be explored experimentally. The process of specifying the component constraints may require two or three iterations, using software to generate, plot, and assess



the vertices of the polyhedral region to see if any may yield property values too far from acceptable. Initial iterations may show the constraints allow some undesirable combinations of component proportions, which leads to revising the constraints or adding new ones until the mixture space specified by constraints (3) and (4) is judged acceptable by subject-matter experts.

Cornell [3] and Smith [4] discuss mixture experiment designs for simplex and polyhedral mixture spaces. Generating experimental designs for polyhedral mixture spaces generally requires using computer software. Design-Expert [5], Minitab [6], and JMP [7] are widely-used software that generate experimental designs for simplex and polyhedral mixture spaces.

Next, product samples are manufactured according to the formulations in the experimental design, and the properties (responses) of interest are measured. The resulting data are then analyzed using mixture experiment methods in software mentioned previously. The type of data analysis depends on the objectives of the experiment, which may include 1) understanding the effects of component and other factors on the response(s), 2) identifying components and other factors with significant and nonsignificant effects on the response(s), 3) developing models for predicting the response(s) as functions of the mixture components and any other factors, and 4) developing products with optimal or desired values and uncertainties of the response(s) [1].

The family of Scheffé canonical polynomial models [3, 4] is widely used when there is no theoretical or first-principle basis for relating the response variable (y) to the component proportions (xi). These models often adequately approximate the true, unknown relationships between each y (or transformation thereof, f(y)) and the xi. This family of models includes:

Linear: f(y) = β

ix

i+ ε

i=1

q

∑ (5)

Quadratic:

f(y) = βix

i+ β

ijx

ix

jj=i+1

q

∑i=1

q−1

∑ + εi=1

q

∑ (6)

Special-cubic:

f(y) = βix

i+ β

ijx

ix

jj=i+1

q

∑i=1

q−1

∑ + βijk

xix

jx

kk=j+1

q

∑j=i+1

q−1

∑i=1

q−2

∑ + εi=1

q

∑ (7)

Full-cubic:

f(y) = βix

i+ β

ijx

ix

jj=i+1

q

∑i=1

q−1

∑ + βijk

xix

jx

kk=j+1

q

∑j=i+1

q−1

∑i=1

q−2

∑i=1

q

∑

+ βijx

ix

jj=i+1

q

∑i=1

q−1

∑ (xi− x

j)+ ε (8)

The partial quadratic mixture (PQM) model [4, 8] is given by:



f(y) = βix

i+ Subset β

iix

i2

i=1

q

∑ + βijx

ix

jj=i+1

q

∑i=1

q−1

∑

+ ε

i=1

q

∑ (9)

The partial cubic mixture (PCM) model:

f(y) = βix

i+ Subset β

iix

i2

i=1

q

∑ + βijx

ix

jj=i+1

q

∑i=1

q−1

∑

i=1

q

∑

+ Subset

βijk

xix

jx

kk=j+1

q

∑j=i+1

q−1

∑i=1

q−2

∑ + βijx

ix

jj=i+1

q

∑i=1

q−1

∑ (xi− x

j)

+ βijx

i2x

j+ β

iix

i3

i=1

q

∑j

q

∑i ≠

q

∑

+ ε

(10)

was developed by the second author (unpublished) and is discussed for the first time in this article. In (9) and (10), only subsets of the terms in brackets are selected by stepwise regression [9] to avoid over-parameterization. Note that PQM models include various reductions of the quadratic mixture model in (6). Similarly, PCM models include various reductions of the special-cubic and full-cubic mixture models in (7) and (8), respectively. In the models (5) to (10), the betas are coefficients estimated using least squares regression and the epsilons represent the random experimental/and measurement uncertainty.

Cornell [3] and Smith [4] comprehensively discuss the main stages of a mixture experiment: planning, experimental design, conducting the experiment, modeling and other data analyses, and selecting and confirming an optimal response. Piepel [10] reviewed the first 50 years of mixture experiment research, summarized experimental design and data analysis methods, and listed references that discuss mixture experiment applications in many different fields.

Mixture Experiment Approach Versus Other Approaches

There are some misconceptions that 1) statistically designed experiments are time-consuming and expensive, and 2) statistical methods require highly qualified staff to implement [11]. On the contrary, non-mixture approaches that are often used to solve formulation problems can be inefficient, yield fewer useful results, develop sub-optimal formulations, and make incorrect conclusions about the system underlying the experimental data. This topic is discussed in subsequent paragraphs.

For the o-ring formulation problem, the mixture experiment approach yielded property-composition models (discussed subsequently) that can be used to



develop modified or new formulations to meet customer requirements and be competitive in the market. Property-composition models also support investigating formulation revisions to 1) reduce the proportions of expensive components, or 2) make trade-offs between property requirements. Models can be applied in this way without additional experimentation or manufacturing changes.

The iterative approach of developing product formulations based on an expert’s knowledge and experience can be inefficient, especially if a new formulation with significantly different property requirements than previous formulations is needed. The iterative approach often requires more experiments, time, and cost to develop an acceptable new formulation (which may be less optimal than a formulation developed using a mixture experiment approach). Even if the iterative, expert-based approach requires the same or less time and cost, the resulting data may not provide a good coverage of the composition space around the new formulation. Hence, that approach may not provide a good basis for developing property models and applying them as discussed previously. For these reasons, the mixture experiment approach (guided by expert involvement) is strongly recommended over the expert-guided, iterative approach to formulation development.

Sometimes factorial-type experiments are performed instead of mixture experiments in formulation development situations. In factorial-type experiments the variables can be changed independently, whereas in mixture experiments the component proportions are dependent because of the constraint (2). One common approach (referred to as the slack-variable (SV) approach) [1, 3, 10] designs the experiment, models the data, and optimizes the formulation ignoring one of the mixture components. The ignored component is the SV and is often the one making up the majority of the product. However, ignoring one of the components in the SV approach can lead to several problems, including making incorrect conclusions and developing sub-optimal formulations [2, 12]. Another approach that uses factorial-type designs is the mathematically-independent variable (MIV) approach. [1, 10] With this approach, ratios or other functions of components are used to reduce the q mixture components to q − 1 mathematically independent variables. Piepel and Cornell [2] discuss when this approach may be appropriate, but often it forces experiments to be designed, data analyzed, and results interpreted in terms of independent variables that are unnatural functions of the components. Anderson and Whitcomb [13] and Cornell [3] also discuss why mixture experiments are preferred over factorial experiments.

