Блог Анкара

In This Chapter

Looking at the advantages of planned experimentation ^ Examining experimental considerations and terminology ^ Exploring the 2* full factorial experiments

The point of Six Sigma is improvement, and you are now at the point in the DMAIC roadmap where you synthesize improvements and/or reconfigure your system or process to be better. Six Sigma offers extremely powerful tools

To aid you in your improvement efforts. Chief among these tools is experimentation. In Six Sigma, you first design an experiment before you carry it out. Then you follow it up with analysis to uncover previously hidden knowledge.

Design of Experiments (or DOE For short) has always been at the technical heart of Six Sigma. As necessity is the mother of invention, the field of DOE

Has matured due to the need to understand, and then improve, the world around you. This chapter gives you the lowdown.

Whij Experiment} The Improvement Porter Of Six Sigma Experiments

How is improvement achieved? The spark of improvement comes from a

Curious mind, trying to figure out what it is that makes things tick.

What is an experiment, amju’auj}

In an Observational study (covered in Chapter 7), you simply act as an outside

Observer, recording data as it happens, trying to glean understanding from

Careful review of the world around you. In these types of studies, you just let

The XS (inputs) of the system or process you are working on take whatever

Values they do. And as this plays out, you record the corresponding process

Output Y Values.

Experiments, On the other hand, are different from observational studies in one fundamental way: In experiments, instead of just letting the Xs of the process you are studying take on whatever values they do, you purposely

Setand control the values that the XS take on. In an experiment, you actively

Control and modify the process being studied.

Experiments offer a greater level of insight and knowledge than observational studies do. Think of the many observational studies performed for decades

In fields like medicine, education, economic policy, diet, and so on. Dozens

Upon dozens of observational studies have only added incremental knowledge to these areas; there is, for example, still a lot of debate about what specific foods are part of a healthy diet. Because you purposefully control the factors, the amount of specific knowledge you get out of an experiment

Almost always exceeds what you gain from observational studies. For that

Reason, designing and analyzing experiments — in spite of the complexity of the topic — has always been one of the core pillars of Six Sigma breakthrough

Improvement.

The purpose of Six Sigma experiments

Every experiment in Six Sigma is targeted at better understanding the Y = f(X) Relational foundation between the inputs and outputs of the process or

System being improved. Better understanding from experimentation includes

Is Knowing which input Xs have a significant effect on the output Y and

Knowing which XS are insignificant.

Formulating and quantifying the mathematical relationship between the

Significant Xs and the output Y.

F Statistically confirming that a change or improvement has been made to a process or system.

Discovering where to set the values of the significant XS so that they combine together to produce the optimal output value of Y.

Few activities in Six Sigma offer as much insight and change horsepower as experiments do. That’s because properly designed experiments reveal, quantify, and confirm the underlying Y = f(X) relationship of a process or system.

Experimenting With Words

The field of planning and analyzing experiments is much older than Six Sigma.

As a result, a few somewhat unique terms are used. Here are some of the interesting terms you need to know — along with their relation to Six Sigma.

Response Is the term used for the output of the process that you investigate in the experiment. In Six Sigma terms, the response is synonymous

With the Y In the Y = F(X) Equation. The whole point of the experiment is to figure out how the Xs combine together to effect the response, or Y.

F The input characteristics, or variables you purposely control during

Theexperiment, are called experimental Factors. Sometimes, they’re also

Called Conditions, variables, Or simply Inputs. In all cases, the experimental factors are the Xs in the Six Sigma Y = f(X) equation.

In your experiment, you choose two or more values for each of the experimental factors. These values are called the Levels For that factor.

Planning your experiment includes deciding how many levels you need to use for each factor.

V Processes and systems have variation. Part of experimentation is

Repeating your whole experiment, or parts of it, to understand how

Much variation there actually is. These types of repetitions are called

Replications. Deciding what part of your experiment needs to be replicated

And deciding how many replications there will be is part of developing

Your experiment plan.

V Every experiment is made up of a series of Runs. Each experimental run consists of a unique, predetermined set of values for each of the factors.

