table of contents - open source six sigma belt manual... · with the six sigma methodology, we use...

Copyright OpenSourceSixSigma.comOSSS LSS Black Belt Manual

Table of Contents

Define PhaseUnderstanding Six Sigma…………………………………………………..….…….…

1Six Sigma Fundamentals………………………………..…………………..……..…. 22Selecting Projects………………………………………………………..……..………

42Elements of Waste………………………………………...……………………………64 Wrap Up and Action Items……………………………………………………….……77

Measure PhaseWelcome to Measure…………………………………………………….……..….....83Process Discovery………………………………………………………………………86Six Sigma Statistics…………………………….………………………………….….135Measurement System Analysis……………………………………………………....168Process Capability …………………………………………………………

……….200Wrap Up and Action Items ………………………………………………………….221

Analyze PhaseWelcome to Analyze……………………………………………………………

.…..227“X”

Sifting………………………………….…………………………….……….….230Inferential Statistics…………………………………………………..………….…….256Introduction to Hypothesis Testing…………………………….…………………….271Hypothesis Testing Normal Data Part 1……………………………..………………285Hypothesis Testing Normal Data Part 2 ……………………………………….……328Hypothesis Testing Non-Normal Data Part 1………………………………….……358Hypothesis Testing Non-Normal Data Part 2……………………………………….384Wrap Up and Action Items ………………………………………………....……..403

Improve PhaseWelcome to Improve…………………………………………………………...…..409Process Modeling Regression……………………………………………………….412Advanced Process Modeling……………………………………………………….431Designing Experiments………………………………………………………………458Experimental Methods………………………………………………………………473Full Factorial Experiments………………………………………………………..…488Fractional Factorial Experiments…………………………………………….……..517Wrap Up and Action Items…………………………………………………………537

Control PhaseWelcome to Control…………………………………………………………………543Lean Controls…………………………………………………………………………546Defect Controls…………………………………………………………….…………561Statistical Process Control…………………………………………………………….573Six Sigma Control Plans………………………………………………………………613Wrap Up and Action Items……………………………………………………….…633

Glossary

Page

Copyright OpenSourceSixSigma.com

409

OSSS LSS Black Belt Manual

Lean Six Sigma

Black Belt Training

Now we are going to continue with the Improve Phase “Designing Experiments”.

Improve Phase Designing Experiments

Improve Phase Designing Experiments


410


Designing Experiments

Overview

Project Status Review

Within this module we will provide an introduction to Design of Experiments, explain what they are, how they work and when to use them.

Graphical AnalysisGraphical Analysis

DOE MethodologyDOE Methodology

Reasons for ExperimentsReasons for Experiments

Full Factorial ExperimentsFull Factorial Experiments

Experimental MethodsExperimental Methods

Designing ExperimentsDesigning Experiments

Advanced Process Modeling: MLR

Advanced Process Modeling: MLR

Process Modeling: RegressionProcess Modeling: Regression

Welcome to ImproveWelcome to Improve

Fractional Factorial Experiments

Fractional Factorial Experiments

Wrap Up & Action ItemsWrap Up & Action Items

• Understand our problem and it’s impact on the business. (Define)

• Established firm objectives/goals for improvement. (Define)• Quantified our output characteristic. (Define)• Validated the measurement system for our output

characteristic. (Measure)• Identified the process input variables in our process.

(Measure)• Narrowed our input variables to the potential “X’s” through

Statistical Analysis. (Analyze)• Selected the vital few X’s to optimize the output response(s).

(Improve)• Quantified the relationship of the Y’s to the X’s with Y=f(x).

(Improve)


411


Six Sigma Strategy

Designs of Experiments help the Belt to understand the cause and

effect between the process output or outputs of interest and the vital few inputs. Some of these causes and effects may include the impact of interactions often referred to synergistic or cancelling effects.