In summary, for product formulation problems the mixture experiment approach should be preferred over the expert-directed, iterative approach. The MIV approach can be appropriate for some problems, but the SV approach should be avoided.



EXPERIMENTAL

Rubber-Compound Formulation for an O-Ring

We now describe a study to develop a new rubber-compound formulation for making o-rings (Figure 1) that would perform well in petroleum-based hydraulic fluid with high-temperature conditions. Ten response variables (denoted R1-R10) and their required ranges (selected by a subject-matter expert based on planned use conditions and a relevant specification) are listed as lower and upper bounds in Table 1. The objective of this study was to formulate and

Table 1. Responses, required ranges, and optimization ranges and goals

Responses Required range

Optimum range and goal

Lower Upper Min Max Goal

Measured after manufacturing

R1 – Hardness (Shore A) 65 80 65 80 Maximize

R2 – Tensile Strength (Psi) 1350 None 1350 2500 Maximize

R3 – Elongation at Break (%) 125a None 125 500 Maximize

R4 – Retraction at Low Temperature (ºC) -45 None -45 -40 Minimize

R5 – Permanent Deformation Test (%) None 70 0 70 Minimize

Measured after aging 70 hours at 135 ± 3ºC in hydraulic fluid

R6 – Hardness Change (Shore A) -10 +5 -10 -5 Maximize

R7 – Tensile Strength Reduction (%) None 50 0 50 Minimize

R8 – Elongation Reduction (%) None 45 0 45 Minimize

R9 – Volume Change (%) 0 15 0.01b 15 Minimize

R10 – Permanent Deformation Test (%) None 45 0 45 Minimizea The way this response is measured, percentage values greater than 100 are possible.b A value greater than zero was necessary because the natural logarithm of R9 was modeled

Figure 1. An o-ring



manufacture an o-ring product that meets all these property requirements. We also illustrate selecting a formulation that makes optimal tradeoffs in trying to minimize or maximize some responses, while maintaining others within their optimum ranges (also listed in Table 1). The components and constraints, experimental design, response measurement and data assessment, property-composition modeling, formulation selection, and formulation confirmation steps of the case study are now discussed.

The Components and Their Constraints

The components of the product were identified by a chemical engineer who is an expert in rubber-based product development. The components include nitril elastomer (N-917), zinc oxide, stearic acid, carbon (N-550), agarite resin D, dioctyl adipate (DOA), dithiodinicotinate (DTDN), santacure, sulfur, and tetramethyl thiuram disulfide (TMTD). The chemical engineer specified proportions of four of these 10 components (N-550 (x1), DOA (x2), TMTD (x3), and santacure (x4)) to be varied in the mixture experiment. Some of the other components are known to affect some responses. However, time and resources available for experimentation did not permit performing a mixture experiment to investigate the effects of all 10 components. The four components selected to vary in the mixture experiment were identified as sufficient to develop a product meeting the requirements. The proportions of all other components in the rubber-compound were held constant in the mixture experiment. Also, other possible factors of the experiment (e.g., material, manufacturing process, people, and equipment) were held stable to prevent potential effects of such nuisance factors on the responses.

The relative proportions (xi) of the four components are subject to the mixture constraint (x1 + x2 + x3 + x4 = 1) and the lower and upper bounds:

0.55 ≤ x1 (N-550) ≤ 0.90 0.09 ≤ x2 (DOA) ≤ 0.40 (11) 0.0062 ≤ x3 (TMTD) ≤ 0.077 0.0012 ≤ x4 (santacure) ≤ 0.027

These bounds were selected to be wide enough to protect against the optimal formulation(s) being outside the region defined by the bounds, but not too wide so that mixture experiment models would be able to adequately approximate the true, unknown property-composition relationships. The constraints (11) specify the three-dimensional polyhedral region shown in Figure 2. All possible combinations of the components permitted by the constraints in (11) were judged to be feasible, so multi-component constraints, as in (4), were unnecessary.



Figure 2. Constrained mixture space and the 20 distinct mixtures in the experimental design. Gray dots denote statistically selected points and black dots denote expert-selected points

Determining the components to vary and the constraints on their proportions is crucial for the experiment to yield relevant and useful data. Consequently, these decisions must be determined thoughtfully by the experts and experimenters. Specifying constraints to define a smaller experimental region increases the chance of mixture experiment models adequately approximating the property-composition relationships. However, care must be taken not to make the experimental region so small that it excludes formulations that may be optimal.

The Experimental Design

Because the experimental region, defined by the component constraints in (11), is an irregular polyhedron, the experimental design was constructed using an optimal design approach. This approach requires specifying 1) a model form that is assumed to adequately approximate the true, unknown property-composition relationship for each response, and 2) a design criterion to optimize. In this study, the experimental design was generated using the D-optimality criterion and assuming a quadratic mixture experiment model, in (6). For further information about optimal design and the D-optimality criterion, see Myers et al. [14].

The JMP software [15] was used to generate 38 possible design points (referred to as candidate points) consisting of the 12 vertices, 18 edge-center points, and 8 face-center points of the polyhedral experimental region. Because of



time and funding limitations, it was decided the experimental design should consist of 20 distinct points (formulations). First, JMP was used to select 15 design points from the 38 candidate points according to the D-optimality criterion. Additionally, five design points were specified by the expert and added to the design. Because of the manufacturing facility’s rules relevant to new product development research, all 20 design points were replicated. Replications enable estimating the experimental and measurement uncertainty for each response variable, which is important for analyzing the data and assessing the lack-of-fit of property-composition models. The 40-run mixture experiment design, consisting of the 20 distinct mixtures and the 20 replicates, is presented in Table 2. The 40 runs were performed in three blocks, with the order of runs randomized within blocks, as shown in Table 2.

