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Quality Technology & Quantitative Management Vol. 8, No. 2, pp. 183-209, 2011

QQTTQQMM© ICAQM 2011

Designed Experiments in Nanotechnology: Reviewing the Statistical Analyses of Five Studies

Johannes Ledolter

Department of Management Sciences, University of Iowa, Iowa City, IA, USA (Received January2010, accepted June2010)

______________________________________________________________________

Abstract: Lu et al. [8] list several papers that utilize design of experiments techniques in the improvement of nanotechnology processes. In this paper we look at five publications in detail, supplement the data analyses that are reported in these studies, and comment on the appropriateness of the published analyses.

Keywords: Contour plots, design of experiments, nanotechnology, regression analysis, response surface methods. ______________________________________________________________________

1. Introduction

ano-particles, which are advanced materials usually in the 1-100 nm grain size range, have become increasingly important to the manufacture of a wide range of components.

The techniques for manufacturing nano-particles can be classified into top-down methods (which break larger-sized particles into smaller nano-sized ones) and bottom-up methods which assemble nano-particles from even smaller building blocks).

Statistical methods have been instrumental to the rapid advances in nanotechnology, and collaborative work between these disciplines has received considerable attention lately. Two recent comprehensive review papers describe instances where statistical methods have contributed to advances in nanotechnology. Jeng et al. [4] describe more than eighty reliability applications in nanotechnology and Lu et al. [8] list more than seventy nanotechnology

studies that utilize statistical design of experiments, employ statistical process/quality control methods, and apply statistical modeling techniques. More recent applications can be found by searching for such studies on Google Scholar.

Lu et al. [8] in their review of statistical methods for quality improvement and control as

currently practiced in nanotechnology, list without detailed discussion several papers that apply design of experiments techniques to improve nanotechnology processes. In this paper we look at five publications in detail, supplement the data analyses that are reported in these papers, comment on the appropriateness of the published analyses, correct several errors and recommend alternative approaches for the collection and analysis of the data. While there are no major changes in the conclusions, our comments provide additional insights that should advance the analyses of subsequent studies. Our paper also demonstrates the type of designs that are found most useful in nanotechnology applications.

The optimization of multi-factor reactions through well-designed experiments is an

important area. Many studies involve the optimization of one or more response variables, and second-order response surface designs (such as central composite, Box-Behnken, and Taguchi orthogonal array designs) for finding the optimum conditions of the input variables

N

184 Ledolter are routinely used. Often the interest is on how several responses vary as functions of input variables, and a joint response surface analysis with several response variables is needed. Montgomery [9] discusses how desirability functions can convert a multi-dimensional optimization problem into a scalar one, and this approach has been implemented in commonly-used statistical software (Minitab’s Response Optimizer).

Alternatively, as is discussed in this paper, one can use constrained nonlinear optimization methods to find a set of operating conditions that optimize all responses or at least keeps them in desired ranges. Nonlinear optimization methods, readily available in Excel SOLVER, can be used to find the solutions of quadratic optimization problems with quadratic constraints that are arise from fitted second-order models. Ill-conditioned situations

where non-linear optimization methods are known to fail are usually not a problem.

Of course, the results of the optimization analysis should always be confirmed with contour plots. A straightforward approach to optimizing several responses in situations where

there are only few process factors is to overlay the contour plots for the studied response

variables. Such plots help investigators and their clients gain a better understanding of the complex relationships between the factors and their responses.

Statistical software to carry out such analyses is readily available. Minitab and JMP, two very flexible packages that can solve most problems, can be recommended highly. Both

packages include excellent contour graphing features for an effective graphical visualization of the results.

All five studies assume that the data originate from completely randomized experiments when calculating the standard errors of estimated effects. Without being involved directly in

these experiments, it is difficult to ascertain whether some of these studies had been blocked, whether some of them were carried out as split-plot designs, or whether the randomization was affected by the presence of “hard-to-change factors.” It is well known that restrictions on the randomization affect the standard errors of estimated effects that are calculated under the assumption of complete randomization (Jones and Nachtsheim [5], Ledolter [6]). Without knowing in detail how the experiments have been conducted, it is difficult to know whether the usual standard errors apply.

Paper 1: “Parameters Optimization of a Nanoparticle Wet Milling Process Using the Taguchi Method, Response Surface Method and Genetic Algorithm” by Hou et al. [3]

This paper discusses a wet-type milling process for the manufacture of titanium oxide nano-particles. The agitator and the grinding ball media are central components of this

process. The stirring and colliding motions in the milling machine generates collisions

between the milling balls and the particles that break the materials into their desired

nanometer size. Several different types of designs for such milling machines – with

proprietary agitators, and different circulation and cooling systems – have been developed. The optimal milling parameters are not known a-priori, and need to be determined through experimentation.

This paper studies five variables: milling time; flow velocity of the circulation system; rotation velocity of the agitator shaft; the solute-to-solvent weight ratio; and the filling ratio of the grinding media.

The authors conduct a 27-run L27 orthogonal array design for five factors with three factor levels. From each experimental run, a random sample of the colloidal solution is taken and measurements on grain size are repeated five times, leading to the means and variances

Designed Experiments in Nanotechnology: Reviewing the Statistical Analyses of Five Studies 185

given in Table 1. The objective is to keep both the mean and the variance of grain size small.

Table 1. Factors, with their levels and units, and the resulting observations [the means for runs 6 and 9 were corrected for errors in the original reference].

The arrangement in Table 1 represents an orthogonal 5 23III design. The 27 runs constitute a full factorial in the three factors A, C, and E. The remaining factors are selected as B CE and

2D CE , using modulus 3 arithmetic; for example, see Wu and Hamada

[12].

Levels Factors

1 2 3

A Milling Time 2 hours 5 hours 8 hours

B Flow velocity of circulation system 1 liter/min 2 liter/min 3 liter/min

C Rotation velocity of agitator shaft 1200 rpm 1800 rpm 2400 rpm

D Solute-to-solvent weight ratio 1 wt% 4 wt% 7 wt%

E Filling ratio of grinding media 5% 15% 25%

Run A

Milling Time

B Flow

Velocity

C Rotation Velocity

D Weight Ratio

E Filling Ratio

1Y

Grain Size (Mean)

2Y

Grain Size (Variance)

1 1 1 1 1 1 232.7 8.5 2 1 2 1 2 2 182.5 21.1 3 1 3 1 3 3 154.0 5.7 4 1 2 2 3 1 181.0 0.8 5 1 3 2 1 2 159.2 1.2 6 1 1 2 2 3 171.6 6.3 7 1 3 3 2 1 169.1 5.1 8 1 1 3 3 2 171.8 5.9 9 1 2 3 1 3 159.7 4.3 10 2 1 1 1 1 217.9 15.9 11 2 2 1 2 2 174.2 16.6 12 2 3 1 3 3 133.0 1.8 13 2 2 2 3 1 164.5 6.1 14 2 3 2 1 2 147.6 5.7 15 2 1 2 2 3 158.8 5.2 16 2 3 3 2 1 156.7 4.2 17 2 1 3 3 2 169.1 6.5 18 2 2 3 1 3 142.6 3.8 19 3 1 1 1 1 213.2 8.3 20 3 2 1 2 2 174.4 10.6 21 3 3 1 3 3 130.3 2.6 22 3 2 2 3 1 151.0 2.9 23 3 3 2 1 2 138.7 5.7 24 3 1 2 2 3 152.3 4.7 25 3 3 3 2 1 148.1 2.5 26 3 1 3 3 2 165.7 5.6 27 3 2 3 1 3 138.6 3.5

184 Ledolter are routinely used. Often the interest is on how several responses vary as functions of input variables, and a joint response surface analysis with several response variables is needed. Montgomery [9] discusses how desirability functions can convert a multi-dimensional optimization problem into a scalar one, and this approach has been implemented in commonly-used statistical software (Minitab’s Response Optimizer).

Alternatively, as is discussed in this paper, one can use constrained nonlinear optimization methods to find a set of operating conditions that optimize all responses or at least keeps them in desired ranges. Nonlinear optimization methods, readily available in Excel SOLVER, can be used to find the solutions of quadratic optimization problems with quadratic constraints that are arise from fitted second-order models. Ill-conditioned situations

where non-linear optimization methods are known to fail are usually not a problem.

Of course, the results of the optimization analysis should always be confirmed with contour plots. A straightforward approach to optimizing several responses in situations where

there are only few process factors is to overlay the contour plots for the studied response

variables. Such plots help investigators and their clients gain a better understanding of the complex relationships between the factors and their responses.

Statistical software to carry out such analyses is readily available. Minitab and JMP, two very flexible packages that can solve most problems, can be recommended highly. Both

packages include excellent contour graphing features for an effective graphical visualization of the results.

All five studies assume that the data originate from completely randomized experiments when calculating the standard errors of estimated effects. Without being involved directly in

these experiments, it is difficult to ascertain whether some of these studies had been blocked, whether some of them were carried out as split-plot designs, or whether the randomization was affected by the presence of “hard-to-change factors.” It is well known that restrictions on the randomization affect the standard errors of estimated effects that are calculated under the assumption of complete randomization (Jones and Nachtsheim [5], Ledolter [6]). Without knowing in detail how the experiments have been conducted, it is difficult to know whether the usual standard errors apply.

Paper 1: “Parameters Optimization of a Nanoparticle Wet Milling Process Using the Taguchi Method, Response Surface Method and Genetic Algorithm” by Hou et al. [3]

This paper discusses a wet-type milling process for the manufacture of titanium oxide nano-particles. The agitator and the grinding ball media are central components of this

process. The stirring and colliding motions in the milling machine generates collisions

between the milling balls and the particles that break the materials into their desired

nanometer size. Several different types of designs for such milling machines – with

proprietary agitators, and different circulation and cooling systems – have been developed. The optimal milling parameters are not known a-priori, and need to be determined through experimentation.

This paper studies five variables: milling time; flow velocity of the circulation system; rotation velocity of the agitator shaft; the solute-to-solvent weight ratio; and the filling ratio of the grinding media.

The authors conduct a 27-run L27 orthogonal array design for five factors with three factor levels. From each experimental run, a random sample of the colloidal solution is taken and measurements on grain size are repeated five times, leading to the means and variances


given in Table 1. The objective is to keep both the mean and the variance of grain size small.

Table 1. Factors, with their levels and units, and the resulting observations [the means for runs 6 and 9 were corrected for errors in the original reference].

The arrangement in Table 1 represents an orthogonal 5 23III design. The 27 runs constitute a full factorial in the three factors A, C, and E. The remaining factors are selected as B CE and

2D CE , using modulus 3 arithmetic; for example, see Wu and Hamada

[12].

Levels Factors

1 2 3

A Milling Time 2 hours 5 hours 8 hours

B Flow velocity of circulation system 1 liter/min 2 liter/min 3 liter/min

C Rotation velocity of agitator shaft 1200 rpm 1800 rpm 2400 rpm

D Solute-to-solvent weight ratio 1 wt% 4 wt% 7 wt%

E Filling ratio of grinding media 5% 15% 25%

Run A

Milling Time

B Flow

Velocity

C Rotation Velocity

D Weight Ratio

E Filling Ratio

1Y

Grain Size (Mean)

2Y

Grain Size (Variance)

1 1 1 1 1 1 232.7 8.5 2 1 2 1 2 2 182.5 21.1 3 1 3 1 3 3 154.0 5.7 4 1 2 2 3 1 181.0 0.8 5 1 3 2 1 2 159.2 1.2 6 1 1 2 2 3 171.6 6.3 7 1 3 3 2 1 169.1 5.1 8 1 1 3 3 2 171.8 5.9 9 1 2 3 1 3 159.7 4.3 10 2 1 1 1 1 217.9 15.9 11 2 2 1 2 2 174.2 16.6 12 2 3 1 3 3 133.0 1.8 13 2 2 2 3 1 164.5 6.1 14 2 3 2 1 2 147.6 5.7 15 2 1 2 2 3 158.8 5.2 16 2 3 3 2 1 156.7 4.2 17 2 1 3 3 2 169.1 6.5 18 2 2 3 1 3 142.6 3.8 19 3 1 1 1 1 213.2 8.3 20 3 2 1 2 2 174.4 10.6 21 3 3 1 3 3 130.3 2.6 22 3 2 2 3 1 151.0 2.9 23 3 3 2 1 2 138.7 5.7 24 3 1 2 2 3 152.3 4.7 25 3 3 3 2 1 148.1 2.5 26 3 1 3 3 2 165.7 5.6 27 3 2 3 1 3 138.6 3.5

186 Ledolter Figure 1 displays the main effects of the five factors, for both 1Y (mean) and 2Y

(standard deviation). Small means and small standard deviations are achieved if the process factors are set at their larger settings.

852

180

170

160

150

321 240018001200

741

180

170

160

150

25155

A(hours)

Mea

n

B(l/min) C(rpm)

D(wt%) E(%)

Main Effects Plot for the Mean of Grain SizeData Means

852

3.00

2.75

2.50

2.25

2.00

321 240018001200

741

3.00

2.75

2.50

2.25

2.00

25155

A(hours)

Mea

n

B(l/min) C(rpm)

D(wt%) E(%)

Main Effects Plot for the Standard Deviation of Grain SizeData Means

Figure 1. Main effects plots for the mean and standard deviation of grain size.