You then conduct the process or system through one cycle with those input values, and the output is recorded. That is a run in an experiment.

The end game of Six Sigma experiments

It has been said, "Knowledge is power." Six Sigma experiments are a confirmation of that statement.

The power of Six Sigma experiments lies in their ability to formulate, quantify and validate the Y = f(X) Relationship of a process or system. Knowing the form and details of Y = f(X) For a system, you literally have a window into the past, present, and — most importantly — the future.

After wrapping up an experiment, you have in your hands a Y = f(X) Equation

That identifies each critical input XAnd quantifies its influence on the output Y. For example, if you are working on a marketing plan to improve brand awareness (that’s the output Y), A Six Sigma experiment provides an equation that tells you which type of advertisements — newspaper, radio, TV, Internet, and so on — and how many of each type to run (the input Xs) to reach a specified

Improvement goal. Or if you are managing the production ofplastic seals that must meet a minimum tear strength requirement (the Y), After proper experimentation, you have an equation that tells you exactly where to set the mold press temperature (X1), how much pigment to add (X2), and the correct operating temperature of the mold press (X3). In all cases, whether they involve

Continuous or attribute data, successful experimentation reveals detailed, specific knowledge of which input Xs influence the output Y — and by how much.

With this level of system or process knowledge, your operational focus immediately switches from passively watching the output and hoping for success to actively monitoring and controlling identified key inputs, knowing that

Your purposeful management and control of these inputs will always lead to the desired process outcome. This is where you open the door to the new world of breakthrough performance.

Look Before \lou Leap: Experimental

Considerations

Trial and error — tinkering with the input knobs of a process or system — is temptingly simple. We all have a desire to jump in and quickly fix a problem. In the long-run, though, careful planning almost universally leads you to a

Quicker and better solution.

Frankenstein should have planned

How should you approach experimentation? Where do you start? The trial-and-error approach

Many people approach experimentation by rolling up their sleeves and jumping into an unstructured exploration of the experimental variables and their resulting output: Tweak the knobs, adjust the settings, and observe the results.

Often, judgment and intuition are the basis for steering the exploration and

Interpreting the findings.

For obvious reasons, however, this unstructured, haphazard approach

Rarely increases knowledge. Every once in a while, you may get lucky, but

Thisapproach, isunreliable.

The one-factor-at-a-time approach

At the other end of the spectrum, there’s a structured approach: Isolate a single input variable and study its effect on the output; carefully hold all other

Factors constant while the selected input variable is incremented across an

Exploratory range of operation. Then repeat this meticulous scan for each of

The input variables.

The downfall of this method is twofold:

The one-factor-at-a-time approach is inefficient and expensive. A scan, conducted one factor at a time, of the possible operating range for each input variable leads to a huge amount of experimental runs. Unless you have only one variable in your system, this approach becomes unwieldy

And wastefully expensive.

The results of one-factor-at-a-time experiments are often misleading.

When you isolate individual variables, you automatically negate the possibility of two or more factors combining together to affect the outcome. But these types of interaction effects are an unavoidable part of reality. Think of baking a cake. A delicious-tasting outcome (the Y) Is a function of several input Xs — like "amount of flour," "number of eggs," "oven temperature," "baking time," and so on. Obviously, the right value for the

Variable of "baking time" depends on the setting for "oven temperature."

How hot the oven is and how long you leave the cake in the oven are two

Input variables that interact with each other. One-factor-at-a-time experiments will never uncover this essential relationship. The danger is that

You draw unfounded conclusions from your experiment — or miss important information altogether.

Use the one-factor-at-a-time approach only when you have a process or

System with a single input variable. This approach works with a single-x

System because there is no possibility of an interaction effect.