Table 2. 40-point mixture experiment design including replicates

Test # and run order

Design point #

& replicates

Point type Block Component proportions

x1 x2 x3 x4

1 1 Vertex 1 0.8060 0.0900 0.0770 0.0270

2 2 EdgeCentb 1 0.8414 0.0900 0.0416 0.0270

3 3 EdgeCent 1 0.7218 0.2450 0.0062 0.0270

4 4 Vertex 1 0.5668 0.4000 0.0062 0.0270

5 5 Vertex 1 0.5500 0.3718 0.0770 0.0012

6 6 EdgeCent 1 0.5500 0.4000 0.0359 0.0141

7 7 Vertex 1 0.8768 0.0900 0.0062 0.0270

8 8 Vertex 1 0.8318 0.0900 0.0770 0.0012

9 9 EdgeCent 1 0.5500 0.3730 0.0500 0.0270

10 10 FaceCentb 1 0.68445 0.22445 0.0770 0.0141

11 11 Vertex 1 0.5500 0.3460 0.0770 0.0270

12 12 EdgeCent 1 0.8884 0.0900 0.0062 0.0154

13 13 Vertex 1 0.9000 0.0926 0.0062 0.0012

14 14 FaceCent 1 0.720733 0.240733 0.037333 0.0012

15 15 Vertex 1 0.5926 0.4000 0.0062 0.0012

16 Rep. 6 EdgeCent 2 0.5500 0.4000 0.0359 0.0141

17 Rep. 3 EdgeCent 2 0.7218 0.2450 0.0062 0.0270

18 Rep. 15 Vertex 2 0.5926 0.4000 0.0062 0.0012

19 Rep. 1 Vertex 2 0.8060 0.0900 0.0770 0.0270

20 Rep. 10 FaceCent 2 0.68445 0.22445 0.0770 0.0141

21 Rep. 14 FaceCent 2 0.720733 0.240733 0.037333 0.0012

22 Rep. 4 Vertex 2 0.5668 0.4000 0.0062 0.0270



Table 2. Cont'd....

Test # and run order

Design point #

& replicates

Point type Block Component proportions

x1 x2 x3 x4

23 Rep. 13 Vertex 2 0.9000 0.0926 0.0062 0.0012

24 Rep. 2 EdgeCent 2 0.8414 0.0900 0.0416 0.0270

25 Rep. 12 EdgeCent 2 0.8884 0.0900 0.0062 0.0154

26 Rep. 7 Vertex 2 0.8768 0.0900 0.0062 0.0270

27 Rep. 11 Vertex 2 0.5500 0.3460 0.0770 0.0270

28 Rep. 8 Vertex 2 0.8318 0.0900 0.0770 0.0012

29 Rep. 5 Vertex 2 0.5500 0.3718 0.0770 0.0012

30 Rep. 9 EdgeCent 2 0.5500 0.3730 0.0500 0.0270

31 16a NearCentc 3 0.7279 0.2230 0.0257 0.0234

32 17a Interior 3 0.6169 0.3340 0.0257 0.0234

33 18a Interior 3 0.6364 0.3340 0.0062 0.0234

34 19a Interior 3 0.5939 0.3340 0.0487 0.0234

35 20a Interior 3 0.6377 0.3340 0.0257 0.0026

36 Rep. 17a Interior 3 0.6169 0.3340 0.0257 0.0234

37 Rep. 20a Interior 3 0.6377 0.3340 0.0257 0.0026

38 Rep. 16a NearCentc 3 0.7279 0.2230 0.0257 0.0234

39 Rep. 19a Interior 3 0.5939 0.3340 0.0487 0.0234

40 Rep. 18a Interior 3 0.6364 0.3340 0.0062 0.0234a Design points 16–20 (and their replicates) were selected by the subject-matter expertb EdgeCent = edge centroid, FaceCent = face centroidc This point is close to the overall center point of the constrained mixture region

The locations of the 20 distinct design points within the polyhedral experimental region are displayed in Figure 2. A column of Table 2 shows the “point type” (e.g., vertex, edge center, face center, interior) for each of the 20 distinct mixtures. The 15 points selected using the D-optimal design approach are listed first in Table 2. They include 8 vertices, 5 edge-center points, and 2 face-center points. The five points selected by the expert include a point close to the overall center of the experimental region (#16) and four points (#17 to #20) in somewhat of a group (see Figure 2).

A scatterplot matrix plot of the proportions of the four mixture components in the design points is given in Figure 3. This figure shows that x1 and x2 are strongly negatively correlated, because x3 and x4 have small values over small ranges. Hence, x1 and x2 must primarily offset changes in each other, resulting in their strong negative correlation.



Figure 3. Scatterplot matrix plot of the 20 distinct mixture design points. Because each plot shows a two-dimension projection of the design points over a three-dimensional region, overplots may occur resulting in fewer than 20 points in a given plot

RESULTS AND DISCUSSION

Measuring the Responses and Assessing the Data

The 10 responses were measured for each of the 40 experimental runs using a standard method for each response. The measured values of R1-R10 for the 40 experimental runs are listed in Table 3. Note that none of the 40 experimental runs have values of R1-R10 that satisfy all of the required ranges in Table 1. Thus, the goals of this work to identify 1) the subregion of the experimental region (formulation space) where the required ranges are satisfied, and 2) an optimal o-ring formulation are challenging goals.

The pooled estimates of experimental testing and response measurement uncertainty, expressed as standard deviations (SD) and percent relative standard deviations (%RSD), are listed in Table 4. These uncertainties are very small, which resulted from the care taken by a single individual who performed the testing and measurement work.