Factors A through E are continuous, and we model their effects through linear and quadratic components, and pair-wise (linear) interactions. Because of the fractional factorial nature of the experiment, we cannot estimate the six interactions BC, BD, BE, CD, CE and DE as they are confounded with the linear and quadratic components of the five factors. We use the results of the 27 experiments to estimate the subsequent second-order model with its 15 parameters. That is, the model


0 A A B B C C D D E E

2 2 2 2 2AA A BB B CC C DD D EE E

AB A B AC A C AD A D AE A E

Y

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) noise,

x x x x x

x x x x x

x x x x x x x x

where A B E, ,...,x x x are the explanatory variables that contain the levels of factors A through E (1 to 3 in coded form, or their respective original units such as 2, 5 and 8 for factor A, and 1200, 1800 and 2400 for factor C).

B(l/min)*A(hours)

7.04.52.0

3

2

1

C(rpm)*A(hours)

7.04.52.0

2200

1700

1200

D(wt%)*A(hours)

7.04.52.0

6

4

2

E(%)*A(hours)

7.04.52.0

24

16

8

C(rpm)*B(l/min)

321

2200

1700

1200

D(wt%)*B(l/min)

321

6

4

2

E(%)*B(l/min)

321

24

16

8

D(wt%)*C(rpm)

220017001200

6

4

2

E(%)*C(rpm)

220017001200

24

16

8

E(%)*D(wt%)

642

24

16

8

A(hours) 8B(l/min) 3C(rpm) 2400D(wt%) 7E(%) 25

Hold Values

>––––––< 110

110 120120 130130 140140 150150 160160 170

170

GrainSize(Mean)Corr

Contour Plots of the Mean of Grain Size

B(l/min)*A(hours)

7.04.52.0

3

2

1

C(rpm)*A(hours)

7.04.52.0

2200

1700

1200

D(wt%)*A(hours)

7.04.52.0

6

4

2

E(%)*A(hours)

7.04.52.0

24

16

8

C(rpm)*B(l/min)

321

2200

1700

1200

D(wt%)*B(l/min)

321

6

4

2

E(%)*B(l/min)

321

24

16

8

D(wt%)*C(rpm)

220017001200

6

4

2

E(%)*C(rpm)

220017001200

24

16

8

E(%)*D(wt%)

642

24

16

8


Hold Values

>–––––––< 0.60

0.60 0.850.85 1.101.10 1.351.35 1.601.60 1.851.85 2.102.10 2.35

2.35

GrainSize(StdDev

Contour Plots of Standard Deviation of the Grain Size

Figure 2. Pair-wise contour plots for the mean and standard deviation of grain size.

186 Ledolter Figure 1 displays the main effects of the five factors, for both 1Y (mean) and 2Y

(standard deviation). Small means and small standard deviations are achieved if the process factors are set at their larger settings.

852

180

170

160

150

321 240018001200

741

180

170

160

150

25155

A(hours)

Mea

n

B(l/min) C(rpm)

D(wt%) E(%)

Main Effects Plot for the Mean of Grain SizeData Means

852

3.00

2.75

2.50

2.25

2.00

321 240018001200

741

3.00

2.75

2.50

2.25

2.00

25155

A(hours)

Mea

n

B(l/min) C(rpm)

D(wt%) E(%)

Main Effects Plot for the Standard Deviation of Grain SizeData Means

Figure 1. Main effects plots for the mean and standard deviation of grain size.

Factors A through E are continuous, and we model their effects through linear and quadratic components, and pair-wise (linear) interactions. Because of the fractional factorial nature of the experiment, we cannot estimate the six interactions BC, BD, BE, CD, CE and DE as they are confounded with the linear and quadratic components of the five factors. We use the results of the 27 experiments to estimate the subsequent second-order model with its 15 parameters. That is, the model


0 A A B B C C D D E E

2 2 2 2 2AA A BB B CC C DD D EE E

AB A B AC A C AD A D AE A E

Y

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) noise,

x x x x x

x x x x x

x x x x x x x x

where A B E, ,...,x x x are the explanatory variables that contain the levels of factors A through E (1 to 3 in coded form, or their respective original units such as 2, 5 and 8 for factor A, and 1200, 1800 and 2400 for factor C).

B(l/min)*A(hours)

7.04.52.0

3

2

1

C(rpm)*A(hours)

7.04.52.0

2200

1700

1200

D(wt%)*A(hours)

7.04.52.0

6

4

2

E(%)*A(hours)

7.04.52.0

24

16

8

C(rpm)*B(l/min)

321

2200

1700

1200

D(wt%)*B(l/min)

321

6

4

2

E(%)*B(l/min)

321

24

16

8

D(wt%)*C(rpm)

220017001200

6

4

2

E(%)*C(rpm)

220017001200

24

16

8

E(%)*D(wt%)

642

24

16

8


Hold Values

>––––––< 110

110 120120 130130 140140 150150 160160 170

170

GrainSize(Mean)Corr

Contour Plots of the Mean of Grain Size

B(l/min)*A(hours)

7.04.52.0

3

2

1

C(rpm)*A(hours)

7.04.52.0

2200

1700

1200

D(wt%)*A(hours)

7.04.52.0

6

4

2

E(%)*A(hours)

7.04.52.0

24

16

8

C(rpm)*B(l/min)

321

2200

1700

1200

D(wt%)*B(l/min)

321

6

4

2

E(%)*B(l/min)

321

24

16

8

D(wt%)*C(rpm)

220017001200

6

4

2

E(%)*C(rpm)

220017001200

24

16

8

E(%)*D(wt%)

642

24

16

8


Hold Values

>–––––––< 0.60

0.60 0.850.85 1.101.10 1.351.35 1.601.60 1.851.85 2.102.10 2.35

2.35

GrainSize(StdDev

Contour Plots of Standard Deviation of the Grain Size

Figure 2. Pair-wise contour plots for the mean and standard deviation of grain size.

188 Ledolter Regression fitting results for 1Y (mean) and 2Y (standard deviation) show that

interaction terms are insignificant, leading us to fit models with linear and quadratic components only. These models (with 11 parameters for the 27 observations) lead to excellent representations, with R-squares of 98.3 (mean) and 77.0 (standard deviation), respectively. We decided not to simplify the model even further even though several insignificant components could have been omitted in a step-wise fashion. Model diagnostics fail to reveal model inadequacies. The residuals are approximately normal, and there are no outliers. Our initial analysis for the mean using the data in Table 2 of the reference did show two unusual residuals; a subsequent inquiry sent to the authors of the paper revealed that the means for runs 6 and 9 had been recorded incorrectly, which led to the corrections in Table 1.

Based on the models with linear and quadratic terms we plot, using the Minitab software, implied contours for all pairs of factors while setting the remaining three factors at their highest level (a strategy that is suggested by the main-effects plots in Figure 1 which show that the response is always lowest for large levels of each factor). The contour plots for the mean size in Figure 2 indicate that it is best to increase each factor to its largest value. Such

a strategy also tends to reduce the variability as shown by the contour plots for the standard

deviation. The recommendation at this stage is to conduct several confirmative runs to check

whether a strategy that sets all factors at their high levels leads to the desired results (that is, low means and low standard deviations). Conducting a second experiment around the high settings would further tell whether the results can be improved even further.

One always hopes for effects sparcity when analyzing experiments with many factors. The response surface can be visualized in lower-dimensional space through a few

well-selected contour plots if the parameter estimates that are associated with several studied factors turn out insignificant. Contour plots can then be overlaid to find a region that is acceptable for all responses. In this particular problem it was not possible to reduce the number of dimensions (factors) and we had to construct contour plots for all pairs of factors. One needs to decide how to set the levels of those factors that are left out in pair-wise contour plots. The main-effects plots in Figure 1 indicated that the omitted factors should be set at their highest values.

Discussion

This study looks for operating conditions that lead to small values, for both level and variability. Our approach is to analyze the two responses individually, and inspect the contour plots that result from estimated models that contain linear and quadratic components.

A strategy of combining the two outcomes into signal-to-noise ratios was pursued by Hou et al. [3] who consider the sum 1 2Y Y and the ratio 2 1Y / Y . However, these measures make little sense as they represent a sum and a ratio of quantities with different units. If one wanted to combine the two outcomes, it would have been more appropriate to analyze

1 2Y Y (which represents the 100(5/6)th percentile, assuming a normal distribution) and 2 1Y / Y (the coefficient of variation).

Working with these two somewhat questionable measures Hou et al. [3] use a genetic

algorithm to optimize the signal-to-noise ratios (also referred to as fitness functions) with respect to the design factors. A genetic algorithm is a general-purpose optimization method with a computational process that mimics the theory of biological evolution. It’s a powerful optimization approach that can be used to solve difficult optimization problems with high complexity and undesirable structure, and it is often used when other nonlinear optimization tools fail. We feel that the complexity of this method is not justified here as much simpler


and more intuitive procedures suffice. For a given signal-to-noise ratio, one can find the explicit algebraic solution for the optimum of a second-order model using standard tools of calculus, or as we have done, can display the nature of the response surface through contour plots. The contour plots indicate that it is best (in terms of level and variability) to increase the level of each factor to the largest value that has been studied. The conclusion by Hou et al. [3] is not that different as their recommend values are all close to the highest settings.

Paper 2: “Optimization of a Self-Nanoemulsified Tablet Dosage Form of Ubiquinone Using Response Surface Methodology: Effect of Formulation Ingredients” by Nazzal et al. [10].

Drug companies are interested in delivering liquid medications in solid form, but their efforts have been hampered by poor water solubility of the products. Several self-emulsifying drug delivery systems (SEDDS) have been developed to solve this problem. These systems generate isotropic mixtures of drug, oil, surfactant, and co-surfactant that form fine

oil-in-water emulsions when introduced into an aqueous medium under gentle agitation. The authors of this paper propose a new method for producing emulsion droplets in nanometer size range. They investigate the release rate of ubiquinome from its adsorbing solid compact and investigate how the cumulative percentage of ubiquinome released after 45 minutes depends on three critical factors in their adopted SEDDS method: the amounts of copolyvidone, maltodextrin, and microcrystalline cellulose. In addition, the resulting formulation is supposed to satisfy several side conditions on variables such as tensile strength, flowability, and friability.

A 15-run Box-Behnken design [1] with 3 center points was carried out, and the results are shown in Table 2. The first 12 runs of a Box-Behnken design in three factors originate

from a balanced incomplete block design with each of the three blocks representing a 22

factorial design for any two of the three factors. The addition of the 3cn center points

leads to the 12 15cn runs in Table 2. Visualizing the process space in three dimensions, one can think of the runs of the Box-Behnken design as the points where a sphere intersects each face of a cube at the midpoint of its edges; see Figure 3(a). Each factor is studied at

three levels and the design is rotatable, which means that the variance of the predicted response at design point 1 2 3( , , )x x x depends only on the distance of the design point from the origin.

The Box-Behnken design allows the regression estimation of the second-order model

2 2 20 1 1 2 2 3 3 11 1 22 2 33 3 12 1 2 13 1 3 23 2 3Y noise.x x x x x x x x x x x x

From the regression estimates we obtain the optimum conditions and determine the settings of the factors that maximize (or minimize) the response. Contour plots illustrate the fitted surface graphically.

An alternative design and close competitor to the Box-Behnken design is the central composite design. For 3k factors, this design consists of a

32 design, six “star” points [given by runs along the first factor axis, ( , 0, 0) and ( , 0, 0); along the second factor axis, (0, , 0) and (0, , 0); along the third factor axis, (0, 0, ) and (0, 0, )], and

cn center points [(0, 0, 0)], for a total of 8 6 14c cn n runs. The design is rotatable for 3/42 1.682; 1w w leads to the face-centered central composite design. Figure 3(b)

shows the design points of the central composite design.

188 Ledolter Regression fitting results for 1Y (mean) and 2Y (standard deviation) show that

interaction terms are insignificant, leading us to fit models with linear and quadratic components only. These models (with 11 parameters for the 27 observations) lead to excellent representations, with R-squares of 98.3 (mean) and 77.0 (standard deviation), respectively. We decided not to simplify the model even further even though several insignificant components could have been omitted in a step-wise fashion. Model diagnostics fail to reveal model inadequacies. The residuals are approximately normal, and there are no outliers. Our initial analysis for the mean using the data in Table 2 of the reference did show two unusual residuals; a subsequent inquiry sent to the authors of the paper revealed that the means for runs 6 and 9 had been recorded incorrectly, which led to the corrections in Table 1.

Based on the models with linear and quadratic terms we plot, using the Minitab software, implied contours for all pairs of factors while setting the remaining three factors at their highest level (a strategy that is suggested by the main-effects plots in Figure 1 which show that the response is always lowest for large levels of each factor). The contour plots for the mean size in Figure 2 indicate that it is best to increase each factor to its largest value. Such

a strategy also tends to reduce the variability as shown by the contour plots for the standard

deviation. The recommendation at this stage is to conduct several confirmative runs to check

whether a strategy that sets all factors at their high levels leads to the desired results (that is, low means and low standard deviations). Conducting a second experiment around the high settings would further tell whether the results can be improved even further.

