Industrial Applications of Industrial Applications of Experimental DesignExperimental Design

John BorkowskiJohn BorkowskiMontana State UniversityMontana State University

University of Economics and FinanceUniversity of Economics and FinanceHCMC, VietnamHCMC, Vietnam

Outline of the PresentationOutline of the Presentation

1. Motivation and the Experimentation Process

2. Screening Experiments

3. 2k Factorial Experiments

4. Optimization Experiments

5. Mixture Experiments

6. Final Comments

MotivationMotivation

In industry (such as manufacturing, pharmaceuticals, agricultural, …), a common goal is to optimize production while maintaining quality and cost of production.

To achieve these goals, successful companies routinely use designed experiments.

Properly designed experiments will provide information regarding the relationship between controllable process variables (e.g., oven temperature, process time, mixing speed) and a response of interest (e.g. strength of a fiber, thickness of a liquid, color, cost…).

The information can then be used to improve the process: making a better product more economically.

The resulting economic benefits of using designed experiments include:

• Improving process yield• Reducing process variability so that products more

closely conform to specifications• Reducing development time for new products• Reducing overall costs • Increasing product reliability• Improving product design

MotivationMotivation

The Experimentation ProcessThe Experimentation Process

Defining Experimental ObjectivesDefining Experimental Objectives The first and most important step in an

experimental strategy is to clearly state the objectives of the experiment.

The objective is a precise answer to the question “What do you want to know when the experiment is complete?

When researchers do not ask this question they may discover after running an experiment that the data are insufficient to meet objectives.

2. Screening Experiments2. Screening Experiments

The experimenter wants to determine which process variables are important from a list of potentially important variables.

Screening experiments are economical because a large number of factors can be studied in a small number of experimental runs.

The factors that are found to be important will be used in future experiments. That is, we have “screened” the important factors from the list.

2. Screening Experiments2. Screening Experiments

Common screening experiments are 1. Plackett-Burman designs2. Two-level full-factorial (2k) designs 3. Two-level fractional-factorial (2k-p) designs

Example: Improve the hardness of a plastic by varying 6 important process variables. Goal: Determine which of the six variables have the greatest influences on hardness.

Example 1: Screening 6 FactorsExample 1: Screening 6 Factors

Response: Plastic Hardness Factor Levels

Factors -1 +1

(X1) Tension control Manual Automatic

(X2) Machine #1 #2

(X3) Throughput (liters/min) 10 20

(X4) Mixing method Single Double

(X5) Temperature 200o 250o

(X6) Moisture level 20 % 30 %

Analysis of the Screening Design DataAnalysis of the Screening Design Data

Interpretation of ResultsInterpretation of Results

• The most influential factor affecting plastic hardness is temperature, followed by throughput and machine type.

• To increase the hardness of the plastic, a higher temperature, higher throughput, and use of Machine type #2 are recommended.

• Tension control, mixing method, and moisture level appear to have little effect on hardness. Therefore, use the most economical levels of each factor in the process.

• A new experiment to further study the effects of temperature, throughput and machine type on plastic hardness is recommended for further improvement.

3.3. 22kk Factorial ExperimentsFactorial Experiments

A 2k factorial design is a design such that k factors each having two levels are studied. Data is collected on all 2k combinations of

factor levels (coded as + and - ).

The 2k experimental combinations may also be replicated if enough resources exist.

You gain information about interactions that was not possible with the Plackett-Burman design.

Example 2:Example 2: 2 233 Design with 3 Replicates Design with 3 Replicates (Montgomery 2005)(Montgomery 2005)

An engineer is interested in the effects of – cutting speed (A) (Low, High rpm)– tool geometry (B) (Layout 1 , 2 )– cutting angle (C) (Low, High degrees)

on the life (in hours) of a machine tool

Two levels of each factor were chosen Three replicates of a 23 design were run

Experimental Design with DataExperimental Design with Data

Factors• A : cutting speed• B : tool geometry• C : cutting angle

ANOVA Results from SASANOVA Results from SASA: cutting speed B: tool geometry C: cutting angleA: cutting speed B: tool geometry C: cutting angle

Maximize Hours at B=+1 C=+1 A= -1Maximize Hours at B=+1 C=+1 A= -1B: tool geometry C: cutting angle B: tool geometry C: cutting angle A: cutting speed A: cutting speed

Layout 2 High LowLayout 2 High Low

3. Optimization Experiments3. Optimization Experiments

The experimenter wants to model (fit a response surface) involving a response y which depends on process input variables V1, V2, … Vk.

Because the exact functional relationship between y and V1, V2, … Vk is unknown, a low order polynomial is used as an approximating function (model).