A scatterplot matrix of the response values in Table 3 is displayed in Figure 4. Relatively strong correlations are seen between some pairs of responses. These



Table 3. Response variable values for the 40-point mixture experiment design

Test # & run order

Design point &

replicates

Responses

R1 R2 R3 R4 R5 R6 R7 R8 R9 R10

1 1 87 2014.6 105.0 -32.1 52.8 -6 48.9 42.4 11.1 25.0

2 2 88 2013.5 121.6 -34.9 72.3 -5 52.4 44.9 14.0 27.9

3 3 72 1955.3 236.8 -39.9 88.2 -7 57.3 43.4 14.8 26.6

4 4 51 1410.0 377.7 -46.1 72.5 -5 54.0 44.4 11.0 38.4

5 5 65 978.8 151.0 -46.1 77.4 -4 25.2 26.8 3.0 33.8

6 6 56 1174.9 239.8 -47.9 85.1 -3 56.2 49.1 9.3 34.1

7 7 88 1696.8 103.8 -38.2 69.4 -10 46.1 36.4 17.3 27.9

8 8 90 1874.8 82.7 -36.1 63.0 -10 53.2 32.7 24.9 19.9

9 9 53 1003.6 259.8 -44.6 90.6 -9 65.4 44.5 26.3 11.5

10 10 65 2002.7 291.3 -40.8 36.7 -12 76.1 48.3 45.8a 5.7

11 11 68 742.1 127.9 -43.9 74.6 -10 30.3 23.2 15.2 17.0

12 12 88 1597.6 127.7 -36.9 77.9 -14 52.7 34.2 22.1 37.4

13 13 90 1533.8 121.0 -38.1 77.8 -13 45.8 34.3 18.8 33.5

14 14 74 1089.9 129.1 -42.0 66.0 -15 44.9 21.9 27.0 9.9

15 15 52 1397.4 395.0 -45.9 97.8 -5 77.1 63.0 20.0 39.5

16 Rep. 6 56 1175.9 240.1 -48.0 85.3 -3 56.0 48.9 9.4 33.8

17 Rep. 3 72 1953.6 237.1 -40.1 88.4 -7 57.2 43.5 14.9 26.7

18 Rep. 15 52 1398.0 394.7 -46.0 98.0 -5 77.0 63.1 19.9 39.7

19 Rep. 1 87 2010.2 104.8 -32.2 52.8 -6 48.7 42.3 11.0 25.1

20 Rep. 10 65 2004.9 291.0 -41.0 36.8 -12 76.3 48.5 45.9a 5.6

21 Rep. 14 74 1088.8 129.0 -42.0 66.1 -15 45.0 22.0 27.1 9.8

22 Rep. 4 51 1411.4 377.4 -46.0 72.3 -5 53.8 44.5 11.0 38.3

23 Rep. 13 90 1535.0 121.1 -38.0 78.0 -13 45.8 34.2 18.7 33.3

24 Rep. 2 87 2016.5 121.7 -35.0 72.1 -5 52.3 45.0 14.0 27.8

25 Rep. 12 88 1599.6 127.6 -36.8 78.1 -14 52.8 34.2 22.0 37.6

26 Rep. 7 89 1694.6 104.0 -38.0 69.5 -10 46.0 36.5 17.3 27.8

27 Rep. 11 68 741.0 127.5 -44.0 74.4 -10 30.2 23.1 15.3 17.2

28 Rep. 8 89 1872.3 83.0 -36.0 62.8 -10 53.1 32.6 24.8 19.6

29 Rep. 5 65 977.6 150.8 -46.0 77.6 -4 25.1 26.8 2.9 34.0

30 Rep. 9 53 1005.4 260.0 -44.7 90.3 -9 65.2 44.6 26.6 11.4

31 16 72 1899.9 244.2 -41.2 79.0 -8 45.5 41.2 14.6 31.0

32 17 72 1087.6 207.7 -43.0 87.0 -8 77.0 55.9 13.2 39.5

33 18 66 1523.8 278.3 -41.0 90.2 -7 75.7 62.6 18.6 45.1

34 19 70 1461.9 173.5 -41.8 79.2 -4 36.4 41.0 6.0 36.8

35 20 66 1773.3 284.0 -41.2 87.5 -7 51.5 48.7 13.4 33.9

36 Rep. 17 72 1088.8 208.1 -43.3 87.2 -8 77.1 56.1 13.3 39.3

37 Rep. 20 66 1774.3 283.0 -41.0 87.3 -7 51.7 48.9 13.5 33.8

38 Rep. 16 72 1902.8 243.6 -41.0 79.8 -8 45.1 41.8 14.5 31.5

39 Rep. 19 70 1461.0 173.0 -42.0 78.9 -4 36.2 40.8 6.1 36.6

40 Rep. 18 66 1525.4 278.1 -41.0 89.9 -7 75.8 62.7 18.7 44.9a These data points were not used to develop the R9 model



Table 4. Pooled estimates of experimental and measurement uncertainty for responses R1–R10 based on replicate testsa

R1 R2 R3 R4 R5 R6 R7 R8 R9 R10

SDP 0.274 1.390 0.289 0.105 0.188 0 0.117 0.130 0.079 0.145

%RSDP 0.309 0.086 0.169 0.257 0.238 0 0.244 0.313 0.730 0.605

a SD

P= [(SD

12 +SD

22 + ...+SD

202 ) / 20]0.5

and

%RSD

P= [(%RSD

12 +%RSD

22 + ...+%RSD

202 ) / 20]0.5

, where %RSDi = SDi/Meani and with SDi and Meani calculated from the replicate pair of the ith distinct design point (i = 1, 2, …, 20). Note these formulas are specific to the o-ring data where there are replicate pairs, so that each SDi or %RSDi is estimated with only one degree of freedom

pairs, with Pearson product-moment correlation values shown in parentheses, are R1 & R4 (0.882), R1 & R3 (-0.837), R7 & R8 (0.822), and R3 & R8 (0.701). Also, test runs #10 and #20 (replicates of design point #10) appear as outliers for several pairs of responses (R3-R5, R4-R5, R5-R6, R5-R7, R5-R8, R5-R9, R5-R10, R6-R9, R7-R9, R8-R9, and R9-R10). The response values for test runs #10 and #20 were checked and are correct. Still, they may be influential in fitting property-composition models.

Figure 4. Scatterplot matrix plot of the R1-R10 response values for the 40 points of the mixture experiment design. For all plots involving R5 and R9, tests #10 and #20 (replicates of design point #10) plotted as solid squares) appear as outliers



Table 3 and Figure 4 also show small sets of data that may be influential in model development. Test pairs 1/19 and 10/20 account for nearly half the range of R5 values, although their R5 values are among the few below the required maximum (in Table 1). Test pair 10/20 also accounts for nearly half the range of R9, with the R9 values of those points being well above the desired maximum. Small numbers of points that account for nearly half the range of a response can be highly influential in developing models, which was considered during the modeling work discussed subsequently.