One always hopes for effects sparcity when analyzing experiments with many factors. The response surface can be visualized in lower-dimensional space through a few

well-selected contour plots if the parameter estimates that are associated with several studied factors turn out insignificant. Contour plots can then be overlaid to find a region that is acceptable for all responses. In this particular problem it was not possible to reduce the number of dimensions (factors) and we had to construct contour plots for all pairs of factors. One needs to decide how to set the levels of those factors that are left out in pair-wise contour plots. The main-effects plots in Figure 1 indicated that the omitted factors should be set at their highest values.

Discussion

This study looks for operating conditions that lead to small values, for both level and variability. Our approach is to analyze the two responses individually, and inspect the contour plots that result from estimated models that contain linear and quadratic components.

A strategy of combining the two outcomes into signal-to-noise ratios was pursued by Hou et al. [3] who consider the sum 1 2Y Y and the ratio 2 1Y / Y . However, these measures make little sense as they represent a sum and a ratio of quantities with different units. If one wanted to combine the two outcomes, it would have been more appropriate to analyze

1 2Y Y (which represents the 100(5/6)th percentile, assuming a normal distribution) and 2 1Y / Y (the coefficient of variation).

Working with these two somewhat questionable measures Hou et al. [3] use a genetic

algorithm to optimize the signal-to-noise ratios (also referred to as fitness functions) with respect to the design factors. A genetic algorithm is a general-purpose optimization method with a computational process that mimics the theory of biological evolution. It’s a powerful optimization approach that can be used to solve difficult optimization problems with high complexity and undesirable structure, and it is often used when other nonlinear optimization tools fail. We feel that the complexity of this method is not justified here as much simpler


and more intuitive procedures suffice. For a given signal-to-noise ratio, one can find the explicit algebraic solution for the optimum of a second-order model using standard tools of calculus, or as we have done, can display the nature of the response surface through contour plots. The contour plots indicate that it is best (in terms of level and variability) to increase the level of each factor to the largest value that has been studied. The conclusion by Hou et al. [3] is not that different as their recommend values are all close to the highest settings.

Paper 2: “Optimization of a Self-Nanoemulsified Tablet Dosage Form of Ubiquinone Using Response Surface Methodology: Effect of Formulation Ingredients” by Nazzal et al. [10].

Drug companies are interested in delivering liquid medications in solid form, but their efforts have been hampered by poor water solubility of the products. Several self-emulsifying drug delivery systems (SEDDS) have been developed to solve this problem. These systems generate isotropic mixtures of drug, oil, surfactant, and co-surfactant that form fine

oil-in-water emulsions when introduced into an aqueous medium under gentle agitation. The authors of this paper propose a new method for producing emulsion droplets in nanometer size range. They investigate the release rate of ubiquinome from its adsorbing solid compact and investigate how the cumulative percentage of ubiquinome released after 45 minutes depends on three critical factors in their adopted SEDDS method: the amounts of copolyvidone, maltodextrin, and microcrystalline cellulose. In addition, the resulting formulation is supposed to satisfy several side conditions on variables such as tensile strength, flowability, and friability.

A 15-run Box-Behnken design [1] with 3 center points was carried out, and the results are shown in Table 2. The first 12 runs of a Box-Behnken design in three factors originate

from a balanced incomplete block design with each of the three blocks representing a 22

factorial design for any two of the three factors. The addition of the 3cn center points

leads to the 12 15cn runs in Table 2. Visualizing the process space in three dimensions, one can think of the runs of the Box-Behnken design as the points where a sphere intersects each face of a cube at the midpoint of its edges; see Figure 3(a). Each factor is studied at

three levels and the design is rotatable, which means that the variance of the predicted response at design point 1 2 3( , , )x x x depends only on the distance of the design point from the origin.

The Box-Behnken design allows the regression estimation of the second-order model

2 2 20 1 1 2 2 3 3 11 1 22 2 33 3 12 1 2 13 1 3 23 2 3Y noise.x x x x x x x x x x x x

From the regression estimates we obtain the optimum conditions and determine the settings of the factors that maximize (or minimize) the response. Contour plots illustrate the fitted surface graphically.

An alternative design and close competitor to the Box-Behnken design is the central composite design. For 3k factors, this design consists of a

32 design, six “star” points [given by runs along the first factor axis, ( , 0, 0) and ( , 0, 0); along the second factor axis, (0, , 0) and (0, , 0); along the third factor axis, (0, 0, ) and (0, 0, )], and

cn center points [(0, 0, 0)], for a total of 8 6 14c cn n runs. The design is rotatable for 3/42 1.682; 1w w leads to the face-centered central composite design. Figure 3(b)

shows the design points of the central composite design.

190 Ledolter Table 2. Factors, with levels and units, and results of the Box-Behnken experiment.

Levels Factors

1 (low) 0 (middle) 1 (high)

1X Amount of copolyvidone added (mg) 30 165 300

2X Amount of maltodextrine added (mg) 300 500 700

3X Amount of microcrystalline cellulose added (mg) 100 250 400

Levels Response

Low High Goal

1Y Weight (mg) 695 1000

2Y Flowability index (Carr) 50 100

3Y Tensile strength (MPa) 0.05 0.32

4Y Friability (%) 0 1

5Y Disintegration time (min) 0 30

6Y Cumulative % ubiquinome released

after 45 minutes Maximize

Order 1X

(coded) 2X

(coded) 3X

(coded) 1Y

Weight 2Y

Flow 3Y

Strength 4Y

Friab 5Y

DisTime 6Y

%Release 14 1 1 0 710 30.5 0.051 0.07 21.2 61.5 9 1 1 0 980 71.0 0.169 0.24 18.2 46.1 12 1 1 0 1110 45.5 0.040 0.10 28.8 49.9 4 1 1 0 1380 66.0 0.127 1.80 27.3 48.5 6 1 0 1 760 43.5 0.020 0.08 16.3 91.1 3 1 0 1 1030 67.0 0.177 0.28 22.8 70.9 1 1 0 1 1060 69.0 0.052 0.10 26.6 61.5 5 1 0 1 1330 67.0 0.125 1.60 6.1 49.2 2 0 1 1 695 76.5 0.328 0.09 20.8 84.1 13 0 1 1 1095 65.0 0.152 0.10 23.8 70.4 7 0 1 1 995 69.5 0.205 0.11 3.9 49.0 15 0 1 1 1395 65.5 0.148 0.50 12.3 54.6 8 0 0 0 1045 65.0 0.156 0.19 19.9 63.9 10 0 0 0 1045 61.5 0.140 0.21 18.1 63.5 11 0 0 0 1045 63.0 0.162 0.14 22.3 64.9

(a) Box-Behnken design. (b) central composite design.

Figure 3. Geometric representation of the (a) Box-Behnken and (b) central composite designs.


For three factors, the Box-Behnken design offers some advantage as it requires fewer runs. For 4 or more factors, this advantage disappears; for 4 factors, both the Box-Behnken

and the central composite designs require 24 cn runs.

The fitted quadratic (second-order) regression model for the release rate of ubiquinome (response 6Y in Table 2) is given by

6 1 2 3

1 2 3

1 2 1 3 2 3

Y 64.1 6.1625X 2.1625X 12.775X

4.475X * *2 8.125X * *2 8.55X * *2

3.5X * X 1.975X * X 4.825X * X .

The model leads to an R-square of 98.7. Apart from perhaps omitting the insignificant interaction 1 3X * X , it is not possible to simplify this model any further.

The three contour plots of 6Y against pairs of factors 1(X and 2 1X , X and 3X , and

2X and 3X ) are quite informative. The third remaining factor in these contour plots is always set at its low value as main-effects plots show that the response can be increased by setting each factor at its lowest setting. The contour plots in Figure 4 show that it is best to

operate the process with 1X (amount of copolyvidone) at 30 mg, 2X (amount of

maltodextrine) at 300 mg, and 3X (amount of microcrystalline cellulose) at 100 mg.

The estimates in the fitted equation can be used to determine the optimum conditions

11ˆ ,

2optx B b with

11 12 13

12 22 23

13 23 33

( / 2) ( / 2)

( / 2) ( / 2)

( / 2) ( / 2)

B and

1

2

3

.b

The eigenvalues of the matrix B tell us about the nature of the optimum. The stationary point is a maximum if all eigenvalues are negative; it is a minimum if all eigenvalues are positive; and it is a saddle point if the eigenvalues have mixed signs. From our estimates,

0.512

ˆ 0.004 ,

0.807optx

with eigenvalues 9.03, 4.02, and 9.00. The optimum value represents a saddle point. This saddle point can be seen on the contour plot in Figure 5 of 6Y against 1X and 3X ,

setting 2X at 0 (or at 500 in uncoded units). The saddle point solution provides little useful information that is pertinent to solving the problem, which is to find regions where 6Y is

large.

Furthermore, in this particular problem we have more than one response variable to contend with. There are five other response variables 1(Y through 5Y ) that need to be kept in acceptable ranges. The relationships between them and the factors 1 2X , X , and 3X are also approximated with quadratic models, and the results are shown in Table 3.

190 Ledolter Table 2. Factors, with levels and units, and results of the Box-Behnken experiment.

Levels Factors

1 (low) 0 (middle) 1 (high)

1X Amount of copolyvidone added (mg) 30 165 300

2X Amount of maltodextrine added (mg) 300 500 700

3X Amount of microcrystalline cellulose added (mg) 100 250 400

Levels Response

Low High Goal

1Y Weight (mg) 695 1000

2Y Flowability index (Carr) 50 100

3Y Tensile strength (MPa) 0.05 0.32

4Y Friability (%) 0 1

5Y Disintegration time (min) 0 30

6Y Cumulative % ubiquinome released

after 45 minutes Maximize

Order 1X

(coded) 2X

(coded) 3X

(coded) 1Y

Weight 2Y

Flow 3Y

Strength 4Y

Friab 5Y

DisTime 6Y

%Release 14 1 1 0 710 30.5 0.051 0.07 21.2 61.5 9 1 1 0 980 71.0 0.169 0.24 18.2 46.1 12 1 1 0 1110 45.5 0.040 0.10 28.8 49.9 4 1 1 0 1380 66.0 0.127 1.80 27.3 48.5 6 1 0 1 760 43.5 0.020 0.08 16.3 91.1 3 1 0 1 1030 67.0 0.177 0.28 22.8 70.9 1 1 0 1 1060 69.0 0.052 0.10 26.6 61.5 5 1 0 1 1330 67.0 0.125 1.60 6.1 49.2 2 0 1 1 695 76.5 0.328 0.09 20.8 84.1 13 0 1 1 1095 65.0 0.152 0.10 23.8 70.4 7 0 1 1 995 69.5 0.205 0.11 3.9 49.0 15 0 1 1 1395 65.5 0.148 0.50 12.3 54.6 8 0 0 0 1045 65.0 0.156 0.19 19.9 63.9 10 0 0 0 1045 61.5 0.140 0.21 18.1 63.5 11 0 0 0 1045 63.0 0.162 0.14 22.3 64.9

(a) Box-Behnken design. (b) central composite design.

Figure 3. Geometric representation of the (a) Box-Behnken and (b) central composite designs.


For three factors, the Box-Behnken design offers some advantage as it requires fewer runs. For 4 or more factors, this advantage disappears; for 4 factors, both the Box-Behnken

and the central composite designs require 24 cn runs.

The fitted quadratic (second-order) regression model for the release rate of ubiquinome (response 6Y in Table 2) is given by

6 1 2 3

1 2 3

1 2 1 3 2 3

Y 64.1 6.1625X 2.1625X 12.775X

4.475X * *2 8.125X * *2 8.55X * *2

3.5X * X 1.975X * X 4.825X * X .

The model leads to an R-square of 98.7. Apart from perhaps omitting the insignificant interaction 1 3X * X , it is not possible to simplify this model any further.

The three contour plots of 6Y against pairs of factors 1(X and 2 1X , X and 3X , and

2X and 3X ) are quite informative. The third remaining factor in these contour plots is always set at its low value as main-effects plots show that the response can be increased by setting each factor at its lowest setting. The contour plots in Figure 4 show that it is best to

operate the process with 1X (amount of copolyvidone) at 30 mg, 2X (amount of

maltodextrine) at 300 mg, and 3X (amount of microcrystalline cellulose) at 100 mg.

The estimates in the fitted equation can be used to determine the optimum conditions

11ˆ ,

2optx B b with

11 12 13

12 22 23

13 23 33

( / 2) ( / 2)

( / 2) ( / 2)

( / 2) ( / 2)

B and

1

2

3

.b

The eigenvalues of the matrix B tell us about the nature of the optimum. The stationary point is a maximum if all eigenvalues are negative; it is a minimum if all eigenvalues are positive; and it is a saddle point if the eigenvalues have mixed signs. From our estimates,

0.512

ˆ 0.004 ,

0.807optx

with eigenvalues 9.03, 4.02, and 9.00. The optimum value represents a saddle point. This saddle point can be seen on the contour plot in Figure 5 of 6Y against 1X and 3X ,

setting 2X at 0 (or at 500 in uncoded units). The saddle point solution provides little useful information that is pertinent to solving the problem, which is to find regions where 6Y is

large.

Furthermore, in this particular problem we have more than one response variable to contend with. There are five other response variables 1(Y through 5Y ) that need to be kept in acceptable ranges. The relationships between them and the factors 1 2X , X , and 3X are also approximated with quadratic models, and the results are shown in Table 3.