The Six Sigma approach — doing more than one thing at a time

Now you know the drawbacks of the haphazard approach and the one-factor-at-a-time approach. Is there a better way? Six Sigma uses a reliable approach

To experimentation that:

Is Efficiently accumulates information about a process or system

Provides valid insights, including knowledge regarding variable interactions

F Quantifies the amount of knowledge discovered about a system as well

As the amount of knowledge that remains unknown

The experimental approach you use in Six Sigma incorporates the best practices from the various disciplines of science. Over the years, scientists have

Developed experiment plans that return a vast amount of knowledge in a very

Efficient way. The key elements of the Six Sigma approach include:

F Planning out the experiment before you conduct it. "Look before you leap" is a mantra of every good experimenter. Careful planning always

Increases the value of your experiment results while minimizing the amount of work and money you have to invest.

F Exploring the effect of more than one input variable at a time. This

Allows you to be efficient while at the same time capturing unsuspected and sometimes hard-to-find interaction effects.

F Minimizing the number of required runs in your experiment. It’s surprising how much you can get out of a small number of properly planned

Experimental runs.

T-" Replicating key experiment conditions to assess variation. A part of every experiment is understanding how much of your system’s or process’s behavior is deterministic and how much is random variation.

Accounting for known and unknown factors that you are not directly including in your experiment. You can never take everything into consideration in your experiment. There are ways, however, to keep these nuisance factors from clouding the results of your experiment.

Simple, sequential, and systematic is best

Rome wasn’t built in a day. Properly planned experiments fit into a larger strategy of iteratively converging to an ideal improvement solution.

The problem ©ith boil-the-ocean super-experiments

The power of designed experiments is intoxicating. Be careful, though, not to

Get carried away. There is a temptation to try to solve everything in one fell

Swoop, using a big, well-designed super-experiment. But putting all your eggs into one experimental basket has some definite drawbacks.

F Creating a single super-experiment based only on the knowledge you

Have Before The experiment begins necessitates that you include all the variables that you suspect are contributing to the situation. This always

Leads to a long list of potential Xs, and consequently, always results in a long, expensive, unwieldy experiment.

As a large super-experiment is carried out over a protracted period of

Time, there is a greater chance of unknown factors creeping in, confounding the experimental conditions and results.

With no prior knowledge, it is difficult to know what values and ranges

To assign to each experimental XInput.

Conducting an experiment takes time and money. If something goes wrong in your one super-experiment or if new information is revealed

That requires a change to your initial assumptions, you will have already

Consumed your experimental budget and resources. The progressive, iterative approach

An efficient and consistently successful approach to experimentation follows

A progressive and iterative approach.

Screening experiments: At this first stage, experiments are designed to handle a large number of factors or variables. When you first start investigating a process or system, you identify all the possible Xs that may be

Influencing the output Y. The whole point of screening experiments is

To quickly verify which of these factors has a significant effect on the output.

Characterizing experiments: When you have screened out the unimportant variables, your experiments focus on characterizing and quantifying the effect of the remaining critical few. These characterization experiments reveal what form and what magnitude the critical factors take in

The Y = f(X) Equation for your process or system.

Optimization experiments: After characterizing your process or system,

The final step is to conduct experiments that teach you what the best settings are for the input variables to meet your desired outcome goal.

Your goal may be to maximize or to minimize the value of the output. Or it may be to hit a certain target level. More often, your goal is simply to

Minimize the amount of variation in the output Y. Optimization experiments find the best settings of the XS to meet your Y Goal.

The purpose of each of these types of experiments — screening, characterizing, and optimizing — are very different. The form and plan of the experiments you conduct at each of these stages, therefore, are necessarily different from

Each other.

Figure 9-1 shows the progressive and iterative approach used in Six Sigma

Experiments.

2k Factorial Experiments

Design and analysis of experiments is a topic large enough for a whole For Dummies Book by itself. To get you quickly up to speed, however, the following section of this book shows you how to plan, conduct, and analyze the most common type of experiment in Six Sigma — the 2" factorial (pronounced two to the k). 2" factorial experiments can be easily adapted to provide screening,

Characterization, or optimization information. Insights into other types of

Experiment designs and variations used in Six Sigma are offered along the way.

Plan your experiment

Like in almost all other endeavors, time spent in planning is rewarded with

Better results in a shorter period of time. Planning 2" factorial experiments

Follows a simple pattern that is outlined in the following sections.