Comparing the response values in Table 3 to the required ranges in Table 1 identified some notable information for R5 and R10. R5 must be below 70, but only test pairs 1/19 and 10/20 have values significantly below 70. Hence, models for R5 may have some limitations in predicting whether compositions satisfy the upper limit for this response. Also, R10 must be below 45, with all test pairs meeting that limit except 18/40 (which have values ~ 45). Hence, the required lower and upper bounds on R10 have little impact on selecting acceptable o-ring formulations.

Property-Composition Models and Other Data Analyses

The Minitab [6] and Design-Expert [5] software packages were used to develop property-composition models for each of the 10 response variables (R1-R10). Preliminary modeling work did not reveal a significant block effect (see Table 2) for any response. Hence, models without block effects were considered subsequently and are reported in this article.

Design-Expert [5] automatically fits and statistically assesses the linear mixture (LM), quadratic mixture (QM), special-cubic mixture (SCM), and full-cubic mixture (FCM) models given in (5) to (8). The statistical assessment includes tests for model lack-of-fit [3]. For all 10 response variables, the LM, QM, and SCM models had statistically significant lack-of-fits with p-values [14] always < 0.0001. All quadratic, special-cubic, and full-cubic terms in the FCM model fit to each response were highly statistically significant (p-values < 0.0001). Normally such results would indicate that at least a FCM model is needed to adequately model each response. In fact, the first and third authors initially used FCM models to select a desirable o-ring formulation. However, the FCM model contains 20 terms, and it exactly fits the averages of the replicate response pairs for the 20 distinct design points. Also, the FCM models may overfit the data without being able to detect it statistically because of the very small experimental and measurement uncertainties for the responses, as discussed previously. Further, because 1) four of the 20 distinct design points selected by the expert are in a group, and 2) there were insufficient



design points on the long edges of the experimental region (see Figure 2), the 20 distinct design points provide less than ideal support for fitting FCM and PCM models. This can lead to extrapolative predictions in areas of the experimental region inadequately covered by the experimental design. On the other hand, QM and SCM models were judged inadequate for modeling some responses. Ultimately, it was decided to develop PCM models for R1–R10 using stepwise regression to add quadratic and/or cubic terms to the linear mixture model. The resulting PCM models for R5, R7, and R9 caused significant extrapolation for portions of the experimental region, so SCMs were developed for those responses to rectify this problem.

Finally, the replicate pair for design point #10 (Tests # 10 and 20) caused problems in developing acceptable models for R9. Values of R9 below 15 are required (per Table 1), but these two points had values greater than 45 and roughly doubled the range of R9 values. The remaining design points had a sufficient range of R9 values below and above 15 to enable models to discriminate, so Tests #10 and 20 were not used to develop the R9 model. Further, the natural logarithm of R9 was modeled based on a Box-Cox analysis [9].

The coefficients and several summary statistics for the PCM and SCM models developed for R1-R10 are listed in Table 5. The summary statistics include the root mean squared error (RMSE), R2, R2

Adj, R2P, and the model

lack-of-fit (LOF) p-value. For a given response 1) RMSE is an estimate of the experimental and measurement standard deviation if the model does not have a significant LOF, 2) R2 represents the fraction of variation in the response variable accounted for by the fitted model, 3) R2

Adj is like R2 but adjusted for the number of data points and model terms, and 4) R2

P is the predicted R2, calculated similar to R2 except leaving out each data point one-at-a-time. Typically 0 ≤ R2

P ≤ R2Adj ≤ R2 ≤ 1, although R2

P can be negative if there are influential data points. A LOF p-value less than 0.05 was used as an indicator of statistically significant model LOF.

The PCM and SCM models in Table 5 were developed using the proportions of the components x1, x2, x3, and x4. Hence, some of the model coefficients listed in Table 5 are very large, for two reasons. First, large coefficients resulted from 1) the small values and narrow ranges for x3 and x4, and 2) the strong negative correlation between x1 and x2. Second, products of component proportions in quadratic and cubic terms are very small numbers, thus requiring large coefficients to yield appropriate contributions to response predictions. Many years ago, such “inflation” of model coefficients could be a sign of computer software yielding inaccurate estimates of model coefficients. However, with modern 32-bit and 64-bit computers, modern regression algorithms, and double-precision software, computational problems in estimating mixture model coefficients are generally no longer an issue. Models



Tab

le 5

. co

effi

cien

ts a

nd s

umm

ary

stat

isti

cs f

or

the

spec

ial-

cub

ic o

r p

arti

al-c

ubic

mix

ture

mo

del

s fo

r re

spo

nses

R1-

R10

Term

R1

R2

R3

R4

R5

R6

R7

R8

ln(R

9)R

10x 1

149.

638

98.7

36.7

-22.

17.

519

.7-1

06.6

63.5

-2.5

135.

4x 2

-895

.3-2

2466

.398

2.2

-290

.5-8

26.9

-517

.9-2

020.

7-4

512.

9-7

6.4

-144

6.5

x 377

2.1

-138

09.3

7223

7.9

34.1

3179

.674

7.7

4307

4.4

-265

.537

9.8

632.

0x 4

87.9

-2.3

02E

0764

206.

4-3

4.1

-218

37.5

-148

2.9

-116

443

272.

4-1

758.

826

92.2

x 1x2

1007

2518

15.9

4100

.014

8.9

x 1x3

-186

5.2

-450

30.6

-316

.2x 1x

42.

319E

0723

066.

012

4244

1983

.3x 2x

3-2

4782

4-1

1685

620

844.

1-2

7950

.6+

627.

8x 2x

42.

223E

0735

352.

119

3511

4353

.8x 3x

42.

347E

0723

5320

2700

.8-5

9852

5-1

.897

E06

1385

93-2

4780

(x1)

2

(x2)

243

53.0

1216

.435

0.9

1592

7.0

(x3)

2-1

9791

.5(x

4)2

-596

884

3635

33x 1x

2x3

4653

11-4

95.2

-479

94.2

-391

25.8

-213

2.9

x 1x2x

41.

952E

06-2

7316

.713

787.

4-1

4167

8-4

835.

2x 1x

3x4

6854

652.

202E

0628

205.

7x 2x

3x4

-703

2.2

7120

42-1

4406

.42.

175E

0631

203.