192 Ledolter

Figure 4. Contour plots of 6Y Cumulative Percent Conversion versus 1X and 2 1X , X and 3X , and 2X and 3X . The remaining factor is always

held at its low value.


Figure 5. Contour plots of 6Y Cumulative Percent Conversion versus 1X and 3X , with the remaining factor 2X held at 500 mg.

Table 3. Regression results for the quadratic models: 1Y through 5Y .

1Y Weight 1 2 31045 135X 200X 150X R-square = 100.0 %

2Y Flowability 1 2 363.2 10.3X 0.69X 2.37X

1 2 3

1 2 1 3 2 3

8.71X * *2 1.21X * *2 7.17X * *2

5.00X * X 6.38X * X 1.87X * X

R-square = 80.4 %

3Y TStrength 1 2 30.153 0.0544X 0.0358X 0.0184X

1 2 3

1 2 1 3 2 3

0.0853X * *2 0.0294X * *2 0.0262X * *2

0.0077X * X 0.0210X * X 0.0297X * X

R-square =93.0 %

4Y Friability 1 2 30.180 0.446X 0.249X 0.220X

1 2 3

1 2 1 3 2 3

0.344X * *2 0.029X * *2 0.009X * *2

0.383X * X 0.325X * X 0.095X * X

R-square =93.2 %

5Y DisTime 1 2 320.1 2.31X 3.51X 4.35X

1 2 3

1 2 1 3 2 3

3.26X * *2 0.51X * *2 5.41X * *2

0.38X * X 6.75X * X 1.35X * X

R-square =88.4 %

Mathematical optimization is used to maximize the quadratic function for 6Y with respect to the decision variables (the factors) 1 2X , X and 3X . The quadratic optimization

6 1 2 3

1 2 3

1 2 1 3 2 3

Y 64.1 6.1625X 2.1625X 12.775X

4.475X * *2 8.125X * *2 8.55X * *2

3.5X * X 1.975X * X 4.825X * X MAX,

is carried out subject to several constraints. The constraints

11 X 1,

21 X 1,

31 X 1,

restrict extrapolations beyond the experimental region. The other constraints guarantee that

192 Ledolter

Figure 4. Contour plots of 6Y Cumulative Percent Conversion versus 1X and 2 1X , X and 3X , and 2X and 3X . The remaining factor is always

held at its low value.


Figure 5. Contour plots of 6Y Cumulative Percent Conversion versus 1X and 3X , with the remaining factor 2X held at 500 mg.

Table 3. Regression results for the quadratic models: 1Y through 5Y .

1Y Weight 1 2 31045 135X 200X 150X R-square = 100.0 %

2Y Flowability 1 2 363.2 10.3X 0.69X 2.37X

1 2 3

1 2 1 3 2 3

8.71X * *2 1.21X * *2 7.17X * *2

5.00X * X 6.38X * X 1.87X * X

R-square = 80.4 %

3Y TStrength 1 2 30.153 0.0544X 0.0358X 0.0184X

1 2 3

1 2 1 3 2 3

0.0853X * *2 0.0294X * *2 0.0262X * *2

0.0077X * X 0.0210X * X 0.0297X * X

R-square =93.0 %

4Y Friability 1 2 30.180 0.446X 0.249X 0.220X

1 2 3

1 2 1 3 2 3

0.344X * *2 0.029X * *2 0.009X * *2

0.383X * X 0.325X * X 0.095X * X

R-square =93.2 %

5Y DisTime 1 2 320.1 2.31X 3.51X 4.35X

1 2 3

1 2 1 3 2 3

3.26X * *2 0.51X * *2 5.41X * *2

0.38X * X 6.75X * X 1.35X * X

R-square =88.4 %

Mathematical optimization is used to maximize the quadratic function for 6Y with respect to the decision variables (the factors) 1 2X , X and 3X . The quadratic optimization

6 1 2 3

1 2 3

1 2 1 3 2 3

Y 64.1 6.1625X 2.1625X 12.775X

4.475X * *2 8.125X * *2 8.55X * *2

3.5X * X 1.975X * X 4.825X * X MAX,

is carried out subject to several constraints. The constraints

11 X 1,

21 X 1,

31 X 1,

restrict extrapolations beyond the experimental region. The other constraints guarantee that

194 Ledolter the remaining five response variables satisfy the bounds that were specified by the

investigators (and shown in Table 2).

1 2 3695 1045 135X 200X 150X 1000 1Y (Weight)

1 2 3 1 2 350 63.2 10.3X 0.69X 2.37X 8.71X * *2 1.21X * *2 7.17X * *2

1 2 1 3 2 35.00X * X 6.38X * X 1.87X *X 100 2Y (Flowability)

1 2 3 1 20.05 0.153 0.0544X 0.0358X 0.0184X 0.0853X * *2 0.0294X * *2

3 1 2 1 3 2 30.0262X * *2 0.0077X * X 0.0210X * X 0.0297X *X 0.32

3Y (TStrength)

1 2 3 1 2 30 0.180 0.446X 0.249X 0.220X 0.344X * *2 0.029X * *2 0.009X * *2

1 2 1 3 2 30.383X * X 0.325X * X 0.095X *X 1 4Y (Friability)

1 2 3 1 2 30 20.1 2.31X 3.51X 4.35X 3.26X * *2 0.51X * *2 5.41X * *2

1 2 1 3 2 30.38X * X 6.75X * X 1.35X *X 30 5Y (DisTime)

The optimization problem looks for a maximum of a quadratic function in three decision variables, with restrictions that are (also) described by quadratic functions. The solution of this “quadratic-quadratic” optimization problem is considerably more difficult than the linear programming problem which looks for an optimum of a linear function of the decision variables subject to linear constraints. For the linear problem, the solution is straightforward. The simplex method can be used, and software such as the EXCEL SOLVER provide the solution quite readily (see Evans [2]). Fortunately, the EXCEL SOLVER optimization routines also include procedures for non-linear models. If the dimensions are not too large (that is, not too many decision variables) and if the problem is not “ill-conditioned” (not many local optima), SOLVER’s iterative non-linear procedures provide prompt answers. For particularly difficult quadratic optimization problems with many decision variables and numerous quadratic constraints, one can consult one of the more advanced solution approaches that are discussed in the optimization literature.

Selecting reasonable starting values for the iterative optimization procedure helps speed up the convergence and usually avoids getting trapped in local optima or saddle points. The contour plots in Figure 4 show that large values of 6Y occur when the three design factors are near their low levels ( 1). With these starting values the nonlinear optimization encountered no difficulties, and found the optimum 6,Y 91.75opt at

1, 2, 3,(X 0.74, X 0.50, X 1.00);opt opt opt see Table 4. In uncoded units, this corresponds to Copolyvidone = 65 mg; Maltodextrin = 400 mg and Microcrystalline Cellulose = 100 mg. The overlaid contour plots in Figure 6 confirm the results of the optimization.

Table 4. Final solution of the optimization (EXCEL output).

Required Bounds Objective Response Variable

Calculated Response Low High

1Y 695 695 1000

2Y 50 50 100

3Y 0.132258 0.05 0.32

4Y 0.121275 0 1

5Y 16.724271 0 30

6Y 91.752177 Max


X1=Copolyvidone(mg)

X2=

Mal

tode

xtri

n(m

g)

30025020015010050

700

600

500

400

300 X3=Microcrystalline(mg) 100Hold Values

6951000

Y1=Weight(mg)Min

5060

Y2=F lowability Max

0.10.15

Y3=TensileStrength(MPa)Max

00.15

Y4=F riability (%)Min

16.5

20

Y5=DisintegrationTime(min)Min

8591.7

Y6=%C oQ 10(%)Max

Contour Plot of Six Responses

Figure 6. Overlaid contour plots from the full quadratic model, with the white area showing a region that satisfies the desired requirements for all six factors.

Discussion

Nazzal et al. [10] fit second-order models to the main response variable 6Y and supplement their analysis with 3-dimesional response surface and 2-dimensional contour

plots, similar to the ones in our paper. They also construct response surface models for the other responses 1(Y through 5Y ) and address the optimization of 6Y subject to the

constraints on 1Y through 5Y . What we provide in our paper is a more detailed discussion

of the optimization and guidance on how the constrained optimization can be implemented

within Excel’s SOLVER routine. Their optimization results are quite similar to ours, with 1X

(copolyvidone) at 67 mg; 2X (maltodextrin) in the midrange at 560 mg, and 3X

(microcrystalline cellulose) at 100 mg.

Paper 3: “Effects of Electrospinning Parameters on Polyacrylonitrile Nanofiber Diameter: An

Investigation by Response Surface Methodology” by Yoerdem et al. [13].

Electro-spinning (also called electrostatic fiber spinning) represents a promising process for producing continuous nano-scale fibers from both synthetic and natural polymers. The main principle in electro-spinning is to apply an electric field to a polymeric solution, generating a charged jet. As the jet travels through the air, the solvent evaporates and a charged fiber is left behind which can then be collected on a grounded plate (collector). This paper studies the effects of the used polymer material 1X (molecular weight of

polyacrylonitrile), the used solution concentration 2X (N-Dimethylformamide), and two process parameter 3(X applied voltage and 4X distance to the collector) on the diameter of the electro-spun polyacrylonitrile fibers. Prior experiments have shown that the molecular weight of polyacrylonitrile 1(X ) has by far the largest impact on the dimensions of the fibers.

The molecular weight 1(X ) was fixed at its lower level (73,000 g/mol). This is because the investigators wanted to learn more about the effects of the other three factors so that they could fine-tune the resultant fiber diameter. At each of the three collector distances

194 Ledolter the remaining five response variables satisfy the bounds that were specified by the

investigators (and shown in Table 2).

1 2 3695 1045 135X 200X 150X 1000 1Y (Weight)

1 2 3 1 2 350 63.2 10.3X 0.69X 2.37X 8.71X * *2 1.21X * *2 7.17X * *2

1 2 1 3 2 35.00X * X 6.38X * X 1.87X *X 100 2Y (Flowability)

1 2 3 1 20.05 0.153 0.0544X 0.0358X 0.0184X 0.0853X * *2 0.0294X * *2

3 1 2 1 3 2 30.0262X * *2 0.0077X * X 0.0210X * X 0.0297X *X 0.32

3Y (TStrength)

1 2 3 1 2 30 0.180 0.446X 0.249X 0.220X 0.344X * *2 0.029X * *2 0.009X * *2

1 2 1 3 2 30.383X * X 0.325X * X 0.095X *X 1 4Y (Friability)

1 2 3 1 2 30 20.1 2.31X 3.51X 4.35X 3.26X * *2 0.51X * *2 5.41X * *2

1 2 1 3 2 30.38X * X 6.75X * X 1.35X *X 30 5Y (DisTime)

The optimization problem looks for a maximum of a quadratic function in three decision variables, with restrictions that are (also) described by quadratic functions. The solution of this “quadratic-quadratic” optimization problem is considerably more difficult than the linear programming problem which looks for an optimum of a linear function of the decision variables subject to linear constraints. For the linear problem, the solution is straightforward. The simplex method can be used, and software such as the EXCEL SOLVER provide the solution quite readily (see Evans [2]). Fortunately, the EXCEL SOLVER optimization routines also include procedures for non-linear models. If the dimensions are not too large (that is, not too many decision variables) and if the problem is not “ill-conditioned” (not many local optima), SOLVER’s iterative non-linear procedures provide prompt answers. For particularly difficult quadratic optimization problems with many decision variables and numerous quadratic constraints, one can consult one of the more advanced solution approaches that are discussed in the optimization literature.

Selecting reasonable starting values for the iterative optimization procedure helps speed up the convergence and usually avoids getting trapped in local optima or saddle points. The contour plots in Figure 4 show that large values of 6Y occur when the three design factors are near their low levels ( 1). With these starting values the nonlinear optimization encountered no difficulties, and found the optimum 6,Y 91.75opt at

1, 2, 3,(X 0.74, X 0.50, X 1.00);opt opt opt see Table 4. In uncoded units, this corresponds to Copolyvidone = 65 mg; Maltodextrin = 400 mg and Microcrystalline Cellulose = 100 mg. The overlaid contour plots in Figure 6 confirm the results of the optimization.

Table 4. Final solution of the optimization (EXCEL output).

Required Bounds Objective Response Variable

Calculated Response Low High

1Y 695 695 1000

2Y 50 50 100

3Y 0.132258 0.05 0.32

4Y 0.121275 0 1

5Y 16.724271 0 30

6Y 91.752177 Max


X1=Copolyvidone(mg)

X2=

Mal

tode

xtri

n(m

g)

30025020015010050

700

600

500

400

300 X3=Microcrystalline(mg) 100Hold Values

6951000

Y1=Weight(mg)Min

5060

Y2=F lowability Max

0.10.15

Y3=TensileStrength(MPa)Max

00.15

Y4=F riability (%)Min

16.5

20

Y5=DisintegrationTime(min)Min

8591.7

Y6=%C oQ 10(%)Max

Contour Plot of Six Responses

Figure 6. Overlaid contour plots from the full quadratic model, with the white area showing a region that satisfies the desired requirements for all six factors.

