Industrial Applications of Experimental Design John Borkowski Montana State University University of Economics and Finance HCMC, Vietnam

Outline of the Presentation 1.Motivation and the Experimentation Process 2.Screening Experiments 3.2 k Factorial Experiments 4.Optimization Experiments 5.Mixture Experiments 6.Final Comments

Motivation  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 Motivation

The Experimentation Process

Defining 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 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 Experiments  Common screening experiments are 1.Plackett-Burman designs 2.Two-level full-factorial (2 k ) designs 3.Two-level fractional-factorial (2 k-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 Factors Response: Plastic Hardness Factor Levels Factors (X 1 ) Tension control Manual Automatic (X 2 ) Machine #1 #2 (X 3 ) Throughput (liters/min) (X 4 ) Mixing method Single Double (X 5 ) Temperature 200 o 250 o (X 6 ) Moisture level 20 % 30 %

Analysis of the Screening Design Data

Interpretation 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.

3.2 k Factorial Experiments  A 2 k factorial design is a design such that  k factors each having two levels are studied.  Data is collected on all 2 k combinations of factor levels (coded as + and - ).  The 2 k 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: 2 3 Design with 3 Replicates (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 2 3 design were run

Experimental Design with Data Factors A : cutting speed B : tool geometry C : cutting angle

ANOVA Results from SAS A: cutting speed B: tool geometry C: cutting angle

Maximize Hours at B=+1 C=+1 A= -1 B: tool geometry C: cutting angle A: cutting speed Layout 2 High Low

3. Optimization Experiments  The experimenter wants to model (fit a response surface) involving a response y which depends on process input variables V 1, V 2, … V k.  Because the exact functional relationship between y and V 1, V 2, … V k is unknown, a low order polynomial is used as an approximating function (model).  Before fitting a model, V 1, V 2, … V k are coded as x 1, x 2, …, x k. For example: V i = x i =

4. Optimization Experiments The experimenter is interested in: 1. Determining values of the input variables V 1, V 2, … V k. 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:

Example 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 3 2 experiment with 2 replicates was run with: Temperature V 1 : Uncoded Levels 100 o 150 o 200 o x 1 Coded Levels Process time V 2 : Uncoded Levels minutes x 2 Coded Levels

True Function: y = 5+ e (.5x 1 – 1.5x 2 ) Fitted function (from SAS)

Predicted Maximum Yield (y) at x 1 = +1, x 2 = -1 (or, Temperature = 200 o, Process Time = 6 minutes)

Central Composite Design Box-Behnken Design (CCD) (BBD) Factorial, axial, and Centers of edges and center points center points

Example 4: Central Composite Design (Myers 1976)  The experimenter wants to study the effects of sealing temperature (x 1 ) cooling bar temperature (x 2 ) polethylene additive (x 3 ) on the seal strength in grams per inch of breadwrapper stock (y).  The uncoded and coded variable levels are -  . x o 225 o 255 o 285 o o x o 46 o 55 o 64 o 70.1 o x 3.09%.5% 1.1% 1.7% 2.11%

Example 4: Central Composite Design

Ridge Analysis of Quadratic Model (using SAS) Predicted Maximum at x 1 =-1.01 x 2 =0.26 x 3 =0.68

Further interpretation:  The predicted maximum occurs at coded levels of x 1 =-1.01 x 2 =0.26 x 3 =0.68. These correspond to sealing temperature of 225 o, cool bar temperature of 57.3 o, 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 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 Experiments  A mixture contains q components where x i is the proportion of the i th component (i=1,2,…, q)  Two constraints exist: 0 ≤ x i ≤ 1 and Σ x i = 1

Mixture Experiment Models  Because the level of the final component can written as x q = 1 – (x 1 + x x q-1 ) any response surface model used for independent factors can be reduced to a Scheffé model. Examples include:

Example of a 3-Component Mixture Design

Analysis of a 3-component Mixture Experiment

4-Component Mixture Experiment with Component Level Constraints (McLean & Anderson 1966) Goal: Find the mixture of Mg, NaNO 3, SrNO 3, and Binder that maximize brightness of the flare.

6. Final Comments  Screening experiments  2 k and 2 k-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.