0-6

2483

.2x 1x

2(x 1-

x 2)40

05.2

x 1x3(

x 1-x 3)

x 1x4(

x 1-x 4)

-764

1.6

x 2x3(

x 2-x 3)

-461

47.5

4878

.5x 2x

4(x 2-

x 4)-1

1747

.2x 3x

4(x 3-

x 4)-6

6529

8040

0213



Tab

le 5

. co

nt'd

....

Term

R1

R2

R3

R4

R5

R6

R7

R8

ln(R

9)R

10(x

1)2 x

2-1

1393

6(x

1)2 x

3-9

0564

.2(x

1)2 x

4

(x2)

2 x1

1375

.062

06.0

(x2)

2 x3

(x2)

2 x4

-141

91.4

(x3)

2 x1

-101

70.4

-213

13.9

(x3)

2 x2

(x3)

2 x4

-223

335

-785

517

(x4)

2 x1

2612

4.7

(x4)

2 x2

(x4)

2 x3

-243

5950

(x1)

3

(x2)

3-5

909.

1-1

728.

9-1

4066

.9-8

59.4

(x3)

315

8318

1282

5911

1699

4(x

4)3

5.47

1E08

1.31

2E08

-953

6060

# Te

rms

813

119

1413

1411

1414

RM

SE

2.98

150.

7319

.14

0.57

2.16

0.96

8.86

2.87

0.21

3.34

R2

0.95

60.

897

0.96

70.

985

0.98

40.

946

0.75

70.

951

0.89

30.

936

R2 A

dj0.

947

0.85

10.

956

0.98

10.

976

0.92

20.

636

0.93

50.

835

0.90

3

R2 P

0.92

90.

814

0.95

20.

979

0.96

70.

919

0.54

20.

922

0.78

60.

865

LOF

p-va

lue

<0.

0001

<0.

0001

<0.

0001

<0.

0001

<0.

0001

− a

<0.

0001

<0.

0001

<0.

0001

<0.

0001

a A

ll of

the

repl

icat

e pa

irs h

ad id

entic

al r

espo

nses

, so

ther

e is

no

pure

err

or te

rm to

det

erm

ine

the

mod

el L

OF

p-va

lue



based on proportions of L-pseudocomponents [3] were also considered, but did not reduce the magnitudes of model coefficients that much. Because subject-matter scientists often do not like pseudocomponent transformations of composition variables (especially if the transformations don’t provide substantive value), we decided to use response models developed in terms of the untransformed component proportions (x1, x2, x3, and x4).

The PCM and SCM models fitted to R1-R10 have R2 values ranging from 0.757 to 0.985 (see Table 5), indicating that the models account for substantial portions of the variation in the response values for the experimental design points. The R2

Adj values are not too much lower than the R2 values, indicating that the models are unlikely to contain unnecessary terms. The LOF p-values of the fitted PCM mixture models for R1-R10 listed in Table 5 are all < 0.0001, suggesting that these models have statistically significant LOFs. The statistically significant LOFs are at least partially due to the very small differences in replicate results (as discussed previously). Hence, some of these LOFs may be statistically significant but not practically significant. Predicted-versus-measured plots (not included in this article) suggest a few PCM models may have some practically significant LOF. Still, the fits of the models in Table 5 were considered good enough to use them and obtain the results discussed in subsequent sections. However, results obtained using the models should be confirmed, given the experimental design did not cover the experimental region as well as would be preferred to support fitting SCM and PCM models.

Ideally, the final response models selected should be validated using data points different from those used to fit the models. It was not possible to collect separate validation data points in this study. In such cases, cross-validation using the model-development data is recommended. The statistic R2

P involves leave-one-out cross-validation, in which the responses for each data point are predicted using a model fit without that data point. The R2

P values are not much lower than the R2

Adj and R2 values for the PCM models in Table 5 (except R7). This indicates 1) relatively good cross-validation performance, and 2) there are not any data points with highly influential response values.

Identifying Product Formulations Meeting Property RequirementsFigure 5 shows the subregion of o-ring formulations predicted to satisfy all of the required constraints on R1–R10 in Table 1 using the PCM and SCM models in Table 5. The size and location of this subregion, in terms of x1, x2, and x3, changes as the value of x4 changes. The largest portion of the subregion occurs when x4 = 0.027 (its maximum value). In the range 0.0098 ≤ x4 ≤ 0.0192 there are no formulations that meet the required response ranges.



Using the FCM models (not reported in this article) that overfit the R1–R10 data, the subregion of formulations meeting the requirements was so small that it could not be visualized in plots similar to those in Figure 5.

Figure 5. Subregion of the mixture experimental region that satisfies all of the “response required ranges” in Table 1, predicted using the Table 5 models. The figure shows slices of the acceptable subregion as unshaded areas (white) inside the experimental region (light gray) at three values of x4. Formulations 1 and 2 discussed in the text are identified



The next step after identifying the subregion of o-ring formulations that meet the required ranges in Table 1 is to select a specific formulation for use in making o-rings. The desirability function approach [3, 4] is one way to optimize multiple responses simultaneously, making tradeoffs among responses if needed. This approach involves specifying an appropriate desirability function dm (where 0 ≤ dm ≤ 1) for each response (m = 1, 2, … , 10) depending on its optimization goal. The optimization goals and optimum ranges for R1–R10 are listed in Table 1. The functional forms of dm used were:

Maximize Rm dm=

0

Rm

- min

max-min

1

if R

m < min

if min ≤Rm≤max

if Rm

> max

for m = 1, 2, 3, 6

(12a)

Minimize Rm dm=

1

max-Rm

max-min

0

if R

m < min

if min ≤Rm≤max

if Rm

> max

for m = 4, 5, 7, 8, 9, 10

(12b)

where the min and max values are listed under the “Optimum Range and Goal” portion of Table 1. The combined desirability (D) was calculated using a function of the form

D = [d1 d2 ... d10]1/10 (13)

which is the geometric mean of the individual desirabilities for the 10 responses. Note that D = 0 if even one response is completely undesirable (i.e., dm = 0 for any m ∈ (1,2,..,10)), and D = 1 only if every response is completely desirable (i.e., dm = 1 for every m ∈ (1,2,..,10)). The expression (13) treats all responses as equally important, but there is a version that allows different importance levels for the responses. Cornell [3] and Smith [4] discuss the desirability function approach in detail, and Design-Expert [5] provides for fully implementing the approach.