Discussion

Nazzal et al. [10] fit second-order models to the main response variable 6Y and supplement their analysis with 3-dimesional response surface and 2-dimensional contour

plots, similar to the ones in our paper. They also construct response surface models for the other responses 1(Y through 5Y ) and address the optimization of 6Y subject to the

constraints on 1Y through 5Y . What we provide in our paper is a more detailed discussion

of the optimization and guidance on how the constrained optimization can be implemented

within Excel’s SOLVER routine. Their optimization results are quite similar to ours, with 1X

(copolyvidone) at 67 mg; 2X (maltodextrin) in the midrange at 560 mg, and 3X

(microcrystalline cellulose) at 100 mg.

Paper 3: “Effects of Electrospinning Parameters on Polyacrylonitrile Nanofiber Diameter: An

Investigation by Response Surface Methodology” by Yoerdem et al. [13].

Electro-spinning (also called electrostatic fiber spinning) represents a promising process for producing continuous nano-scale fibers from both synthetic and natural polymers. The main principle in electro-spinning is to apply an electric field to a polymeric solution, generating a charged jet. As the jet travels through the air, the solvent evaporates and a charged fiber is left behind which can then be collected on a grounded plate (collector). This paper studies the effects of the used polymer material 1X (molecular weight of

polyacrylonitrile), the used solution concentration 2X (N-Dimethylformamide), and two process parameter 3(X applied voltage and 4X distance to the collector) on the diameter of the electro-spun polyacrylonitrile fibers. Prior experiments have shown that the molecular weight of polyacrylonitrile 1(X ) has by far the largest impact on the dimensions of the fibers.

The molecular weight 1(X ) was fixed at its lower level (73,000 g/mol). This is because the investigators wanted to learn more about the effects of the other three factors so that they could fine-tune the resultant fiber diameter. At each of the three collector distances

196 Ledolter

4(X ), experiments with changed 2X and 3X were carried out. The thirteen runs at each collector distance consisted of the nine runs of a

23 factorial experiment (with levels for the solution concentration 2X at 8, 12, and 16 percent, and levels for the voltage 3X at 10, 20 and 30 kV), and a smaller

22 factorial inside the region of the 23 (with levels for 2X at

10 and 14, and levels for 3X at 15 and 25). Such a design results in five different levels for each of the two factors 2X and 3X . Fibers from each experimental set-up were sampled, and the means and standard deviations of fiber diameter in Table 5 were calculated from representative samples of approximately 50 fibers each.

The interest here isn’t so much to increase or decrease the mean diameter of a fiber, but to be able to manufacture fibers with a certain specified mean diameter. Of course, the objective is to do so with as little variability as possible.

Since the experiments at the three studied collector distances were carried out in three distinct blocks, it is preferably to analyze the results for each collector distance separately. Alternatively, one could have analyzed the results for all three collector distances jointly according to a split-plot design with 4X acting as the whole plot factor. However, the absence of replications on the whole plot factor would have made it difficult to assess the significance of 4X . We approximate the response surfaces with quadratic models,

2 20 2 2 3 3 22 2 33 3 23 2 3Y noise.x x x x x x

The response surfaces are well approximated by quadratic surfaces. The addition of

cubic terms is statistically insignificant for all combinations of response (mean, standard deviation) and collector distance, except for the mean diameter at 4X 8. The contour plots

in Figure 7 show that low solution concentration 2(X ) leads to small means and standard deviations. The results, especially for the standard deviation, are not particularly sensitive to changes in voltage 3(X ).

Overlaid contour plots of the mean (in red) and the standard deviation (in green) are shown in Figure 8. The overlaid contour plots are almost identical because the mean and standard deviation are highly correlated. For 4X distance to the collector 16, fibers with very small mean diameter (less than 0.05) and very small standard deviation (less than 0.05) can be produced with voltage 3(X ) around 20 and solution 2(X ) less than 9-10 percent. For 4X distance to the collector 12, the diameter of the fibers becomes larger. Fibers with mean diameter less than 0.2 and standard deviation less than 0.1 can be produced with solution 2(X ) less than 9 percent; the results are not sensitive to the used voltage 3(X ).

For 4X distance to the collector 8, the mean size of the fibers and their standard deviation are even larger; fibers with mean diameter less than 0.5 and standard deviation less than 0.2 can be produced with low voltage 3(X ) and low solution 2(X ).

Collector distance has a great influence on the results. The fiber diameter, as well as the fiber variability, can be reduced by increasing the collector distance. The control of fiber diameter in micrometer scale is further affected by concentration 2X (with low values leading to smaller size), whereas voltage 3X has a lesser impact (with levels in the low to midrange leading to smaller size). These conclusions are strengthened even further when adjusting for two suspicious observations (mean response =1.592 for 2 3X 8, X 30 and

4X 8; and mean response =1.348 for 2 3X 16, X 20 and 4X 16). However, without being involved in the experiments directly, it is impossible to establish whether these observations are questionable.


Table 5. Factors, with levels and units, and the response variables of the experiment.

Block Run 2X

Solution Concentration

(%)

3X

Applied Voltage (kV)

4X

Collector Distance (cm)

1Y

Mean Fiber

Diameter ( m)

2Y

StdDev Fiber

Diameter ( m)

1 1 8 10 8 0.083 0.039 1 2 8 20 8 0.109 0.065 1 3 8 30 8 1.592 0.473 1 4 10 15 8 0.773 0.403 1 5 10 25 8 0.955 0.440 1 6 12 10 8 0.920 0.360 1 7 12 20 8 0.556 0.226 1 8 12 30 8 0.548 0.243 1 9 14 15 8 0.804 0.350 1 10 14 25 8 0.942 0.409 1 11 16 10 8 1.933 0.655 1 12 16 20 8 2.782 0.742 1 13 16 30 8 1.771 0.413 2 14 8 10 12 0.166 0.058 2 15 8 20 12 0.093 0.040 2 16 8 30 12 0.084 0.033 2 17 10 15 12 0.244 0.086 2 18 10 25 12 0.326 0.107 2 19 12 10 12 0.625 0.305 2 20 12 20 12 0.548 0.209 2 21 12 30 12 0.635 0.383 2 22 14 15 12 1.608 0.731 2 23 14 25 12 1.569 0.474 2 24 16 10 12 1.756 0.586 2 25 16 20 12 1.643 0.625 2 26 16 30 12 1.361 0.386 3 27 8 10 16 0.137 0.065 3 28 8 20 16 0.162 0.111 3 29 8 30 16 0.124 0.041 3 30 10 15 16 0.171 0.101 3 31 10 25 16 0.339 0.127 3 32 12 10 16 0.714 0.205 3 33 12 20 16 0.540 0.173 3 34 12 30 16 0.561 0.166 3 35 14 15 16 0.880 0.222 3 36 14 25 16 1.127 0.378 3 37 16 10 16 2.694 0.696 3 38 16 20 16 1.348 0.419 3 39 16 30 16 2.564 1.167

Discussion

The cubic terms in the response surface models fit by Yoerdem et al. [13] are not needed; second-order models describe the response surfaces quite adequately. The two suspicious observations mentioned in our analysis may have led them to consider third-order models. Our fitted models are more parsimonious and they imply smoother contours, thus simplifying the interpretation of the results. Yoerdem et al. [13] analyze the mean and the coefficient of

variation while our analysis addresses the mean and the standard deviation. Our overlaid

contour plots for the mean and standard deviation provide an effective summary of the

196 Ledolter

4(X ), experiments with changed 2X and 3X were carried out. The thirteen runs at each collector distance consisted of the nine runs of a

23 factorial experiment (with levels for the solution concentration 2X at 8, 12, and 16 percent, and levels for the voltage 3X at 10, 20 and 30 kV), and a smaller

22 factorial inside the region of the 23 (with levels for 2X at

10 and 14, and levels for 3X at 15 and 25). Such a design results in five different levels for each of the two factors 2X and 3X . Fibers from each experimental set-up were sampled, and the means and standard deviations of fiber diameter in Table 5 were calculated from representative samples of approximately 50 fibers each.

The interest here isn’t so much to increase or decrease the mean diameter of a fiber, but to be able to manufacture fibers with a certain specified mean diameter. Of course, the objective is to do so with as little variability as possible.

Since the experiments at the three studied collector distances were carried out in three distinct blocks, it is preferably to analyze the results for each collector distance separately. Alternatively, one could have analyzed the results for all three collector distances jointly according to a split-plot design with 4X acting as the whole plot factor. However, the absence of replications on the whole plot factor would have made it difficult to assess the significance of 4X . We approximate the response surfaces with quadratic models,

2 20 2 2 3 3 22 2 33 3 23 2 3Y noise.x x x x x x

The response surfaces are well approximated by quadratic surfaces. The addition of

cubic terms is statistically insignificant for all combinations of response (mean, standard deviation) and collector distance, except for the mean diameter at 4X 8. The contour plots

in Figure 7 show that low solution concentration 2(X ) leads to small means and standard deviations. The results, especially for the standard deviation, are not particularly sensitive to changes in voltage 3(X ).

Overlaid contour plots of the mean (in red) and the standard deviation (in green) are shown in Figure 8. The overlaid contour plots are almost identical because the mean and standard deviation are highly correlated. For 4X distance to the collector 16, fibers with very small mean diameter (less than 0.05) and very small standard deviation (less than 0.05) can be produced with voltage 3(X ) around 20 and solution 2(X ) less than 9-10 percent. For 4X distance to the collector 12, the diameter of the fibers becomes larger. Fibers with mean diameter less than 0.2 and standard deviation less than 0.1 can be produced with solution 2(X ) less than 9 percent; the results are not sensitive to the used voltage 3(X ).

For 4X distance to the collector 8, the mean size of the fibers and their standard deviation are even larger; fibers with mean diameter less than 0.5 and standard deviation less than 0.2 can be produced with low voltage 3(X ) and low solution 2(X ).

Collector distance has a great influence on the results. The fiber diameter, as well as the fiber variability, can be reduced by increasing the collector distance. The control of fiber diameter in micrometer scale is further affected by concentration 2X (with low values leading to smaller size), whereas voltage 3X has a lesser impact (with levels in the low to midrange leading to smaller size). These conclusions are strengthened even further when adjusting for two suspicious observations (mean response =1.592 for 2 3X 8, X 30 and

4X 8; and mean response =1.348 for 2 3X 16, X 20 and 4X 16). However, without being involved in the experiments directly, it is impossible to establish whether these observations are questionable.


Table 5. Factors, with levels and units, and the response variables of the experiment.

Block Run 2X

Solution Concentration

(%)

3X

Applied Voltage (kV)

4X

Collector Distance (cm)

1Y

Mean Fiber

Diameter ( m)

2Y

StdDev Fiber

Diameter ( m)

1 1 8 10 8 0.083 0.039 1 2 8 20 8 0.109 0.065 1 3 8 30 8 1.592 0.473 1 4 10 15 8 0.773 0.403 1 5 10 25 8 0.955 0.440 1 6 12 10 8 0.920 0.360 1 7 12 20 8 0.556 0.226 1 8 12 30 8 0.548 0.243 1 9 14 15 8 0.804 0.350 1 10 14 25 8 0.942 0.409 1 11 16 10 8 1.933 0.655 1 12 16 20 8 2.782 0.742 1 13 16 30 8 1.771 0.413 2 14 8 10 12 0.166 0.058 2 15 8 20 12 0.093 0.040 2 16 8 30 12 0.084 0.033 2 17 10 15 12 0.244 0.086 2 18 10 25 12 0.326 0.107 2 19 12 10 12 0.625 0.305 2 20 12 20 12 0.548 0.209 2 21 12 30 12 0.635 0.383 2 22 14 15 12 1.608 0.731 2 23 14 25 12 1.569 0.474 2 24 16 10 12 1.756 0.586 2 25 16 20 12 1.643 0.625 2 26 16 30 12 1.361 0.386 3 27 8 10 16 0.137 0.065 3 28 8 20 16 0.162 0.111 3 29 8 30 16 0.124 0.041 3 30 10 15 16 0.171 0.101 3 31 10 25 16 0.339 0.127 3 32 12 10 16 0.714 0.205 3 33 12 20 16 0.540 0.173 3 34 12 30 16 0.561 0.166 3 35 14 15 16 0.880 0.222 3 36 14 25 16 1.127 0.378 3 37 16 10 16 2.694 0.696 3 38 16 20 16 1.348 0.419 3 39 16 30 16 2.564 1.167

Discussion

The cubic terms in the response surface models fit by Yoerdem et al. [13] are not needed; second-order models describe the response surfaces quite adequately. The two suspicious observations mentioned in our analysis may have led them to consider third-order models. Our fitted models are more parsimonious and they imply smoother contours, thus simplifying the interpretation of the results. Yoerdem et al. [13] analyze the mean and the coefficient of

variation while our analysis addresses the mean and the standard deviation. Our overlaid

contour plots for the mean and standard deviation provide an effective summary of the

198 Ledolter information that is useful to the engineer who wants to manufacture fibers with certain specified mean dimensions and as little variation as possible.

Figure 7. Contour plots of standard deviations (left panel, with contours 0.1, 0.2, 0.3, 0.4, 0.5) and mean (right panel, with contours 0.1, 0.5, 1.0, 1.5) for factors 2X (concentration) and 3X (applied voltage).


