Experiment - DOE

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Designed Experiment (DOE)


Purpose

Experiments (DOE) are used to identify, verify, and optimize the influence of the leverage variables associated with a process.


Anatomy

Design of Experiments

Reference: The Vision of Six Sigma: Supplier Breakthrough

Terminology

A. The first step is to determine whether or not a statistical experiment is actually justified or needed.

B. Next, determine the objectives of the experiment.

C. Determine the Factors, or independent variables, to be studied.

D. If the number of Factors is large, it may necessitate a screening experiment.

E. Conduct a screening experiment of the number of Factors is greater than 8.

F. If a high degree of resolution is not required, then a Characterization experiment may be conducted.

G. Conduct a Characterization experiment.

H. If a high degree of resolution is required, conduct an optimization experiment.

I. If no practical effects are observed from the experiment, then the Factors should be re-considered and the experiment re-run.

J. If the objectives have been met, then the experiment is concluded, otherwise the assumptions should be re-visited and the experiment re-run.


Major Considerations

Time, desired accuracy, and cost are major factors in determining whether to proceed with an experiment (DOE), or whether to conduct a fractional experiment.

Careful selection of the experimental factors is critical to the success or failure of the experiment.

Application Cookbook

1. Determine the nature of the problem to be investigated by the experiment.

2. Establish the goals and objectives of the experiment.

3. Select the response variable(s) for the experiment.

4. Select the independent variable(s), or factors.

5. Choose the factor levels .

6. Select the experimental design.

7. Conduct the experiment and collect the data.

8. Analyze the data.

9. Draw the experimental conclusions.

10. Achieve the objective.

DOE Factors and Levels


Purpose

To describe the independent variables (X1, X2, …, XN) of a process when conducting a study of their effect on a dependent variable (Y) with tools such as ANOVA and Design of Experiments (DOE).

Anatomy

Designed Experiment Factors

Reference: Juran’s Quality Control Handbook

Terminology

A. Process inputs (independent variables – see concept Variables – Dependent and Independent).

B. Factor – Input or independent variable name such as Operator, Material type, Pressure, Policy, etc. The objective of the Breakthrough Strategy is to identify and contain the factors that have the greatest effect on the CT characteristic.

C. Levels – Set values for a factor in a study. In most cases the factors are studied when set at two levels. It is possible to increase the number of levels (3, 4, etc.), but this increases the complexity of the study.

A factor is said to be random when the levels are randomly selected from a larger set of levels. In this case, the resultant analysis accounts for random effects of the factor, and conclusions can be drawn about those levels not considered in the analysis.

A factor is said to be fixed when the levels are specifically assigned. In this case, subsequent analysis of the response would report on the fixed effects of the levels, and the conclusions constrained to only those levels present in the analysis.

D. Process output (dependent variable – see concept Variables – Dependent and Independent

Full Factorial Experiment


Purpose

A Full Factorial Experiment is a rigorous experiment where all possible combinations of the Factors at each of the chosen levels are tested. If “k” Factors are tested at “m” levels, then there would be mk experimental runs made.

Anatomy

Full Factorial Experiment

Reference: Juran

Terminology

A. The experimental run number. For the example shown, 3 Factors (A,B, and C) and 2 levels (-1 and 1), means 23, or 8, experimental runs.

B. The Yates designation of the experimental run.

C. The different contrasts, representing the main effects (A, B, and C) and the second order interactions (AB, BC, and AC), and the full interaction ABC. The example is shown in Yates Standard Order.

D. The experimental Response, giving the value of the dependent variable at each run of the experiment. The experimental treatments for each run.

E. Factor levels.

Fractional Factorial Experiment


Purpose

A Fractional Factorial Experiment is a sub-set of a Full Factorial DOE, where only a selected fraction of all the possible combinations of design factor levels are run.

It is typically performed when it is impractical or too expensive to run a Full Factorial DOE, such as when there are a relatively large number of Factors.

While not as precise as Full Factorial Experiment, because a certain amount of information will be lost, the impact is minimal. The trade-off is that information regarding third-order and higher interactions are lost, though these interactions are usually considered negligible.

Anatomy

Fractional Factorial Experiment

Terminology

A. Full Factorial DOE – 3 Factors at 2 Levels, requiring 23 = 8 runs.

B. Fractional Factorial Experiment, using the third order interaction ABC at its highest level as the generator, requiring 23-1=4 runs.

C. Second order interaction effects which are confounded with the main effects, and whose effects cannot be separated.

D. Main Factor effects.

E. Design generator, chosen to define the Fractional Factorial Experiment. In this example, the generator is ABC at its high (+1) setting. Typically, generators are chosen from the highest-order interactions, because the action of selecting the generator means its effect will be lost in the analysis.

Notation – Two-Level Factorial Experiment


Purpose

To provide a standard notation for describing a Two-Level Full or Fractional Factorial Experiment.

Anatomy

DOE Notation

Reference: Statistics for Experimenters

Terminology

A. Number of DOE Factor Levels. This example represents a two-level factorial experiment, but other factor levels are possible.

B. Resolution of Experiment, generally expressed as a Roman numeral (see Concept – Resolution).

C. Fraction of the DOE Experiment:

  • P = 1 corresponds to a half-fraction.
  • P = 2 corresponds to a quarter-fraction.
  • P = 3 corresponds to an eighth-fraction, etc.
    Number of experimental Factors.