Select the experiment factors

The first thing to do in your planning is to identify the input variables, the

XS, that you will include in your experimental investigation. The factors youinclude should all be potential contributors to the output Y You are investigating.

How many factors you want to include in your experiment guides you in

Choosing the right experimental design. 2" factorial experiments work best

When you have between two and five Xs. But if you have over five Xs in your experiment, full 2" 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 are 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.

Experience with experiments verifies the Pareto Principle Introduced in Chapter 7 — that even if you include dozens of contributing factors in your experiment, only a small number of these Xs have a significant effect on the output response. When you initially have more than four or five factors, your experiment purpose is to screen out the "trivial many" factors from the "critical few." After that, you then run characterization experiments to provide the

Detailed knowledge about the remaining critical few.

Plac"ettBurman 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 2" 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

2" Factorial experiments all have one thing in common — they use only two levels for each input factor. (That’s what the "2" in 2" stands for! The " Represents the number of factors included in your experiment.) For each X In your

Experiment you select a "high" and a "low" value that bounds the scope of your investigation.

For example, suppose you are 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 contributing 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. Table 9-1 provides a

Summary of the selected experiment variables and their values.

Table 9-1 Variable Values for the Ice Cream _Carton Filler Experiment_

Variable_Symbol_"Low" Setting "High" Setting

Ice cream flavor X Vanilla Strawberry

Fill time (seconds) X2 0.5 1.1

Pressure (psi) X3 120 140

2" experiments are intended to provide knowledge only Within The bounds

Ofyour chosen variable settings. Be careful not to put too much credence on information extrapolated outside these original boundaries.

Experimental codes and the design matrix

With the experiment variables selected and their "low" and "high" levels set, you are now ready to outline the plan for the runs of your experiment. For 2"

Factorial experiments, there will be 2" number of unique runs, where " Is the

Number of variables included in your experiment. For the ice cream carton

Filler example, then, there will be 23 = 2 X 2 X 2 = 8 runs in the experiment,

Because there are three input variables. For an experiment with two variables

There will be 22 = 2 X 2 = 4 runs, and so on.

Each of these 2" experimental runs corresponds to a unique combination of

The variable settings. In a full 2" 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 22 = 4 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

F The first factor at its "low" setting and the second factor at its "high"

Setting

Both factors at their "high" setting

There are no other ways that these two factors can combine with their two levels. For a three-factor experiment, there are eight such unique variable setting combinations.

A quick, shorthand way to create a complete table of an experiment’s unique run combinations is to create a column for each of the experiment variables and a row for each of the 2" runs. Then, using -1s as a code for the "low" variable settings and +1s as a code for the "high" settings, start with the left-most variable column, and fill in the column cells with alternating -1s and +1s.

With the left-most column filled in, move on to the next column to the right

And repeat the process — but 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. This table of patterned +1s and -1s is called the Coded design matrix. Table 9-2 shows the coded design matrix for a three-factor experiment, such as the ice cream carton filler.

Table 9-2 Coded Design Matrix for a Three-Factor Experiment

RunX, 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 in the table; when you see a under the X1 column, it really represents a discrete value, such as "Vanilla" in the ice cream experiment; and a really represents the other

Value, like "Strawberry."

Conduct your experiment

With your experiment well planned, the act of carrying it out is easy — it’s like falling off a log. Now it’s time to roll up your sleeves and get into the scientific trenches.

Randomize: Safeguard against unknown nuisance factors

Despite your best efforts, external factors beyond the control of your selected experiment variables may creep in and influence the outcome of your experiment. These are factors (called Nuisance factors) That you haven’t foreseen,

But they have the potential to blur the clarity of your analysis and insights.

For example, in the ice cream carton filling process discussed in the preceding

Section, a rise in the ambient factory temperature during the duration of the experiment may affect the experiment outcomes and be falsely construed as a

Real effect from your selected experimental factors.

One way to compensate for these unknown nuisance variables is to Randomize The order of your experimental runs. This spreads out the otherwise concentrated or confounding potential for nuisance effects evenly and fairly over all of the experimental runs and preserves the clarity of your results.