The optimal solution from Design-Expert [5] for this optimization problem is (x1, x2, x3, x4) = (0.6352, 0.2608, 0.0770, 0.0270), denoted Formulation 2. In the original work completed before this paper was written, the selected formulation was (x1, x2, x3, x4) = (0.6620, 0.2448, 0.0738, 0.0194), denoted Formulation 1. Even though Formulation 1 was selected based on FCM models originally fitted to the data by the first and third authors, it falls within the composition



subregion predicted to meet the response requirements in Table 1 based on the PCM and SCM models in Table 5. Formulations 1 and 2 are shown in the second and third panels of Figure 5, respectively. The required-range subregion for (x1, x2, x3) when x4 = 0.0194 (corresponding to Formulation 1) is much smaller than the subregion when x4 = 0.0270 (corresponding to Formulation 2). Also, the overall desirabilities of Formulations 1 and 2 are 0.112 and 0.506, respectively. Hence, while Formulation 1 is acceptable, Formulation 2 is more desirable in that it makes better tradeoffs in the optimality constraints and is more robust to manufacturing deviations from the target formulation.

Confirming the Selected Formulation

After the initial study (before the additional modeling and optimization work in this article), Formulation 1 was manufactured to verify it satisfied all of the response requirements. Table 6 lists measured and predicted values of R1–R10 based on the fitted FCM models (not shown in the article) and the PCM and SCM models (in Table 5). There are small to moderate discrepancies between measured and predicted values of R1, R2, R3, R4, R6, R9, and R10 for Formulation 1, whereas the discrepancies are larger for R5, R7, and R8. For Formulation 1, the 1) R5 model underpredicts because of limited experimental data near the upper limit of R5, and 2) R7 and R8 models overpredict because of limited experimental data near the lower limits of R7 and R8. However, because all measured values of the Ta

ble

6.

Mod

el-

pre

dic

ted

an

d m

easu

red

resp

onse

valu

es

for

two f

orm

ula

tions

Valu

esa

R1

R2

R3

R4

R5

R6

R7

R8

R9

R10

F1 F

CM

Pre

dict

ed76

.818

02.2

157.

6-4

0.7

40.5

-6.6

26.4

28.8

14.0

20.5

F1 P

CM

/SC

M P

redi

cted

68.6

1748

.918

9.2

-40.

939

.9-8

.749

.341

.23.

219

.1

F1 M

easu

red

74.0

1720

.615

9.7

-40.

564

.8-4

.024

.725

.76.

628

.2

F2 P

CM

/SC

M P

redi

cted

69.6

2153

.327

4.0

-40.

629

.6-7

.6-1

.7(b

)16

.71.

711

.8a

F1 =

For

mul

atio

n 1

(0.6

620,

0.2

448,

0.0

738,

0.0

194)

and

F2

= F

orm

ulat

ion

2 (0

.635

2, 0

.260

8, 0

.077

0, 0

.027

0), w

here

form

ulat

ions

are

lis

ted

as (x

1, x

2, x

3, x

4)b

The

R7

mod

el p

redi

cts

nega

tive

valu

es fo

r a

very

sm

all s

ubre

gion

of t

he e

xper

imen

tal r

egio

n, b

ut th

ese

are

trea

ted

as z

ero

or n

eglig

ible

va

lues



responses are within their required ranges given in Table 1, Formulation 1 was confirmed. Table 6 also lists the values of R1–R10 predicted for Formulation 2 using the PCM and SCM models in Table 5. It was not possible to confirm in manufacturing that Formulation 2 satisfies the required and optimum ranges of R1–R10 in Table 1, but all of the predicted responses are within those ranges.

Discussion

As mentioned previously, there are 10 components in the rubber compound used to make o-rings. Because of cost and time constraints, only four components were varied in the experiment, with the remaining six components held at constant values. Because at least some of these six components would be expected to affect responses if varied, it may be possible to further improve o-ring formulations beyond the accomplishments in this article. Mixture experiments having 10 or more components have been performed [16], although they require larger experimental designs to adequately model the dependence of the responses on the component proportions. Still, the desirability function approach can be applied even with larger numbers of components. However, financial, time, and labor constraints exist in practice, and so it is reasonable to perform mixture experiments varying a subset of the components (e.g., the components believed to have the largest effects on the responses).

The o-ring study using mixture experiment methods provided some “lessons learned” that can be used to improve future mixture experiments.

1. Because PCM models were necessary to adequately model some responses, it would have been better to develop the D-optimal design for a FCM model rather than a QM model as was done. For the case study discussed in this article, that would have required a larger experimental design, which could have addressed the less-than-ideal coverage of the experimental region by the experimental design in Table 2. For future studies, the likelihood of special-cubic or full-cubic terms being needed to model responses should be assessed, and an appropriate mixture model (quadratic, special-cubic, or full-cubic) should be used to generate the optimal experimental design. If in doubt what model may be adequate, it is better to design the experiment for the largest model that might be needed for at least one response.

2. The D-optimal design approach tends to pick all design points on the boundary of the experimental region. However, it is good to include some points on the interior of the region, including the center point. One way



to do this a hybrid design, in which 1) the center point is specified as the first design point, 2) boundary points are selected by D-optimal design to augment the center point, and 3) interior points are selected to augment the center point and boundary points using a space-filling optimality criterion. Another way to include a center point and interior points in a design is by using a layered design [17]. In the o-ring mixture experiment, design point #16 (chosen by the expert) was close to the center point of the region. Design points #17 − 20 selected by the expert were grouped in one area of the interior of the region. A hybrid approach would have given a better coverage of the interior of the formulation space of interest.

3. When possible, it is nice to replicate all design points as was done in the o-ring mixture design. However, if experimental resources and time are limited, only a subset of the design points need be replicated. At least five replicates should be included in the experimental design to provide minimally sufficient degrees of freedom for estimating the experimental and measurement uncertainty, and for performing statistical tests for model LOF. Without increasing the total number of experiments in this study, it would have been much more beneficial to replicate fewer points and investigate more distinct formulations with better coverage of the boundary and interior of the experimental region.