Factorial Experiment - Runs


Purpose

A Factorial Experiment run is the determination of the process Response, or output, for a specific combination of Factor, or process input, levels. An Experiment consists of multiple runs, whose number will depend upon the number of Factors and the type of experimental design.

Anatomy

DOE Runs

Reference: The Vision of Six Sigma

Terminology

A. The experimental Run number.

B. The Yates designation for the Run.

C. The settings (high or low) for the main process Factors.

D. The response measured for the experimental run.

Factorial Experiment - Resolution


Purpose

An experimental design of Resolution R is one in which no p-Factor effect is confounded with any other effect containing less than R-p Factors. The Resolution of a design is represented by a Roman numeral appended as a subscript.

Anatomy

DOE Resolution

Reference: Statistics for Experimenters

Terminology

A. The number of runs in the experimental design, for a 2k-p Factorial Design.

B. The number of Factors under study.

C. The Resolution of the design, where:

  • A design of Resolution III does not confound main effects with one another, but does confound main effects with two-Factor interactions.
  • A design of Resolution IV does not confound main effects and two-Factor interactions, but does confound two-Factor interactions with other two-Factor interactions.
  • A design of Resolution V does not confound main effects and two-Factor interactions with each other, but does confound two-Factor interactions with three-Factor interactions.

Factorial Experiment - Blocking


Purpose

To divide an experimental design into a series of experimental spaces or periods of time, in such a way that bias effects are negligible within the block. In other words, the variation due to noise is minimized within the block.

Anatomy

DOE Blocking

Reference: Statistics for Experimenters

Terminology

A. Experimental run number in Yates Standard Order.

B. Block number. In this example, the design is divided into two blocks, runs conducted during the day and runs conducted at night.

C. The randomized sequence of 8 runs for Block number 1 (runs conducted during the day).

D. The randomized sequence of 8 runs for Block number 2 (runs conducted during the night).

Major Considerations

Blocking can increase the precision of the DOE for factor effects by reducing the size of the error term. Blocking is also very useful when running an experiment on a process that is out of control.

Blocking should only be used for reducing error from unavoidable sources, and should not be used for dealing with avoidable sources of error which could be dealt with during the initial design of the experiment.

Minitab has the capability of generating a blocked design.

Application Cookbook

1. Select the DOE design based upon the number of Factors, levels, and desired resolution.

2. Select number of blocks based upon the experimental situation, by selecting portions of the experiment and grouping runs where certain Factors are expected to be more homogeneous than others.

3. Divide the experimental runs into the number of blocks.

4. Randomize the experiment by ensuring that the runs are conducted in random order.

Factorial Experiment - Confounding


Purpose

Main and/or interaction effects are confounded if only their combined effects, and not their individual effects, can be determined from the experimental design.

In other words, the unique effect of one contrast cannot be separated from another. Confounding is also known as Alias Structure.

Anatomy

DOE Confounding

Reference: The Vision of Six Sigma

Terminology

A. Main effect contrast Factor A.

B. Interaction of contrast Factors B and C, confounded with the main effect.

Confounded contrasts can be seen in an experimental design when both columns have the same signs at each experimental run.

Factorial Experiment - Randomization


Purpose

Factorial DOE experiment runs, and the allocation of the experimental resources, should be done in random order to average out the effects of extraneous sources of variation, placing the effects of noise throughout the experiment.

Anatomy

DOE Randomization

Reference: Statistics for Experimenters

Terminology

A. Three factor, two-level Full Factorial Experiment, shown unrandomized, in Yates Standard Order.

B. Same Full Factorial Experiment, randomized.


Major Considerations

The consequences of an erroneous conclusion based upon a non-randomized design, often justify the cost and complexity of performing randomization.

Completely randomizing an experiment, changing Factor settings each time, can add significant cost to the experiment. Randomization of run order is best performed using Minitab, which allows variation of the base number for the random number generation.

Application

1. Select desired experimental design, based upon desired results, number of Factors, number of levels, etc.

2. In Minitab, select Stat>DOE>Create Factorial Design, and enter desired “Type of Design” and “Number of Factors”.

3. Click on “Designs…” button, and select the specific design, and the number of center points, replicates and blocks.

4. Back at the “Factorial Designs” window, click on “Options…” button, and ensure that the “Randomize Runs” button is set to on. This will ensure that when Minitab generates the specific design requested, it will list the runs in completely random order, as shown by the random Yates order column. Minitab will still show the “Run Order” in numerical sequence.

Factorial Experiment - Replication


Purpose

DOE "Replication" is the systematic duplication of a series of experimental runs, in order to increase precision or to provide the means for measuring precision by calculating the experimental error. For Robust Design, replication allows us to analyze the response mean and variance.

Anatomy

DOE Replication

Reference: Statistics for Experimenters

Terminology

A. First experimental replicate.

B. Second experimental replicate.

C. Experimental Factors are duplicated exactly from one replicate to the next.


Major Considerations

Replication should be carried out in such a way that variation among replicates can provide an accurate measure of errors that affect comparisons between runs. Replication is best performed using Minitab.

Application Cookbook

1. Select desired experimental design, based upon desired results, number of Factors, number of levels, etc.

2. In Minitab, select Stat>DOE>Create Factorial Design, and enter desired “Type of Design” and “Number of Factors”.

3. Click on “Designs…” button, and select the specific design, and the number of center points, replicates and blocks.

4. If the runs are not randomized, then replicating the design n times will cause n identical designs to be generated in order, “stacked” one after the other. If randomization is selected, then the replicated runs will be randomized.

From DOE to Six Sigma Tools.

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