Reasons for Experiments

This is reoccurring awareness. By using tools we filter the variables of defects. When talking of the Improve Phase in the Six Sigma methodology we are confronted by many designed experiments; transactional, manufacturing, research.


(X5)(X6) (X7)(X9)(X1) (X8) (X11)(X4)

(X10)(X3)(X2)

SuppliersContractors

Employees

CustomersInputsOutputs

(X3)

(X2)(X4)

(X5)(X1)

(X8)(X11)

(X5)

(X4)

(X11)

(X3)

SIPOCVOCProject Scope

P-Map, XY, FMEACapability

Box Plot, ScatterPlots, Regression

Fractional FactorialFull FactorialCenter Points

The Analyze Phase narrowed down the many inputs to a critical few, now it is necessary to determine the proper settings for the vital few inputs because:

– The vital few potentially have interactions.– The vital few will have preferred ranges to achieve optimal results.– Confirm cause and effect relationships among factors identified in

analyze phase (e.g. regression)

Understanding the reason for an experiment can help in selectingthe design and focusing the efforts of an experiment.Reasons for experimenting are:

– Problem Solving (Improving a process response)– Optimizing (Highest yield or lowest customer complaints)– Robustness (Constant response time)– Screening (Further screening of the critical few to the vital

few X’s)

Design where you’re going - be sure you get there!


412


Desired Results of Experiments

Here we have models that are the result of designed experiments.

Many have difficulty determining DOE models from that of physical models. A physical model includes: biology, chemistry, physics and usually many variables, typically using complexities and calculus to describe. The DOE model doesn’t include any variables or complex calculus: it includes most important variables and shows variation of data collected. DOE will focus on the specific region of interest.

DOE Models vs. Physical Models


Problem Solving– Eliminate defective products or services.– Reduce cycle time of handling transactional processes.

Optimizing– Mathematical model is desired to move the process response.– Opportunity to meet differing customer requirements (specifications or

VOC).Robust Design

– Provide consistent process or product performance. – Desensitize the output response(s) to input variable changes including

NOISE variables.– Design processes knowing which input variables are difficult to maintain.

Screening– Past process data is limited or statistical conclusions prevented good

narrowing of critical factors in Analyze Phase

When it rains it PORS!

Designed experiments allows us to describe a mathematical relationship between the inputs and outputs. However, often the mathematical equation is not necessary or used depending on the focus of the experiment.

What are the differences between DOE modeling and physical models?– A Physical model is known by theory using concepts of

physics, chemistry, biology, etc...– Physical models explain outside area of immediate project

needs and include more variables than typical DOE models.– DOE describes only a small region of the experimental

space.

The objective is to minimize the response. The physical model is not important for our business objective. The DOE Model will focus in the region of interest.


413


Definition for Design of Experiments

One Factor at a Time is NOT a DOE

Design of Experiment shows the cause and effect relationship of variables of interest X and Y. By way of input variables, designed experiments have been noted within the Analyze Phase then are executed in the Improve Phase. DOE tightly controls the input variables and carefully monitors the uncontrollable variables.

Let’s assume a Belt has found in the Analyze Phase that pressure and temperature impact his process and no one knows what yield is achieved for the possible temperature and pressure combinations.

If a Belt inefficiently did a One Factor at a Time experiment (referred to as OFAT), one variable would be selected to change first while the other variable is held constant, once the desired result was observed,


Design of Experiments (DOE) is a scientific method of planning and conducting an experiment that will yield the true cause-and-effect relationship between the X variables and the Y variables of interest.

DOE allows the experimenter to study the effect of many input variables that may influence the product or process simultaneously, as well as possible interaction effects (for example synergistic effects).

The end result of many experiments is to describe the results as a mathematical function.

y = f (x)

The goal of DOE is to find a design that will produce the information required at a minimum cost.

Properly designed DOE’s are more efficient experiments.