Always randomize the order of your experiment runs. This reduces the risk of extraneous variables skewing the results of your analysis.

Randomize materials being used in your experiment, your personnel, or your

Equipment. The idea is to guarantee that only the effect of your selected factors is purposely concentrated during your experiment.

Blocking: Safeguard against known nuisance factors

When you know the source of nuisance variation that is not part of your

Selected experimental factors, you can purposely include this nuisance effect

In All Your experimental runs. In this way, you guarantee that there will be no

Bias on only a portion of your experimental settings.

In the ice cream carton filling example, you may decide to perform each

Experimental run at the same time each day. This way, the influences from

Different times of day are blocked from impacting only some of the experimental runs.

A catchy phrase may help you remember the roles of randomizing and blocking in your experiments: Block what you can and randomize against what you can’t block.

Perform the experiment and gather the data

Running the experiment is the fun part. All you have to do is follow your experimental plan, like the one shown in Table 9-3 for the ice cream carton

Filler project.

Table 9-3 Plan and Results for the Ice Cream _Carton Filler Experiment_

Run OrderX,: FlavorX2: TimeX3: PressureY

1 7-1-1 -1 1,238

2 2 +1 -1 -1 1,252

35-1 +1 -1 1,228

48+1 +1 -1 1,237

53-1 -1 +1 1,223

66+1 -1 +1 1,234

71 -1 +1 +1 1,238

84+1 +1 +1 1,250

In Table 9-3, the coded design matrix is augmented with a column showing the random order in which the experimental runs are conducted. Also, on the

Far right, a column is added to capture the outcome Y Variable for each experimental run. In Table 9-3, recorded values for the ice cream carton filling example experiment are provided.

Analyze your experiment

The purpose of analyzing your experiment is to take the experiment results

And piece together the Y = F(X) Puzzle for your process or system. How much effect does x1 have on Y? What mathematical form does this relationship take on? These are the questions that your analysis will answer.

Visualize and calculate the main effects

A Main effect Is the quantitative influence a single experiment factor has on

The response Y. There will be a main effect for each factor in your experiment.

For example, how much effect does ice cream flavor — going from "Vanilla" to "Strawberry" — have on the resulting filled weight of the carton?

The main effect of the X ice cream flavor factor is the average response of the experiment runs with X! at its "high" or "Strawberry" setting, minus the average response of the experiment runs with X at its "low" or "Vanilla" setting. To find the answer, refer to the captured values in Table 9-3. Runs 2, 4, 6, and 8 are where X is at its "high" setting. Runs 1, 3, 5, and 7 are where X is at its "low" setting. So the main effect of ice cream flavor (called Ј1) can be written mathematically as

E 1 _ 4 4

E _ 1,252 + 1,237 + 1,234 + 1,250 1,238 + 1,228 + 1,223 + 1,238 E 1 _ 4 4

E1 _ 1,243.25 _ 1,231.75

E1 _ 11.5

Figure 9-2 shows the main effect of ice cream flavor graphically. You can see

That as the ice cream flavor changes from "Vanilla" to "Strawberry," the

Carton weight changes by 11.5 grams.

Ј,: Main Effect

12441242124012381236-

Figure 9-2:

Main effect Ј1 On carton weight due to the ice cream flavor.

Vanilla

Strawberry

JT,: Ice Cream Flavor

To calculate the main effect E2 of fill time on the filled carton weight Y you can leverage the coded setting values for factor X2 in Table 9-3. Call these coded values c21, c22, and so on through c28, for each of the experimental runs. Another way to write the equation for the main effect of fill time, then, is

E 2 _ E 2 _

C2,1 Y + C2,2 Y2 + C2,3 Y3 + C2,4 Y4 + C2,5 Y5 + C2,6 Y + C2,7 Y7 + C2,8 Y.

