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How to Plan a Six Sigma 2k Factorial Experiment

Like in most other endeavors, time spent planning for Six Sigma is rewarded with better results in a shorter period of time. Planning 2k factorial experiments follows a simple pattern: choosing the factors you want to experiment with, establishing the high and low levels for those factors, and creating the coded design matrix.

Select the experiment factors

The first thing to do is identify the input variables, the Xs, that you’ll include in your experimental investigation. The factors you include should be potential contributors to the output Y you’re investigating and should be ones that are critical. How many factors you want in your experiment guides you in choosing the right experimental design. 2k factorial experiments work best when you have between two and five Xs.

If you have over five Xs in your experiment, full 2k factorial experiments become relatively inefficient and can be replaced with pared down versions called fractional factorials, or with other screening designs. One good strategy is to include all potential Xs in a first screening experiment — even the ones you’re skeptical about.

You then use the analysis of the experiment results to tell you objectively, without any guessing, which variables to keep pursuing and which ones to set aside. Remember, in Six Sigma, you let the data do the talking.

Plackett-Burman experiment designs are an advanced method you may hear about for efficiently screening dozens of potential Xs. Although they don’t reveal all the detailed knowledge provided by a 2k factorial design, Plackett-Burman experiments quickly identify which experimental variables are active in your system or process. You then follow these screening studies up with more detailed characterization experiments.

Set the factor levels

2k factorial experiments all have one thing in common: They use only two levels for each input factor. For each X in your experiment, you select a high and a low value that bound the scope of your investigation.

For example, suppose you’re working to improve an ice cream carton filling process. Each filled half-gallon carton needs to weigh between 1,235 and 1,290 grams. Your Six Sigma work up to this point has identified ice cream flavor, the time setting on the filling machine, and the pressure setting on the filling machine as possible critical Xs to the Y output of weight.

For each of these three factors, you need to select a high and a low value for your experiment. With only two values for each factor, you want to select high and low values that bracket the expected operating range for each variable. For the ice cream flavor variable, for example, you may select vanilla and strawberry to book-end the range of possible ice cream consistencies.

Variable Symbol Low Setting High Setting
Ice cream flavor X1 Vanilla Strawberry
Fill time (seconds) X2 0.5 1.1
Pressure (psi) X3 120 140

2k experiments are intended to provide knowledge only within the bounds of your chosen variable settings. Be careful not to put too much credence on information inferred outside these original boundaries.

Explore experimental codes and the design matrix

With the experiment variables selected and their low and high levels set, you’re ready to outline the plan for the runs of your experiment. For 2k factorial experiments, you have 2k number of unique runs, where k is the number of variables included in your experiment.

For the ice cream carton filler example, then, you have 23 = 2 × 2 × 2 = 8 runs in the experiment because you have three input variables. For an experiment with two variables, you have 22 = 2 × 2 = 4 runs, and so on.

Each of these 2k experimental runs corresponds to a unique combination of the variable settings. In a full 2k factorial experiment, you conduct a run or cycle of your experiment at each of these unique combinations of factor settings. In a two-factor, two-level experiment, the four unique setting combinations are with

  • Both factors at their low setting

  • The first factor at its high setting and the second factor at its low setting

  • The first factor at its low setting and the second factor at its high setting

  • Both factors at their high setting

These groupings are the only ways that these two factors can combine. For a three-factor experiment, eight such unique variable setting combinations exist.

A quick way to create a complete table of an experiment’s run combinations is to create a table called the coded design matrix. Make a column for each of the experiment variables and a row for each of the 2k runs. Using –1s as code for the low variable settings and +1s as code for the high settings, fill in the left-most variable column cells with alternating –1s and +1s.

Repeat the process with the next column to the right, this time with alternating pairs of –1s and +1s. Fill in the next column to the right with alternating quadruplets of –1s and +1s, and so on, repeating this process from left to right until, in the right-most column, you have the first half of the runs marked as –1s and the bottom half listed as +1s.

Run X1 X2 X3
1 –1 –1 –1
2 +1 –1 –1
3 –1 +1 –1
4 +1 +1 –1
5 –1 –1 +1
6 +1 –1 +1
7 –1 +1 +1
8 +1 +1 +1

Remember that these three factors are coded values; when you see a –1 under the X1 column, it really represents a discrete value, such as “vanilla” in the ice cream experiment; a +1 really represents the other value, like “strawberry.”

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