Before fitting a model, V1, V2, … Vk are coded as x1, x2, …, xk. For example:

Vi = 100 150 200

xi = -1 0 +1

4. Optimization Experiments4. Optimization Experiments

The experimenter is interested in:

1. Determining values of the input variables V1, V2, … Vk. that optimize the response y (known as the optimum operating conditions). OR

2. Finding an operating region that satisfies product specifications for response y.

A common approximating function is the quadratic or second-order model:

2

1i1

1

110

)(i

k

iiji

k

ijij

k-

ii

k

ii

xbxxbxbbf

x

Example 3: Approximating FunctionsExample 3: Approximating Functions

The experimental goal is to maximize process yield (y).

By maximizing yield, the company can save a lot of money by reducing the amount of waste.

A two-factor 32 experiment with 2 replicates was run with:

Temperature V1: Uncoded Levels 100o 150o 200o x1 Coded Levels -1 0 +1

Process time V2: Uncoded Levels 6 8 10 minutes x2 Coded Levels -1 0 +1

True Function: True Function: yy = 5+ e = 5+ e((.5x.5x11 – 1.5x– 1.5x22))

Fitted function Fitted function (from SAS) (from SAS)

Predicted Maximum Yield (y) at xPredicted Maximum Yield (y) at x11= +1 , x= +1 , x22= -1= -1

(or, Temperature = 200(or, Temperature = 200oo , Process Time = 6 minutes) , Process Time = 6 minutes)

Central Composite Design Box-Behnken DesignCentral Composite Design Box-Behnken Design

(CCD) (BBD)(CCD) (BBD)

Factorial, axial, and Centers of edges andFactorial, axial, and Centers of edges andcenter points center pointscenter points center points

Example 4: Central Composite Design Example 4: Central Composite Design (Myers 1976)(Myers 1976)

The experimenter wants to study the effects of

• sealing temperature (x1) • cooling bar temperature (x2) • polethylene additive (x3)

on the seal strength in grams per inch of breadwrapper stock (y).

The uncoded and coded variable levels are

- -1 0 1 .

x1 204.5o 225o 255o 285o 305.5o

x2 39.9o 46o 55o 64o 70.1o

x3 .09% .5% 1.1% 1.7% 2.11%

Example 4: Central Composite DesignExample 4: Central Composite Design

Ridge AnalysisRidge Analysis of Quadratic Modelof Quadratic Model (using SAS)(using SAS)

Predicted Maximum atPredicted Maximum at x x11=-1.01 x=-1.01 x22=0.26 x=0.26 x33=0.68=0.68

Further interpretation:Further interpretation:The predicted maximum occurs at coded levels

of x1=-1.01 x2=0.26 x3=0.68. These correspond to

sealing temperature of 225o, cool bar temperature of 57.3o, and polyethelene additive of 1.51%.Note how flat the maximum ridge is around this

maximum. That implies there are other choices of sealing temperature, cool bar temperature, and additive % that will also give excellent seal strength for the breadwrapper.

Pick that combination that minimizes cost.

5. Mixture Experiments5. Mixture Experiments

Goal: Find the proportions of ingredients (components) of a mixture that optimize a response of interest.

3-in-1 coffee mix has 3 components: coffee,

sugar, creamer. What are the proportions of

the components that optimize the taste? Major applications: formulation of food and drink

products, agricultural products (such as fertilizers), pharmaceuticals.

Mixture ExperimentsMixture Experiments

A mixture contains q components where xi is the proportion of the ith component (i=1,2,…, q)

Two constraints exist: 0 ≤ xi ≤ 1 and Σ xi = 1

Mixture Experiment ModelsMixture Experiment Models

Because the level of the final component can written as

xq = 1 – (x1 + x2 + + xq-1)

any response surface model used for independent factors can be reduced to a Scheffé model. Examples include:

Example of a 3-Component Mixture DesignExample of a 3-Component Mixture Design

Analysis of a 3-component Mixture ExperimentAnalysis of a 3-component Mixture Experiment

4-Component Mixture Experiment with Component 4-Component Mixture Experiment with Component Level Constraints Level Constraints (McLean & Anderson 1966)(McLean & Anderson 1966)

Goal:Goal: Find the mixture of Mg, NaNO Find the mixture of Mg, NaNO33, SrNO, SrNO33, and , and

Binder that maximize brightness of the flare.Binder that maximize brightness of the flare.

6. Final Comments6. Final Comments

Screening experiments 2k and 2k-p experiments Optimization experiments Mixture experiments Other applications:

• Path of steepest ascent (descent) to locate a process maximum (minimum).

• Experiments with mixture and process variables.• Repeatability and reproducability designs for statistical

quality and process control studies.