One Factor at a Time (OFAT) is an experimental style but not a planned experiment or DOE.The graphic shows yield contours for a process that are unknown to the experimenter.

Pres

sure

(psi

)

75

80

85

90

95

Yield Contours AreUnknown To Experimenter

30 31 32 33 34 35

120

125130

135

Temperature (C)

Trial Temp Press Yield1 125 30 742 125 31 803 125 32 854 125 33 925 125 34 866 130 33 857 120 33 90

7

21 43

6

5 Optimum identifiedwith OFAT

True Optimum availablewith DOE

the first variable is set at that level and the second variable is changed. Basically, you pick the winner of the combinations tested.

The curves shown on the graph above represent a constant process

yield if the Belt knew the theoretical relationships of all the variables and the process output of pressure. These contour lines are familiar if you’ve ever done hiking in the mountains and looked at an elevation map which shows contours of constant elevation. As a test we decided to increase

temperature to achieve a higher yield. After achieving a maximum yield with temperature, we then decided to change the other factor, pressure. We then came to the conclusion the maximum yield is near 92% because it was the highest yield noted in our 7 trials.

With the Six Sigma methodology, we use DOE which would have found a higher yield using equations. Many sources state that OFAT experimentation is inefficient when compared with DOE methods. Some people call it hit or miss. Luck has a lot to do

with results using OFAT methods.


414


Types of Experimental Designs

Value Chain

DOE is iterative in nature and may require more than one experiment at times.

As we learn more about the important variables, our approach will change as well. If we have a very good understanding of our process maybe we will only need one experiment, if not we very well may need a series of experiments.

Fractional Factorials or screening designs are used when the process or product knowledge is low. We may have a long list of possible input variables (often referred to as factors) and need to screen them down to a more reasonable or workable level.

Full Factorials are used when it is necessary to fully understand the effects of interactions and when there are between 2 to 5 input variables.

Response surface methods (not typically applicable) are used to optimize a response typically when the response surface has significant curvature.


The most common types of DOE’s are:– Fractional Factorials

• 4-15 input variables – Full Factorials

• 2-5 input variables – Response Surface Methods (RSM)

• 2-4 input variables

Fractional Factorials

Full Factorial

ResponseSurface

The general notation used to designate a full factorial design is given by:

•Where k is the number of input variables or factors.– 2 is the number of “levels” that will be used for each

factor.• Quantitative or qualitative factors can be used.

DOE is iterative in Generally noted is 2 to the k and k is number of input variables or factors and 2 is the number of levels all factors used. If the experiment called for 3 factors, each with levels, it would be 2 cubed designs; as the number of experimental runs are shown by the MATH denoted. Two levels and four factors are shown at the bottom of our slide; by using the notation, how many runs would be involved in this design? 16 is the answer, of course.


415


Visualization of 2 Level Full Factorial

Let’s consider a 2 squared design which means we have 2 levels for 2 factors. The factors of interest are temperature and pressure. There are several ways to visualize this 2 level Full Factorial design. In experimenting we often use what’s called coded variables. Coding simplifies the notation. The low level for a factor is minus one, the high level is plus one. Coding is not very friendly when trying to run an experiment so we use uncoded or actual variable levels. In our example 300 degrees is the low

Graphical DOE Analysis -

The Cube Plot

The representation here has two cubed designs and 2 levels of three factors and shows a treatment combination table using coded inputs level settings. The table has 8 experimental runs. Run 5 shows start angle, stop angle very low and the fulcrum relatively high.


T P T*P-1+1-1+1

-1+1-1+1

-1-1+1+1

-1-1+1+1

+1-1-1

+1

+1-1-1

+1

Temp300

350

Press500

600

22

Temp

Press

350F300F

(+1,-1)(-1,-1)500

600 (+1,+1)(-1,+1)

Four experimental runs:• Temp = 300, Press = 500• Temp = 350, Press = 500• Temp = 300, Press = 600• Temp = 350, Press = 600

Coded levels for factors

Uncoded levels for factors

level, 500 degrees is the high level for temperature.