Figure 8. Overlaid contour plots of mean (red) and standard deviation (green): 0.2; 0.5s x for 4X 8; 0.1; 0.2s x for 4X 12; 0.05; 0.05s x for

4X 16.

198 Ledolter information that is useful to the engineer who wants to manufacture fibers with certain specified mean dimensions and as little variation as possible.

Figure 7. Contour plots of standard deviations (left panel, with contours 0.1, 0.2, 0.3, 0.4, 0.5) and mean (right panel, with contours 0.1, 0.5, 1.0, 1.5) for factors 2X (concentration) and 3X (applied voltage).


Figure 8. Overlaid contour plots of mean (red) and standard deviation (green): 0.2; 0.5s x for 4X 8; 0.1; 0.2s x for 4X 12; 0.05; 0.05s x for

4X 16.

200 Ledolter

Paper 4: “Optimization of Polylactic-Co-Glycolic Acid Nanoparticles Containing Itraconazole Using 23 Factorial Design” by Prakobvaitayakit and Nimmannit [11].

PLGA (polylactic-co-glycolic acid) polymers are commonly used in micro-particulate

drug delivery systems, and they play a large role in pharmacology. PLGA nano-particles containing itraconazole were prepared using a solvent displacement technique. Polymers and benzyl benzoate were dissolved in acetone, and the organic phase (itraconazole) and a

surfactant were added to this solution under mechanical stirring. This process results in the formation of nano-particles and induces the deposition of the polymer around the oily

droplets.

The resulting particle size and the amount of itraconazole entrapped in the nano-particles are of interest. They depend on various process parameters, in particular the amount of PLGA 1(X ), the amount of benzyl benzoate 2(X ), and the amount of

itraconazole 3(X ) in the base solution.

A 23 factorial experiment with two replications and five replications at the center point was carried out. The three levels of the three process factors and the design are shown in Table 6. The results of the experiments are listed in the last three columns. The amount of itraconazole that is entrapped in the nano-particles is expressed both as amount 2(Y ) and as a percentage of the amount of drug in the organic phase 3 2 3(Y Y /X ).

Table 6. Factors, with levels and units, and the response variables of the 23 factorial experiment; runs 16 to 20 are center points.

Run 1X

(mg/mL) 2X

( g/mL) 3X

( g/mL) 1Y

Size (nm)

2Y

Encaps Efficiency (mg/mL)

3 2 3Y Y /X

Encaps Efficiency (percent)

1 10 5.0 200 193.9 98.28 49.1400 2 10 20.0 1800 425.6 976.30 54.2389 3 10 20.0 1800 420.6 983.34 54.6300 4 100 5.0 1800 539.6 628.87 34.9372 5 10 20.0 200 304.3 159.97 79.9850 6 100 5.0 200 249.6 123.97 61.9850 7 10 5.0 1800 306.9 355.60 19.7556 8 10 20.0 200 305.6 154.22 77.1100 9 10 5.0 200 190.0 102.96 51.4800 10 10 5.0 1800 310.8 365.20 20.2889 11 100 20.0 1800 639.5 1256.80 69.8222 12 100 20.0 1800 643.9 1249.60 69.4222 13 100 20.0 200 337.6 176.40 88.2000 14 100 20.0 200 329.6 179.30 89.6500 15 100 5.0 1800 543.6 631.97 35.1094 16 55 12.5 1000 461.2 701.30 70.1300 17 55 12.5 1000 458.3 712.60 71.2600 18 55 12.5 1000 460.5 706.20 70.6200 19 55 12.5 1000 467.7 699.30 69.9300 20 55 12.5 1000 466.9 705.60 70.5600 21 100 5.0 200 238.9 124.12 62.0600


The replications at the design points and at the center point can be used to obtain an estimate of the natural variability that allows us to obtain standard errors of the main effects and interactions of the estimated model. Such calculations assume that the 21 runs are fully randomized, and that the replications are “genuine” and not split-plot type replications that are carried out by repeating runs at each design point in short sequence. Without being involved in the experiment, one doesn’t know for sure how the replications were carried out.

Whereas observations at the center point have no influence on the estimates of main effects and interactions (they only affect the intercept), they provide a useful omnibus check for curvature. Table 7 shows the fitting results for a regression model that includes main effects and interactions, as well as an indicator variable for the center point observations. The pooled standard deviation 4.1264,s with 12 degrees of freedom, is obtained from the 8 replications of the factorial experiment and the five measurements at the center point. The variability among the replications at the 8 factorial experiments and among the

replications at the center point can be pooled as no difference in the variance can be detected

[ 4.12s at the factorial observations versus 4.15s at the center point]. Furthermore, the estimated coefficient for the indicator variable (89.17) indicates that the estimated level at the center point using only results of the center point observations is 89.17 units higher than the estimate using only results of the factorial experiment (373.75). The difference is highly

significant indicating that there is considerable curvature in the response surface. Similar

results are found for the other two response variables (encapsulation of itraconazole, in weight as well as percent). These results confirm the findings of the (non-linear) main effects plots shown in Figure 9.

The results in Table 7 show that interactions are needed. For 1Y (size) and 2Y

(encapsulation efficiency, in mg/mL) only the 1 2X * X and 1 3X * X interactions are

required. For 3Y (encapsulation efficieny in percent) 1 3X * X and 2 3X * X are required.

Table 7. Regression fitting results (full model with all possible interactions and restricted model after removing insignificant higher-order interactions) for 1Y (size), 2Y (encapsulation efficiency in mg/mL) and 3Y (encapsulation efficiency in percent). Regression Analysis: 1Y Size (nm)

Predictor Coef SE Coef T P Constant 373.750 1.032 362.30 0.000

1X coded 66.538 1.032 64.50 0.000

2X coded 52.087 1.032 50.49 0.000

3X coded 105.062 1.032 101.84 0.000

1 2X * X 4.725 1.032 4.58 0.001

1 3X * X 46.300 1.032 44.88 0.000

2 3X * X 1.500 1.032 1.45 0.172

1 2 3X * X * X 1.187 1.032 1.15 0.272 Indicator(cp) 89.170 2.114 42.18 0.000

s 4.12642 R-Sq = 99.9% R-Sq(adj) = 99.9%

Analysis of Variance

Source DF SS MS F P Regression 8 355861 44483 2612.42 0.000

Residual Error 12 204 17 Total 20 356065

200 Ledolter

Paper 4: “Optimization of Polylactic-Co-Glycolic Acid Nanoparticles Containing Itraconazole Using 23 Factorial Design” by Prakobvaitayakit and Nimmannit [11].

PLGA (polylactic-co-glycolic acid) polymers are commonly used in micro-particulate

drug delivery systems, and they play a large role in pharmacology. PLGA nano-particles containing itraconazole were prepared using a solvent displacement technique. Polymers and benzyl benzoate were dissolved in acetone, and the organic phase (itraconazole) and a

surfactant were added to this solution under mechanical stirring. This process results in the formation of nano-particles and induces the deposition of the polymer around the oily

droplets.

The resulting particle size and the amount of itraconazole entrapped in the nano-particles are of interest. They depend on various process parameters, in particular the amount of PLGA 1(X ), the amount of benzyl benzoate 2(X ), and the amount of

itraconazole 3(X ) in the base solution.

A 23 factorial experiment with two replications and five replications at the center point was carried out. The three levels of the three process factors and the design are shown in Table 6. The results of the experiments are listed in the last three columns. The amount of itraconazole that is entrapped in the nano-particles is expressed both as amount 2(Y ) and as a percentage of the amount of drug in the organic phase 3 2 3(Y Y /X ).

Table 6. Factors, with levels and units, and the response variables of the 23 factorial experiment; runs 16 to 20 are center points.

Run 1X

(mg/mL) 2X

( g/mL) 3X

( g/mL) 1Y

Size (nm)

2Y

Encaps Efficiency (mg/mL)

3 2 3Y Y /X

Encaps Efficiency (percent)

1 10 5.0 200 193.9 98.28 49.1400 2 10 20.0 1800 425.6 976.30 54.2389 3 10 20.0 1800 420.6 983.34 54.6300 4 100 5.0 1800 539.6 628.87 34.9372 5 10 20.0 200 304.3 159.97 79.9850 6 100 5.0 200 249.6 123.97 61.9850 7 10 5.0 1800 306.9 355.60 19.7556 8 10 20.0 200 305.6 154.22 77.1100 9 10 5.0 200 190.0 102.96 51.4800 10 10 5.0 1800 310.8 365.20 20.2889 11 100 20.0 1800 639.5 1256.80 69.8222 12 100 20.0 1800 643.9 1249.60 69.4222 13 100 20.0 200 337.6 176.40 88.2000 14 100 20.0 200 329.6 179.30 89.6500 15 100 5.0 1800 543.6 631.97 35.1094 16 55 12.5 1000 461.2 701.30 70.1300 17 55 12.5 1000 458.3 712.60 71.2600 18 55 12.5 1000 460.5 706.20 70.6200 19 55 12.5 1000 467.7 699.30 69.9300 20 55 12.5 1000 466.9 705.60 70.5600 21 100 5.0 200 238.9 124.12 62.0600


The replications at the design points and at the center point can be used to obtain an estimate of the natural variability that allows us to obtain standard errors of the main effects and interactions of the estimated model. Such calculations assume that the 21 runs are fully randomized, and that the replications are “genuine” and not split-plot type replications that are carried out by repeating runs at each design point in short sequence. Without being involved in the experiment, one doesn’t know for sure how the replications were carried out.

Whereas observations at the center point have no influence on the estimates of main effects and interactions (they only affect the intercept), they provide a useful omnibus check for curvature. Table 7 shows the fitting results for a regression model that includes main effects and interactions, as well as an indicator variable for the center point observations. The pooled standard deviation 4.1264,s with 12 degrees of freedom, is obtained from the 8 replications of the factorial experiment and the five measurements at the center point. The variability among the replications at the 8 factorial experiments and among the

replications at the center point can be pooled as no difference in the variance can be detected

[ 4.12s at the factorial observations versus 4.15s at the center point]. Furthermore, the estimated coefficient for the indicator variable (89.17) indicates that the estimated level at the center point using only results of the center point observations is 89.17 units higher than the estimate using only results of the factorial experiment (373.75). The difference is highly

significant indicating that there is considerable curvature in the response surface. Similar

results are found for the other two response variables (encapsulation of itraconazole, in weight as well as percent). These results confirm the findings of the (non-linear) main effects plots shown in Figure 9.

The results in Table 7 show that interactions are needed. For 1Y (size) and 2Y

(encapsulation efficiency, in mg/mL) only the 1 2X * X and 1 3X * X interactions are

required. For 3Y (encapsulation efficieny in percent) 1 3X * X and 2 3X * X are required.

Table 7. Regression fitting results (full model with all possible interactions and restricted model after removing insignificant higher-order interactions) for 1Y (size), 2Y (encapsulation efficiency in mg/mL) and 3Y (encapsulation efficiency in percent). Regression Analysis: 1Y Size (nm)


1X coded 66.538 1.032 64.50 0.000

2X coded 52.087 1.032 50.49 0.000

3X coded 105.062 1.032 101.84 0.000

1 2X * X 4.725 1.032 4.58 0.001

1 3X * X 46.300 1.032 44.88 0.000

2 3X * X 1.500 1.032 1.45 0.172

1 2 3X * X * X 1.187 1.032 1.15 0.272 Indicator(cp) 89.170 2.114 42.18 0.000

s 4.12642 R-Sq = 99.9% R-Sq(adj) = 99.9%




202 Ledolter

Table 7. (Continued) Regression Analysis: 1Y Size (nm); simplified model


1X coded 66.538 1.083 61.42 0.000

2X coded 52.087 1.083 48.08 0.000

3X coded 105.062 1.083 96.98 0.000

1 2X * X 4.725 1.083 4.36 0.001

1 3X * X 46.300 1.083 42.74 0.000 Indicator(cp) 89.170 2.220 40.16 0.000

s = 4.33335 R-Sq = 99.9% R-Sq(adj) = 99.9%




Regression Analysis: 2Y Encapsulation Efficiency (mg/mL) Predictor Coef SE Coef T P Constant 472.931 1.116 423.86 0.000

1X coded 73.447 1.116 65.83 0.000

2X coded 169.060 1.116 151.52 0.000

3X coded 333.029 1.116 298.48 0.000

1 2X * X 0.086 1.116 0.08 0.940

1 3X * X 62.403 1.116 55.93 0.000

2 3X * X 141.490 1.116 126.81 0.000

1 2 3X * X * X 0.754 1.116 0.68 0.512 Indicator(cp) 232.069 2.287 101.49 0.000

4.46304s R-Sq = 99.99% R-Sq(adj) = 99.99%




Regression Analysis: 2Y Encapsulation Efficiency (mg/mL); simplified model


1X coded 73.447 1.053 69.77 0.000

2X coded 169.060 1.053 160.60 0.000

3X coded 333.029 1.053 316.36 0.000

1 2X * X 62.403 1.053 59.28 0.000

2 3X * X 141.490 1.053 134.41 0.000 Indicator(cp) 232.069 2.157 107.57 0.000

s 4.21082 R-Sq = 99.99% R-Sq(adj) = 99.99%





Table 7. (Continued)

Regression Analysis: 3Y Encapsulation Efficiency (percent) Predictor Coef SE Coef T P Constant 57.3634 0.2200 260.70 0.000