-1)1,238-

-1) 1,252-

4

+1) 1,228-

-1) 1,237 + (-1) 1,223 -

1)1,234

1)1,238

1)1,250

4

-1,238 – 1,252 + 1,228 + 1,237 – 1,223 – 1,234 + 1,238 + 1,250

E2_1.5

Which gives a main effect of fill time of 1.5 grams.

Then using the coded setting values for X3 — C31, c32, c38 — the same procedure can be used to calculate the main effect’ of pressure E3:

E3 E3

1,238 _ 1,252 _ 1,228 _ 1,237 + 1,223 + 1,234 + 1,238 + 1,250

2.5

With the main effect of pressure being -2.5 grams.

In fact, the coded setting values can be leveraged to create a generalized equation to compute Any Effect in a 2* full factorial experiment.

E._ 1 yC. Y

2 / _ 1

Where k is the number of experiment factors and / designates which effect you’re calculating.

Figure 9-3 shows all three main effects on a single plot for comparison.

Figure 9-3:

Graphical – 1244 comparison

Of main

Effects for

The ice

Cream

Carton filling

Example.

=: 1240

■=T 1236

1232

Vanilla

Strawberry

0.5

1.1

120

140

Visually, it is easy to see that X1, the flavor of the ice cream, has the largest

Main effect on the filled weight of the cartons. (See Chapter 5 for a more detailed discussion of main effects plots.)

Visualize and calculate the interaction effects

One input variable interacting with another is always a possibility. Are there any of these type of interaction effects in the ice cream carton filling example? How do you find out?

Call the interaction effect between ice cream flavor (X) and fill time (X2) E12. What you do next is create a new column of coded setting variables that represents the interaction of factors X and X2. You do this by multiplying the coded values of X and X2 together for each experiment run. For example, c121 = c11 x c21, c122 = c12 x c22, and so on up through c128 = c18 x c2 8. Table 9-4 shows the ‘ new coded setting values for the two-variable and the three-variable interactions possible in the 23 ice cream carton filler experiment.

Table 9-4 Interaction Coded Variables for the _Ice Cream Carton Filler Experiment

Rune, c2 c3 c,2 c,3 c23 c123 Y

1 -1 -1 -1 +1 +1 +1 -1 1,238

2 +1 -1 -1 -1 -1 +1 +1 1,252

3-1 +1 -1 -1 +1 -1 +1 1,228

4 +1 +1 -1 +1 -1 -1 -1 1,237

5-1 -1 +1 +1 -1 -1 +1 1,223

6 +1 -1 +1 -1 +1 -1 -1 1,234

7-1 +1 +1 -1 -1 +1 -1 1,238

8 +1 +1 +1 +1 +1 +1 +1 1,250

With the coded values for the interaction effects, you can now use the general formula to calculate each of the possible two-variable interaction effects.

For example, the interaction effect between ice cream flavor (X1) and fill time

(X2) Is calculated as

E12 = 2T-T 2 C kjYJ

_ ( +1)1,238 + (-1)1,252 + (-1) 1,228 + ( +1) 1,237 + ( +1) 1,223 + (-1)1,234 + (-1) 1,238 + ( +1) 1,250

E12 4

1,238 – 1,252 – 1,228 + 1,237 + 1,223 – 1,234 – 1,238 + 1,250

E12 _-4-

E12 _-1.0

Or -1.0 grams effect when the X and the X2 factors are combined together.

Using the same procedure, you can calculate interaction effects for E13 and E23. You should get values of 0.0 grams and 14.0 grams, respectively. Figure 9-4

Shows all three two-variable interaction effects.

V, : Flavor

Figure 9-4:

Two-factor interactions in the ice cream carton filler example.

0.5

Interaction Effects

1.1 120

140

1242

1236

1230

1242

1236

1230

Flavor Vanilla Strawberry

Fill Time

0.5 1.1

.V3 : Pressure

In the grid layout of Figure 9-4 for the X2 – X3 interaction, you can see that the plotted effect lines have very different slopes. This is your graphical clue to know that E23 is very strong. The plotted effect lines for X1 – X2 and X1 – X3, however, have very similar slopes. It is no surprise that their calculated interaction effects, E12 and E13, are rather small.