Back when we had to calculate the effects of experiments by hand

it was much simpler to use coded variables. Also when you look at the prediction equation generated you could easily tell which variable had the largest effect. Coding also helps us explain some of the math involved in DOE.

Fortunately for us, MINITABTM

calculates the equations for both coded and uncoded data.

Consider a 23 design on a catapult...

Sto

p A

ngle

Start Angle 0.92.1

2.45.15

3.35

8.2 4.55

1.5

Fulcrum

Run Start Stop MetersNumber Angle Angle Fulcrum Traveled

1 -1 -1 -1 2.102 1 -1 -1 0.903 -1 1 -1 3.354 1 1 -1 1.505 -1 -1 1 5.156 1 -1 1 2.407 -1 1 1 8.208 1 1 1 4.55

A B C Response

Sto

p A

ngle

Start Angle 0.92.1

2.45.15

3.35

8.2 4.55

1.5

Fulcrum

Sto

p A

ngle

Start Angle 0.92.1

2.45.15

3.35

8.2 4.55

1.5

Fulcrum


1 -1 -1 -1 2.102 1 -1 -1 0.903 -1 1 -1 3.354 1 1 -1 1.505 -1 -1 1 5.156 1 -1 1 2.407 -1 1 1 8.208 1 1 1 4.55

A B C Response


1 -1 -1 -1 2.102 1 -1 -1 0.903 -1 1 -1 3.354 1 1 -1 1.505 -1 -1 1 5.156 1 -1 1 2.407 -1 1 1 8.208 1 1 1 4.55

A B C Response

What are the inputs being manipulated in this design?How many runs are there in this experiment?


416



The Cube Plot (cont.)


The Main Effects Plot

boxes on the corners of the cube. If we set the stop angle high, start angle low and fulcrum high we would expect to launch a ball about 8.2 meters with the catapult.

Make sense?

MINITABTM

generates various plots, the cube plot is one. Open the MINITABTM

worksheet “Catapult.mtw”.

This cube plot is a 2 cubed design for a catapult using three variables:

Start AngleStop AngleFulcrum

Here we used coded variable level settings so we do not know what the actual process setting were in uncoded units. The data means for the response distances are the

The Main Effects Plots shown here display the effect that the input values have on the output response.

The y axis is the same for each of the plots so they can be compared side by side.

Which has the steepest Slope? What has the largest impact on the output?

Answer: Fulcrum


This graph is used by the experimenter to visualize how the response data is distributed across the experimental space.

How do you read or interpret this plot?

What are these?

Stat>DOE>Factorial>Factorial Plots … Cube, select response and factors

Catapult.mtw

1

-1

1

-11-1

Fulcrum

Stop Angle

Start Angle

4.55

2.405.15

8.20

1.50

0.902.10

3.35

Cube Plot (fitted means) for Distance

1

-1

1

-11-1

Fulcrum

Stop Angle

Start Angle

4.55

2.405.15

8.20

1.50

0.902.10

3.35

Cube Plot (fitted means) for Distance

This graph is used to see the relative effect of each factor on the output response.

Stat>DOE>Factorial>Factorial Plots … Main Effects, select response and factors

Start Angle

Mea

n of

Dis

tanc

e

1-1

5.0

4.5

4.0

3.5

3.0

2.5

2.0

Main Effects Plot (data means) for Distance

Stop Angle

Mea

n of

Dis

tanc

e

1-1

5.0

4.5

4.0

3.5

3.0

2.5

2.0


Fulcrum

Mea

n of

Dis

tanc

e

1-1

5.0

4.5

4.0

3.5

3.0

2.5

2.0


Which factor has the largest impact on the output?

Hint: Check the slope!