1X coded 6.5349 0.2200 29.70 0.000

2X coded 15.5189 0.2200 70.53 0.000

3X coded 12.5878 0.2200 57.21 0.000

1 2X * X 0.1435 0.2200 0.65 0.526

1 3X * X 1.0124 0.2200 4.60 0.001

2 3X * X 1.7339 0.2200 7.88 0.000

1 2 3X * X * X 0.1902 0.2200 0.86 0.404 Indicator(cp) 13.1366 0.4509 29.13 0.000

s 0.880133 R-Sq = 99.9% R-Sq(adj) = 99.8%


Source DF SS MS F P Regression 8 7794.73 974.34 1257.81 0.000

Residual Error 12 9.30 0.77 Total 20 7804.02

Regression Analysis: 3Y Encapsulation Efficiency (percent); simplified model


1X coded 6.5349 0.2134 30.62 0.000

2X coded 15.5189 0.2134 72.71 0.000

3X coded 12.5878 0.2134 58.98 0.000

1 2X * X 1.0124 0.2134 4.74 0.000

2 3X * X 1.7339 0.2134 8.12 0.000 Indicator(cp) 13.1366 0.4374 30.03 0.000

s = 0.853737 R-Sq = 99.9% R-Sq(adj) = 99.8%




Assume that one would like to achieve an encapsulation efficiency of between 60 and 70 percent 3(60 Y 70), and at the same time achieve a size in the 300 to 400 range

1(300 Y 400). The main-effects plots in Figure 9 indicate that such a desired efficiency requires 3X at its lowest setting; on the other hand, achieving a desired size between 300 and 400 may not be possible with the low setting of 3X as for that one needs higher levels of

3X . A mid-level for 3X may be able to achieve both requirements. Overlaid contours of

1Y (size) and 3Y (encapsulation efficiency, in percent) for 1X and 2X when 3X is set at its midlevel (1000 g/mL) are shown in Figure 10. The graph shows that there is a region for 1X and 2X (with low settings for 1X and high settings for 2X ) that can satisfy both requirements. However note that this contour plot comes from the model with just main effects and interactions. As we find considerable curvature in the model, it may be misleading to plot the response surfaces that are implied by just main effects and interactions. In order to do better, we would need the coefficients of the quadratic components, but unfortunately the design that was run does not allow us to estimate these coefficients (the omnibus test for curvature is all we can get).

202 Ledolter

Table 7. (Continued) Regression Analysis: 1Y Size (nm); simplified model


1X coded 66.538 1.083 61.42 0.000

2X coded 52.087 1.083 48.08 0.000

3X coded 105.062 1.083 96.98 0.000

1 2X * X 4.725 1.083 4.36 0.001

1 3X * X 46.300 1.083 42.74 0.000 Indicator(cp) 89.170 2.220 40.16 0.000

s = 4.33335 R-Sq = 99.9% R-Sq(adj) = 99.9%




Regression Analysis: 2Y Encapsulation Efficiency (mg/mL) Predictor Coef SE Coef T P Constant 472.931 1.116 423.86 0.000

1X coded 73.447 1.116 65.83 0.000

2X coded 169.060 1.116 151.52 0.000

3X coded 333.029 1.116 298.48 0.000

1 2X * X 0.086 1.116 0.08 0.940

1 3X * X 62.403 1.116 55.93 0.000

2 3X * X 141.490 1.116 126.81 0.000

1 2 3X * X * X 0.754 1.116 0.68 0.512 Indicator(cp) 232.069 2.287 101.49 0.000

4.46304s R-Sq = 99.99% R-Sq(adj) = 99.99%




Regression Analysis: 2Y Encapsulation Efficiency (mg/mL); simplified model


1X coded 73.447 1.053 69.77 0.000

2X coded 169.060 1.053 160.60 0.000

3X coded 333.029 1.053 316.36 0.000

1 2X * X 62.403 1.053 59.28 0.000

2 3X * X 141.490 1.053 134.41 0.000 Indicator(cp) 232.069 2.157 107.57 0.000

s 4.21082 R-Sq = 99.99% R-Sq(adj) = 99.99%





Table 7. (Continued)

Regression Analysis: 3Y Encapsulation Efficiency (percent) Predictor Coef SE Coef T P Constant 57.3634 0.2200 260.70 0.000

1X coded 6.5349 0.2200 29.70 0.000

2X coded 15.5189 0.2200 70.53 0.000

3X coded 12.5878 0.2200 57.21 0.000

1 2X * X 0.1435 0.2200 0.65 0.526

1 3X * X 1.0124 0.2200 4.60 0.001

2 3X * X 1.7339 0.2200 7.88 0.000

1 2 3X * X * X 0.1902 0.2200 0.86 0.404 Indicator(cp) 13.1366 0.4509 29.13 0.000

s 0.880133 R-Sq = 99.9% R-Sq(adj) = 99.8%




Regression Analysis: 3Y Encapsulation Efficiency (percent); simplified model


1X coded 6.5349 0.2134 30.62 0.000

2X coded 15.5189 0.2134 72.71 0.000

3X coded 12.5878 0.2134 58.98 0.000

1 2X * X 1.0124 0.2134 4.74 0.000

2 3X * X 1.7339 0.2134 8.12 0.000 Indicator(cp) 13.1366 0.4374 30.03 0.000

s = 0.853737 R-Sq = 99.9% R-Sq(adj) = 99.8%




Assume that one would like to achieve an encapsulation efficiency of between 60 and 70 percent 3(60 Y 70), and at the same time achieve a size in the 300 to 400 range

1(300 Y 400). The main-effects plots in Figure 9 indicate that such a desired efficiency requires 3X at its lowest setting; on the other hand, achieving a desired size between 300 and 400 may not be possible with the low setting of 3X as for that one needs higher levels of

3X . A mid-level for 3X may be able to achieve both requirements. Overlaid contours of

1Y (size) and 3Y (encapsulation efficiency, in percent) for 1X and 2X when 3X is set at its midlevel (1000 g/mL) are shown in Figure 10. The graph shows that there is a region for 1X and 2X (with low settings for 1X and high settings for 2X ) that can satisfy both requirements. However note that this contour plot comes from the model with just main effects and interactions. As we find considerable curvature in the model, it may be misleading to plot the response surfaces that are implied by just main effects and interactions. In order to do better, we would need the coefficients of the quadratic components, but unfortunately the design that was run does not allow us to estimate these coefficients (the omnibus test for curvature is all we can get).

204 Ledolter

Figure 9. Main effects plots for 1Y (size), 2Y (encapsulation efficiency in mg/mL) and 3Y (encapsulation efficiency in percent).


X1

X2

100908070605040302010

20.0

17.5

15.0

12.5

10.0

7.5

5.0

X3 1000Hold Values

300400

(nm)Size

6070

EncapEff(percent)

Contour Plot of Size (nm) and Encapsulation Efficiency (percent)

Figure 10. Overlaid contours of 1Y (size) and 3Y (encapsulation efficiency, in percent) for 1X and 2X , when 3X is set at its midlevel (1000 g/mL), from the full model without center point observations.

Discussion

The assessment of the replication error and its use in establishing standard errors of estimated main and interaction effects, and our test for model curvature are achieved within a linear regression formulation. Our results are identical to the ones in the original paper, but are easier to understand. Prakobvaitayakit and Nimmannit [11] use desirability functions

to transform the three response variables, 1Y (size), 2Y and 3Y (encapsulation efficiencies)

into a single response. But not all components of their desirability functions are fully

described which makes it difficult to replicate their analysis. Furthermore, the functional relationship between the two measures of encapsulation efficiency 2Y and 3 2 3Y Y /X reduces the dimensionality of the problem and it is not obvious whether the desirability

approach is appropriate. We believe that it is more instructive to explore the joint

relationships by overlaying contour plots for 1Y and 3Y .

Paper 5: “Surface Topographical Characterization of Silver-Plated Film on the Wedge Bondability of Leaded IC Packages” by Lin et al. [7].

A strong bonding between the gold wire and the silver-plated lead frame is essential for any high-quality integrated circuit manufacturing process. Experiments show that the surface roughness of the lead frame (LF) is important for achieving a good bond. Two types of lead frames (A and B) were compared in the experiment which also varied the process conditions (current, bond time and bond force) for attaching the silver lining. A wedge pull test was then applied to the attached silver linings. The force (in grams) needed to pull the lining from the frame was measured on 80 boards, all manufactured under the same conditions. The

mean and standard deviation of the required wedge pull force for each of the sixteen runs of a

42 factorial experiment are shown in Table 8. The objective is to achieve a large mean (it takes a large force to pry loose) and a small standard deviation (consistent results).

Estimates of main and interaction effects are obtained, and they are plotted on a normal probability plot and a Pareto chart (Figure 11 for the mean and Figure 12 for the

204 Ledolter

Figure 9. Main effects plots for 1Y (size), 2Y (encapsulation efficiency in mg/mL) and 3Y (encapsulation efficiency in percent).


X1

X2

100908070605040302010

20.0

17.5

15.0

12.5

10.0

7.5

5.0

X3 1000Hold Values

300400

(nm)Size

6070

EncapEff(percent)

Contour Plot of Size (nm) and Encapsulation Efficiency (percent)

Figure 10. Overlaid contours of 1Y (size) and 3Y (encapsulation efficiency, in percent) for 1X and 2X , when 3X is set at its midlevel (1000 g/mL), from the full model without center point observations.

Discussion

The assessment of the replication error and its use in establishing standard errors of estimated main and interaction effects, and our test for model curvature are achieved within a linear regression formulation. Our results are identical to the ones in the original paper, but are easier to understand. Prakobvaitayakit and Nimmannit [11] use desirability functions

to transform the three response variables, 1Y (size), 2Y and 3Y (encapsulation efficiencies)

into a single response. But not all components of their desirability functions are fully

described which makes it difficult to replicate their analysis. Furthermore, the functional relationship between the two measures of encapsulation efficiency 2Y and 3 2 3Y Y /X reduces the dimensionality of the problem and it is not obvious whether the desirability

approach is appropriate. We believe that it is more instructive to explore the joint

relationships by overlaying contour plots for 1Y and 3Y .

Paper 5: “Surface Topographical Characterization of Silver-Plated Film on the Wedge Bondability of Leaded IC Packages” by Lin et al. [7].

A strong bonding between the gold wire and the silver-plated lead frame is essential for any high-quality integrated circuit manufacturing process. Experiments show that the surface roughness of the lead frame (LF) is important for achieving a good bond. Two types of lead frames (A and B) were compared in the experiment which also varied the process conditions (current, bond time and bond force) for attaching the silver lining. A wedge pull test was then applied to the attached silver linings. The force (in grams) needed to pull the lining from the frame was measured on 80 boards, all manufactured under the same conditions. The

mean and standard deviation of the required wedge pull force for each of the sixteen runs of a

42 factorial experiment are shown in Table 8. The objective is to achieve a large mean (it takes a large force to pry loose) and a small standard deviation (consistent results).

Estimates of main and interaction effects are obtained, and they are plotted on a normal probability plot and a Pareto chart (Figure 11 for the mean and Figure 12 for the

206 Ledolter standard deviation of wedge pull force). Only the effect of LF type turns out significant. This conclusion is confirmed by the results in Table 9 which pool 3- and 4-factor interactions into a standard error of estimated effects to be used in the significance tests for main and interaction effects.

Table 8. Factors, with levels and units, and the response variables of the

42 factorial experiment.

Run

1X

Current (mA)

2X

Bond Time (ms)

3X

Bond Force

(g)

4X

Lead Frame Type

A (Smooth)

1Y

Wedge Pull Force

Mean (g)

2Y

Wedge Pull Force Std Dev (g)

1 100 35 70 B 6.091 0.592

2 100 35 70 A 6.103 0.549

3 100 35 30 B 6.048 0.547

4 100 35 30 A 6.041 0.621

5 100 20 70 B 5.990 0.539

6 100 20 70 A 6.358 0.622

7 100 20 30 A 6.129 0.658

8 100 20 30 B 5.807 0.599

9 65 35 70 A 6.341 0.626

10 65 35 70 B 5.918 0.561

11 65 35 30 B 5.667 0.523

12 65 35 30 A 6.331 0.618

13 65 20 70 B 5.766 0.628

14 65 20 70 A 6.120 0.610

15 65 20 30 B 5.517 0.619

16 65 20 30 A 6.022 0.685

The negative sign for the coefficient of LF type (with coded units 1 for A and 1 for B) in Table 9 indicates that the required mean wedge pull force for lead frame B is 0.33g lower than the one for lead frame A. Lead frame A performs better than lead frame B. Although LF has a statistically significant impact on mean strength, its effect is rather small. The estimated effect 0.33g (compared to the average strength of 6g) implies that the

strength of lead frame A is only 5% larger than that of type B. Whether such a 5% increase

in product strength is also practically significant needs to be assessed by the engineering team.

Lead frame A has a slightly larger standard deviation than lead frame B. The estimate

0.047g, barely significant at the 0.05 level, expresses that lead frame A increases the standard deviation by 100(0.047) / (0.6) 7.8%. The engineering team needs to decide

whether a 5% increase in the level compensates for a slight increase in the standard deviation (which, furthermore, is of questionable significance). A strategy that increases the average will increase the variability in the results. An approach that favors more consistent results will lower the average strength.