Despite things that could have been done to improve the o-ring experiment and the challenging nature of the problem faced, it was a success in that it yielded an acceptable o-ring formulation (Formulation 1), a proposed better formulation (Formulation 2), and property-composition models that provide opportunities to revise the formulation to meet other needs.

CONCLUSIONS

Today, many companies use an iterative approach based on an expert’s knowledge and experience to develop formulations for new or improved products. This approach can be inefficient, not only requiring more time and money, but also the formulation obtained may not be optimum. In contrast, the mixture experiment approach uses the expert’s knowledge and experience, as well as statistical experimental design, modeling, and optimization methods, to achieve better results. Mixture experiment design provides for efficiently covering the formulation region of interest. The resulting experimental data are used to develop property-composition models, which can be used to develop formulations that are optimized with respect to multiple product responses. The models can also be used to 1) computationally investigate the effects of changing component proportions to improve product properties, 2) make



trade-offs in product properties, and 3) reduce cost. Being able to do this computationally provides for meeting fast-changing customer requirements without additional experimentation and manufacturing costs. In contrast, the iterative, expert-based approach is generally finished when a product with desired properties is obtained, and often must be done again when product changes are needed.

For the mixture experiment approach to be successful, collaboration of subject-matter experts and those with some knowledge in mixture experiment methods is essential. A key step is to select the 1) components of the formulation to hold fixed (if any), 2) components to vary experimentally, and 3) component constraints that specify the experimental region. When there are more than a few components in a product, and it is not clear which have the most important effects on product properties, it can be useful to perform a screening mixture experiment3 to identify the most important components. Then, those components can be varied in a more comprehensive mixture experiment (such as discussed in this article) to support developing models for optimizing and modifying product formulations.

When designing a mixture experiment where the goal is to develop an optimal formulation as well as adequate property-composition models, the design should be developed to support fitting the largest mixture model that may be applicable to at least one response (considering financial and time constraints). At a minimum, the design should support fitting quadratic mixture models (6), and if possible, special-cubic models (7). If full-cubic blending behavior is thought to be possible by experts, and resources allow, then a mixture design to support fitting FCM models (8) should be developed. In addition to formulations on the boundary of the mixture space (e.g., vertices, edge centers, face centers), it is also desirable to include in mixture experiment designs some formulations on the interior of the mixture space. Interior formulations provide additional support for fitting special-cubic, partial cubic, or full-cubic mixture models.

In performing a mixture experiment, all possible factors such as environmental conditions, process factors, labor, equipment, and raw materials must be held constant if not being varied as part of the experiment. Otherwise, such factors should be controlled to the extent possible and values recorded during each experiment.

In summary, mixture experiment design, modeling, and optimization methods applied using inputs from subject-matter experts are powerful tools that can be used to efficiently develop new product formulations or modify existing formulations to meet customer requirements and/or reduce the cost of manufacturing the product.



ACKNOWLEDGMENTS

The first and third authors performed this work as unsponsored research. The second author got involved after the completion of the initial study (experimental design, modeling, and selecting Formulation 1) and was involved in developing the new models and Formulation 2. The portion of his work represented in this article was performed on his personal time and was not a part of any project for Battelle or Pacific Northwest National Laboratory (PNNL). The authors acknowledge Mehmet Kavak for selecting the components and their constraints and Alejandro Heredia-Langner (PNNL) for a technical peer review of the manuscript.

REFERENCES

1. Piepel G.F., Mixture experiments. Encycl. Stat. Qual. Reliabil., John Wiley & Sons, Ltd., Chichester, UK (2007) 1099-1112.

2. Piepel G.F. and Cornell J.A., Mixture experiment approaches: examples, discussion, and recommendations. J. Qual. Technol., 26 (1994) 177-196.

3. Cornell J.A., Experiments With Mixtures: Designs, Models, and The Analysis of Mixture Data. 3rd Edition, John Wiley & Sons, Inc., New York (2002).

4. Smith W.F., Experimental Design for Formulation, ASA-SIAM Series on Statistics and Applied Probability, American Statistical Association, Alexandria, VA and Society for Industrial and Applied Mathematics, Philadelphia (2005).

5. Stat-Ease, Design-Expert Version 8, Stat-Ease Inc., Minneapolis, MN (2010).

6. Minitab, Minitab Version 16, Minitab Inc., State College, PA (2010).

7. JMP, JMP 9, SAS Institute, Inc., Cary, NC (2011).

8. Piepel G.F., Szychowski J.M., and Loeppky J.L., Augmenting Scheffé linear mixture models with squared and/or crossproduct terms. J. Qual. Technol., 34 (2002) 297-314.

9. Draper N.R. and Smith H., Applied Regression Analysis, Third Edition, John Wiley & Sons, New York (1998).

10. Piepel G.F., 50 years of mixture experiment research: 1955-2004. Chapter 12 in Response Surface Methodology and Related Topics, A. I. Khuri, Ed., World Scientific Press, Singapore (2006) 283-327.

11. Borosy A.P., Quantitative composition–property modelling of rubber mixtures by utilising artificial neural networks. Chemomet. Intell. Lab. Systems, 47 (1999) 227-238.

12. Piepel G.F. and Landmesser S.M., Mixture experiment alternatives to the slack variable approach. Qual. Engineering, 21, (2009) 262-276.



13. Anderson M.J. and Whitcomb P.J., Mixture DOE uncovers formulations quicker. Rubber Plastics News, (October 21, 2002) 16-18.

14. Myers R.H., Montgomery D.C., and Anderson-Cook C., Response Surface Methodology: Process and Product Optimization Using Designed Experiments, 3rd Edition, John Wiley & Sons, New York (2009).

15. JMP, JMP 7, SAS Institute, Inc., Cary, NC (2007).

16. Piepel G.F. and Cornell J.A., A catalog of mixture experiment examples, BN-SA-3298, Rev. 16.0, Battelle−Pacific Northwest Division, Richland, WA (2008).

17. Piepel G.F., Anderson C.M., and Redgate P.E., Response surface designs for irregularly-shaped regions (Parts 1, 2, and 3). 1993 Proc. Section Physical Engin. Sciences, American Statistical Association, Alexandria, VA (1993) 205-227.



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