417


Interaction Definition

Main Effects Plot’s Creation

In order to create the Main Effects Plot we must be able to calculate the average response at the low and high levels for each Main Effect. The coded values are used

to show which responses must be used to calculate the average.

Let’s review what is happening here. How many experimental runs were operated with the start angle at the high level or 1. The answer is 4 experimental runs shows the process to run with the start angle at the high level. The 4 experimental runs running with the start angle at the high level are run number 2, 4, 6 and 8. If we take the 4 distances or process output and

take the average, we see the average distance when the process had the start angle running at the high level was 2.34 meters. The second dot from the left in the Main Effects Plots shows the distance of 2.34 with the start angle at a high level.


Run # Start Angle Stop Angle Fulcrum Distance1 -1 -1 -1 2.102 11 -1 -1 0.900.903 -1 1 -1 3.354 11 1 -1 1.501.505 -1 -1 1 5.156 11 -1 1 2.402.407 -1 1 1 8.208 11 1 1 4.554.55

Avg Distance at Low Setting of Start Angle: 2.10 + 3.35 + 5.15 + 8.20 = 18.8/4 = 4.70

Avg. distance at High Setting of Start Angle: 0.900.90 + 1.501.50 + 2.402.40 + 4.554.55 = 9.40/4 = 2.34Start Angle Stop Angle Fulcrum


2.0

2.8

3.6

4.4

5.2

Dis

t

-1 1 -1 1 -1 1

Interactions occur when variables act together to impact the output of the process. Interactions plots are constructed by plotting both variables together on the same graph. They take the form of the graph below. Note that in this graph, the relationship between variable “A” and Y changes as the level of variable “B” changes. When “B” is at its high (+) level, variable “A” has almost no effect on Y. When “B” is at its low (-) level, A has a strong effect on Y. The feature of interactions is non-parallelism between the two lines.

A- +

Y

Lower

HigherB-

B+

Out

put

A- +

Y

Lower

HigherB-

B+

Out

put

When B changes from low to high, the output drops dramatically.When B changes

from low to high, the output drops very little.


418


Degrees of Interaction Effect

with the green arrow above we must find the response data when the fulcrum is set low and start angle is set high (notice the color coding MINITABTM

uses in the upper right hand corner of the plot for the second factor). The point indicated with the purple arrow is where fulcrum is set high and start angle is high. Take a few moments to verify the remaining two points plotted.

Let’s review what is happening here. The dot indicated by the green

arrow is the mean distance when the fulcrum is at the low level as indicated by a -1 and when the start angle is at the high level as indicated by a 1. Earlier we said the point indicated by the green arrow had the fulcrum at the low level and the start angle at the high level. Experimental runs 2 and 4 had the process running at those conditions so the distance from those two experimental runs is averaged and plotted in reference to a value of 1.2 on the vertical axis. You can note the red dotted line shown is for when the start angle is at the high level as indicated by a 1.

Degrees of interaction can be related to non-

parallelism and the more non-parallel the lines are the stronger the interaction.

A common misunderstanding is that that the lines must actually cross each other for an interaction to exist but that’s NOT true. The lines may cross at some level OUTSIDE of the experimental region, but we really don’t know that.

Interaction Plot Creation


A- +

Y

Low

High

B-

B+

No Interaction

A- +

Y

Low

High B-

B+

Strong InteractionA- +

Y

Low

High B-

B+

Full Reversal

B+

A- +

Y

Low

High B-

Moderate Reversal

B+

A- +

Y

Low

High B-

Some Interaction

B+

B+

A- +

Y

Low

High

B-

B+

No Interaction

A- +

Y

Low

High B-

B+


Y

Low

High B-

B+

Full Reversal

B+

A- +

Y

Low

High B-

Moderate Reversal

B+

A- +

Y

Low

High B-

Some Interaction

A- +

Y

Low

High

B-

B+

No Interaction

A- +

Y

Low

High B-

B+


Y

Low

High B-

B+

Full Reversal

B+

A- +

Y

Low

High B-

Moderate Reversal

B+

A- +

Y

Low

High B-

Some Interaction

B+

B+

Parallel lines show absolutely no interaction and in all likelihood will never cross.