Discussion

Lin et al. [7] were mostly interested in comparing the two lead frames, and when assessing the effect they looked at differences in the response (either mean or standard deviation) from runs at the same process conditions (current, bond time, bond force). They


calculated the difference between runs 1 and 2 (6.091 6.103), between runs 3 and 4 (6.048 6.048),..., between runs 15 and 16 (5.517 6.022), and they obtained the average, 0.3301, as an estimate of the main effect of lead frame. They never looked at the effects of the three process factors and more importantly, the interactions between the main factor of interest (the lead frame) and the process factors. With such interactions, the comparison between the two lead frames would have to be conditioned on the process factors. Fortunately, as our analysis shows, none of the interactions matter.

Table 9. Estimates and significance of main effects and 2-factor interactions. Significance tests are carried out by pooling 3- and 4-factor interactions into an estimated standard error of an effect. Only LF type is significant. Factorial Fit: Wedge Pull Mean versus Current, Time, Force, LF Type

Estimated Effects and Coefficients for Wedge Pull Mean (g) (coded units) Term Effect Coef SE Coef T P

Constant 6.0156 0.02943 204.38 0.000 Current 0.1106 0.0553 0.02943 1.88 0.119

Time 0.1039 0.0519 0.02943 1.76 0.138 Force 0.1406 0.0703 0.02943 2.39 0.062

LF Type 0.3301 0.1651 0.02943 5.61 0.002 Current*Time 0.1041 0.0521 0.02943 1.77 0.137 Current*Force 0.0114 0.0057 0.02943 0.19 0.854

Current*LF Type 0.1564 0.0782 0.02943 2.66 0.045 Time*Force 0.0491 0.0246 0.02943 0.83 0.442

Time*LF Type 0.0571 0.0286 0.02943 0.97 0.376 Force*LF Type 0.0409 0.0204 0.02943 0.69 0.518

R-Sq = 91.82% R-Sq(pred) = 16.26% R-Sq(adj) = 75.47%


Source DF Seq SS Adj SS Adj MS F P

Main Effects 4 0.60714 0.60714 0.15179 10.95 0.011 2-Way Interactions 6 0.17109 0.17109 0.02851 2.06 0.223

Residual Error 5 0.06931 0.06931 0.01386 Total 15 0.84754

Factorial Fit: Wedge Pull Standard Deviation versus Current, Time, Force, LF Type Estimated Effects and Coefficients for Wedge Pull Standard Deviation (g) (coded units)

Term Effect Coef SE Coef T P Constant 0.59981 0.008733 68.69 0.000 Current 0.01787 0.00894 0.008733 1.02 0.353

Time 0.04037 0.02019 0.008733 2.31 0.069 Force 0.01788 0.00894 0.008733 1.02 0.353









206 Ledolter standard deviation of wedge pull force). Only the effect of LF type turns out significant. This conclusion is confirmed by the results in Table 9 which pool 3- and 4-factor interactions into a standard error of estimated effects to be used in the significance tests for main and interaction effects.

Table 8. Factors, with levels and units, and the response variables of the


Run

1X

Current (mA)

2X

Bond Time (ms)

3X

Bond Force

(g)

4X

Lead Frame Type

A (Smooth)

1Y

Wedge Pull Force

Mean (g)

2Y

Wedge Pull Force Std Dev (g)

1 100 35 70 B 6.091 0.592

2 100 35 70 A 6.103 0.549

3 100 35 30 B 6.048 0.547

4 100 35 30 A 6.041 0.621

5 100 20 70 B 5.990 0.539

6 100 20 70 A 6.358 0.622

7 100 20 30 A 6.129 0.658

8 100 20 30 B 5.807 0.599

9 65 35 70 A 6.341 0.626

10 65 35 70 B 5.918 0.561

11 65 35 30 B 5.667 0.523

12 65 35 30 A 6.331 0.618

13 65 20 70 B 5.766 0.628

14 65 20 70 A 6.120 0.610

15 65 20 30 B 5.517 0.619

16 65 20 30 A 6.022 0.685

The negative sign for the coefficient of LF type (with coded units 1 for A and 1 for B) in Table 9 indicates that the required mean wedge pull force for lead frame B is 0.33g lower than the one for lead frame A. Lead frame A performs better than lead frame B. Although LF has a statistically significant impact on mean strength, its effect is rather small. The estimated effect 0.33g (compared to the average strength of 6g) implies that the

strength of lead frame A is only 5% larger than that of type B. Whether such a 5% increase

in product strength is also practically significant needs to be assessed by the engineering team.

Lead frame A has a slightly larger standard deviation than lead frame B. The estimate

0.047g, barely significant at the 0.05 level, expresses that lead frame A increases the standard deviation by 100(0.047) / (0.6) 7.8%. The engineering team needs to decide

whether a 5% increase in the level compensates for a slight increase in the standard deviation (which, furthermore, is of questionable significance). A strategy that increases the average will increase the variability in the results. An approach that favors more consistent results will lower the average strength.

Discussion

Lin et al. [7] were mostly interested in comparing the two lead frames, and when assessing the effect they looked at differences in the response (either mean or standard deviation) from runs at the same process conditions (current, bond time, bond force). They


calculated the difference between runs 1 and 2 (6.091 6.103), between runs 3 and 4 (6.048 6.048),..., between runs 15 and 16 (5.517 6.022), and they obtained the average, 0.3301, as an estimate of the main effect of lead frame. They never looked at the effects of the three process factors and more importantly, the interactions between the main factor of interest (the lead frame) and the process factors. With such interactions, the comparison between the two lead frames would have to be conditioned on the process factors. Fortunately, as our analysis shows, none of the interactions matter.

Table 9. Estimates and significance of main effects and 2-factor interactions. Significance tests are carried out by pooling 3- and 4-factor interactions into an estimated standard error of an effect. Only LF type is significant. Factorial Fit: Wedge Pull Mean versus Current, Time, Force, LF Type

Estimated Effects and Coefficients for Wedge Pull Mean (g) (coded units) Term Effect Coef SE Coef T P

Constant 6.0156 0.02943 204.38 0.000 Current 0.1106 0.0553 0.02943 1.88 0.119

Time 0.1039 0.0519 0.02943 1.76 0.138 Force 0.1406 0.0703 0.02943 2.39 0.062









Factorial Fit: Wedge Pull Standard Deviation versus Current, Time, Force, LF Type Estimated Effects and Coefficients for Wedge Pull Standard Deviation (g) (coded units)

Term Effect Coef SE Coef T P Constant 0.59981 0.008733 68.69 0.000 Current 0.01787 0.00894 0.008733 1.02 0.353

Time 0.04037 0.02019 0.008733 2.31 0.069 Force 0.01788 0.00894 0.008733 1.02 0.353









208 Ledolter

Figure 11. Normal probability plot and Pareto chart of the estimated effects of the mean of the pull force in the


Figure 12. Normal probability plot and Pareto chart of the estimated effects of the standard deviation of the pull force in the


Acknowledgments

I would like to thank the two referees for their thoughtful and detailed comments which improved the paper greatly.

References

1. Box, G. E. P. and Behnken, D. W. (1960). Some new three level designs for the study of quantitative variables. Technometrics, 2, 455-475.

2. Evans, J. R. (2010). Statistics, Data Analysis, and Decision Modeling, th4 edition. Upper Saddle River, Prentice Hall, New Jersey.

3. Hou, T. H., Su, C. H. and Liu, W. L. (2007). Parameters optimization of a nanoparticle wet milling process using the Taguchi method, response surface method and genetic algorithm. Powder Technology, 173(3), 153-162.

4. Jeng, S. L., Lu, J. C. and Wang, K. (2007). A review of reliability research on nanotechnology, IEEE Transactions on Reliability, 56, 401-410.

5. Jones, B. and Nachtsheim, C. J. (2009). Split-Plot designs: what, why, and how. Journal of Quality Technology, 41, 340-361.


6. Ledolter, J. (2010). Split-plot designs: discussion and examples. International Journal of Quality Engineering and Technology, 1(4), 441-457.

7. Lin, T. Y., Davison, K. L., Leong, W. S., Chua, S., Yao, Y. F., Pan, J. S., Chai, J. W., Toh, K. C. and Tjiu, W. C. (2003). Surface topographical characterization of silver-plated film on the wedge bondability of leaded IC packages. Microelectronics Reliability, 43, 803-809.

8. Lu, J. C., Jeng, S. L. and Wang, K. (2009). A review of statistical methods for quality improvement and control in nanotechnology. Journal of Quality Technology, 41, 148-164.

9. Montgomery, D. C. (2008). Design and Analysis of Experiments, th7 edition. Wiley, New York.

10. Nazzal, S., Nutan, M., Palamakula, A., Shah, R., Zaghloul, A. A. and Khan, M. A. (2002). Optimization of a self-nanoemulsified tablet dosage form of ubiquinone using response surface methodology: effect of formulation ingredients. International Journal of Pharmaceutics, 240(1-2), 103-114.

11. Prakobvaitayakit, M. and Nimmannit, U. (2003). Optimization of polylactic-co-glycolic

acid nanoparticles containing itraconazole using 32 factorial design. AAPS

PharmSciTech, 4(4), 1-9.

12. Wu, C. F. J. and Hamada, M. (2000). Experiments: Planning, Analysis, and Parameter Design Optimization. Wiley, New York.

13. Yoerdem, O. S., Papila, M. and Menceloglu, Y. Z. (2008). Effects of electrospinning parameters on polyacrylonitrile nanofiber diameter: an investigation by response

surface methodology. Materials and Design, 29(1), 34-44.

Author’s Biography:

Johannes Ledolter is the C. Maxwell Stanley Professor of International Operations Management at the University of Iowa and a Professor of statistics at the Vienna University of Economics and Business. He received a Ph.D. in statistics from the University of Wisconsin-Madison. He is an elected fellow of the American Statistical Association and the International Statistical Institute. He is the author of books on time series analysis and forecasting, engineering statistics, statistical methods for process improvement, regression, and design of experiments.

208 Ledolter

Figure 11. Normal probability plot and Pareto chart of the estimated effects of the mean of the pull force in the


Figure 12. Normal probability plot and Pareto chart of the estimated effects of the standard deviation of the pull force in the


Acknowledgments

I would like to thank the two referees for their thoughtful and detailed comments which improved the paper greatly.

References

1. Box, G. E. P. and Behnken, D. W. (1960). Some new three level designs for the study of quantitative variables. Technometrics, 2, 455-475.

2. Evans, J. R. (2010). Statistics, Data Analysis, and Decision Modeling, th4 edition. Upper Saddle River, Prentice Hall, New Jersey.

3. Hou, T. H., Su, C. H. and Liu, W. L. (2007). Parameters optimization of a nanoparticle wet milling process using the Taguchi method, response surface method and genetic algorithm. Powder Technology, 173(3), 153-162.

4. Jeng, S. L., Lu, J. C. and Wang, K. (2007). A review of reliability research on nanotechnology, IEEE Transactions on Reliability, 56, 401-410.

5. Jones, B. and Nachtsheim, C. J. (2009). Split-Plot designs: what, why, and how. Journal of Quality Technology, 41, 340-361.


6. Ledolter, J. (2010). Split-plot designs: discussion and examples. International Journal of Quality Engineering and Technology, 1(4), 441-457.

7. Lin, T. Y., Davison, K. L., Leong, W. S., Chua, S., Yao, Y. F., Pan, J. S., Chai, J. W., Toh, K. C. and Tjiu, W. C. (2003). Surface topographical characterization of silver-plated film on the wedge bondability of leaded IC packages. Microelectronics Reliability, 43, 803-809.

8. Lu, J. C., Jeng, S. L. and Wang, K. (2009). A review of statistical methods for quality improvement and control in nanotechnology. Journal of Quality Technology, 41, 148-164.

9. Montgomery, D. C. (2008). Design and Analysis of Experiments, th7 edition. Wiley, New York.

10. Nazzal, S., Nutan, M., Palamakula, A., Shah, R., Zaghloul, A. A. and Khan, M. A. (2002). Optimization of a self-nanoemulsified tablet dosage form of ubiquinone using response surface methodology: effect of formulation ingredients. International Journal of Pharmaceutics, 240(1-2), 103-114.

11. Prakobvaitayakit, M. and Nimmannit, U. (2003). Optimization of polylactic-co-glycolic

acid nanoparticles containing itraconazole using 32 factorial design. AAPS

PharmSciTech, 4(4), 1-9.

12. Wu, C. F. J. and Hamada, M. (2000). Experiments: Planning, Analysis, and Parameter Design Optimization. Wiley, New York.

13. Yoerdem, O. S., Papila, M. and Menceloglu, Y. Z. (2008). Effects of electrospinning parameters on polyacrylonitrile nanofiber diameter: an investigation by response

surface methodology. Materials and Design, 29(1), 34-44.

Author’s Biography:

Johannes Ledolter is the C. Maxwell Stanley Professor of International Operations Management at the University of Iowa and a Professor of statistics at the Vienna University of Economics and Business. He received a Ph.D. in statistics from the University of Wisconsin-Madison. He is an elected fellow of the American Statistical Association and the International Statistical Institute. He is the author of books on time series analysis and forecasting, engineering statistics, statistical methods for process improvement, regression, and design of experiments.

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