For a three-factor experiment, there is one more interaction effect you need to compute. It is the possible interaction when all three variables are combined (E123). This may sound tricky, but it’s not because you’re using the

Coded setting values and the same general formula for calculating the effects.

E123 _ TJT-T2 C123,/ Y/ 2 / _ 1

E -1,238 + 1,252 + 1,228 – 1,237 + 1,223 – 1,234 – 1,238 – 1,250

E 123 _ 4

E123_1.5

Or 1.5 grams effect when all three factors are combined.

Which effects are significant}

Even though you can calculate all the main and interaction effects of the variables, are they all significant? Are they all necessary? The Pareto Principle (see Chapter 7) tells you that a relatively small subset of all the possible effects explains the vast majority of the output responses. So how do you

Know which effects to hold on to and which ones to cast aside?

If the factors you select for your experiment have no impact on the outcome

Y the calculated main and interaction effects will just be random — they’ll be normally distributed and centered around zero. But if any one of the effects is

Significant, it will depart from the random cluster of the rest.

The easiest way to detect this departure is graphically, by plotting all the calculated effects against a line representing a normal distribution. If a plotted effect doesn’t fit this line, you know that it is not part of the random noise, but instead is significant.

To create this graph for the ice cream carton filler example, you list all the calculated effects in rank order from smallest to largest and write down the

Rank i next to each effect. In case of ties, like between E2 and E123, you assign

The average rank to the tied effects. You can see this in Table 9-5.

As an intermediate step, you have to calculate the expected probability for

Each rank. This is called P and is in the fourth column of Table 9-5. The formula calculating the P for each row in the table is

So for the E13 effect, its expected probability, P is

PI _ ‘l1^ _ T5 _ °.357

The final step in creating the values of Table 9-5 is looking up the Z value for each intermediate P value. Using a look-up table for Z, you can see that the Z score corresponding to the P of °.357 on E13 Is -°.366.

Having filled in all the values of Table 9-5, you now simply plot the calculated Z value against each of the corresponding effect values. This is shown for the ice cream carton filler example in Figure 9-5.

Looking at Figure 9-5, it is obvious that effects E1 and E23 are very different from the rest of the effects. While E1 and E23 are not centered around zero and clearly don’t fit the expected normal probability line, all the others do.

The more complicated a potential interaction is, the less likely it is to be significant in reality. Very often, for example, two-factor interaction effects are found to be significant. Much less often, three-factor interactions are determined to be important. It is a real rarity to uncover a legitimate interaction

Effect that includes four or more factors. The more complicated an interaction effect is, the more skeptical you should be about it being real.

With just an eight-run experiment, you have determined that there are really

Only two effects that significantly effect the performance of the ice cream

Carton filler. The first is the type or flavor of ice cream being produced. Also,

The combined, interactive effect of filler time and pressure definitely impacts performance. But filler time and pressure, acting by themselves, don’t have a

Significant effect.

This is the power of Six Sigma. Rather than guessing or fumbling in the dark

For the answer, you let the data and the analysis show what is important and

What is not. In return, you look like the hero! The general form of the equation

2* factorial experiments not only reveal which factors effect the output y but they also allow you to understand the form of the Y = F(X) Equation for the system or process you are improving. At the onset, a 2* experiment investigates the possibility of all main and interaction effects being significant. (Subsequent analysis shows you which ones you can safely ignore.)

Picture in your mind a general y = F(X) Equation with a term for each main effect, a term for each interaction effect, and an overall offset effect. For the

Three-factor ice cream carton filler example, this general equation takes the

Form:

Y _ F3 ° + Ј 1 x1 + P 2 X 2 + P 3 X 3 + P12 X1 X 2 + P13 X1 X 3 + /3 23 X 2 X 3 + /3123 X1 X 2 X 3

In this general equation, each combination of the input X Variables is prefixed with a multiplier coefficient represented by the jtfs (the Greek letter beta; pronounced BAY-tah). The little subscripts at the lower right of each jO tell you which effect it corresponds to. In stuffy mathematical terms, these jtfs are called coefficients.