-11

-1-1 11

1.5

2.5

3.5

4.5

5.5

6.5

Fulcrum

Start Angle

Mea

n

Interaction Plot (data means) for Distance

-11

-1-1 11

1.5

2.5

3.5

4.5

5.5

6.5

Fulcrum

Start Angle

Mea

n


Run # Start Angle Stop Angle Fulcrum Distance1 -1 -1 -1 2.102 1 -1 -1 0.903 -1 1 -1 3.354 1 1 -1 1.505 -1 -1 1 5.156 1 -1 1 2.407 -1 1 1 8.208 1 1 1 4.55

(0.90 + 1.50)/2 = 1.20

(4.55 + 2.40)/2 = 3.48

Calculating the points to plot the interaction is not as straight forward as it was in the Main Effects Plot. Here we have four points to plot and since there are only 8 data points each average will be created using data points from 2 experimental runs. This plot is the interaction of Fulcrum with Start Angle on the distance. Starting with the point indicated


419



The Interaction Plots

Based on how many factors you select MINITABTM

will create a number of interaction plots.

Here there are 3 factors selected so it generates the 3 interaction plots. These are referred to as 2-

way interactions.

MINITABTM

will also plot the mirror images, just in case it is easier to interpret with the variables flipped. If you care to create the mirror image of the interaction plots, while creating interaction plots, click on “Options”

and choose “Draw full interaction plot matrix”

with a checkmark in the box. These mirror images present the same data but visually may be easier to understand.


Start Angle

1-1 1-1

6

4

2

Stop Angle

6

4

2

Fulcrum

StartAngle

-11

StopAngle

-11


Start A ngle

1-1 1-1

6

4

2

Stop A ngle

6

4

2

Fulcrum

StartAngle

-11

StopAngle

-11


Stat>DOE>Factorial>Factorial Plots … Interactions, select response and factors

When you select more than two variables, MINITABTM generates an Interaction Plot Matrix which allows you to look at interactions simultaneously. The plot at the upper right shows the effects of Start Angle on Y at the two different levels of Fulcrum. The red line shows theeffects of Fulcrum on Y when Start Angle is at its high level. The black line represents the effects of Fulcrum on Y when Start Angle is at its low level.

Stat>DOE>Factorial>Factorial Plots … Interactions, select response and factors

The plots at the lower left in the graph above (outlined in blue) are the “mirror image” plots of those in the upper right. It is often useful to lookat each interaction in both representations.

Choose this optionfor the additional

plots.

Start Angle

6

4

2

1-1

Stop Angle

6

4

2

1-1

6

4

2

Fulcrum

1-1

StartAngle

-11

StopAngle

-11

Fulcrum-11


Start A ngle

6

4

2

1-1

Stop A ngle

6

4

2

1-1

6

4

2

Fulcrum

1-1

StartAngle

-11

StopAngle

-11

Fulcrum-11



420


DOE Methodology

Generate Full Factorial Designs in MINITABTM

It is easy to generate full factorial designs in MINITABTM. Follow the command path shown here. These are the designs that MINITABTM

will create. They are color coded using the Red, Yellow and Green. Green are the “go”

designs, yellow are the “use caution”

designs and red are the “stop, wait and think”

designs. It has a similar meaning as do street lights.


1. Define the practical problem2. Establish the experimental objective3. Select the output (response) variables4. Select the input (independent) variables5. Choose the levels for the input variables6. Select the experimental design7. Execute the experiment and collect data8. Analyze the data from the designed experiment and

draw statistical conclusions9. Draw practical solutions10.Replicate or validate the experimental results 11.Implement solutions

table of contents - open source six sigma belt manual... · with the six sigma methodology, we use...

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