A two-factor system would have a general equation of

Y _ / ° + /1×1 + / 2 x 2 + P12 x1x 2

While a four-factor system would include additional terms for all the three-variable and four-variable interactions.

The A, term in all these equations represents the overall level of the process or system you are working on. No matter what you do to the setting of any of

The system variables, the system will take on at least this value. That’s why it

Is often called an Offset Or Constant Term.

Define §our \t = f(K) equation

For the system or process you are working on, the only terms of the general

Equation you need to hang on to are the ones that correspond to the effects you have found to be significant. For example, in the ice cream carton filler process, only the ice cream flavor x1 and the filler time-pressure interaction x2x3 effects were found to be significant. That leads to a simplified equation form of

Y = P „ + P X X X + P 23 X 2 X 3

But what are the values of the jHs? Again, finding these values is easier than you may think.

The value for the offset ft is simply the computed average for all the 2" Experiment runs. For the ice cream carton filler example, the average output rfor the eight experiment runs is 1,237.5, so

Ft = 1,237.5

The /lvalue for all other significant factors is found by dividing the corresponding effect value in half. That means that

Pi = T = ^ = 5.75 p 23 = ^ = 140 = 7.0

Why are the {i Coefficients half the effect value instead of the full effect value? It’s because the effect value is calculated over a span of +1 to -1 for the variable. That’s an effective distance of two, not one. Therefore, to get back to the right equation coefficient, you have to divide the calculated effect value by two.

With these coefficients calculated, you can write the Y = ice cream carton filler system:

F(X) Equation for the

Y = 1,237.5 + 5.75 X + 7.OX2X3

Armed with this equation, you can now go out to the ice cream production line and immediately correct the problem situation of this example.

4?

4E

Suppose that the weight of the filled ice cream cartons is required to be between 1,225 and 1,280 grams. If you are producing a batch of vanilla ice cream, you can plug that coded value into the equation (X = -1), and then

Plug in various coded values for X2 and X3 to calculate what your fill time and pressure settings should be on the ice cream filler machine. When you switch

Over to making strawberry ice cream, you can then pull out your equation again and know exactly how to alter your fill time and pressure settings to

Maintain the correct filled carton weight.

Be careful to plug only Coded Values into your derived Y = F(X) Equation.

\[ou’Ve Ontu lust Begun — More Topics in Experimentation

2k Full factorial experiments give you a powerful jump start into the world of improvement through DOE. But really, they are just the tip of the iceberg. As

You gain experience, you want to discover how to address more advanced topics.

Curvature: The assumption of 2* experiments is that the effects of your experimental factors is linear. Although this is often a good first approximation, there are many times when a line doesn’t fit your process or

System. For those cases, you need to design your experiment to reveal the curved nature of reality. This is usually done by including more than

Two levels for each of your experimental factors.

Replications: If you repeat your experiment, you get slightly different results. This shouldn’t surprise you. Variation, as always, is a part of everything — including your experiment. Repeating runs of your experiment (called Replications) Allows you to estimate how much of the

Observed variation in your process or system is explained by the

Derived y = F(X) Equation and how much remains unexplained.

Analysis of variance (ANOVA): Almost all experiments involve exploring, investigating, and comparing the sources of observed variations.

ANOVA is an advanced method that allows you to categorize and quantify all the various sources of variation.

F Robustness: The ability of a process or system to perform consistently in the face of variation is called Robustness. Taguchi and other experiment designs allow you to investigate and optimize your process or system so that it is as immune as possible to the ravages of variation.

Response surface methods (RSM) and optimization: The purpose of

Many experiments is to find out the best values to set the input variables at. A whole branch of the field of DOE focuses on designing and

Analyzing experiments to find the local or global optimal operation settings.

Fractional factorial experiments: 2* full factorial experiments can be adapted to more efficiently search through a large number of experimental factors. What you give up in increasing the number of experimental factors is analysis accuracy. Fractional factorial experiments teach how and where to adapt your experiment to get the most out of your search efforts.

Chapter 10