Business process improvement_6 potx

Three-level, mixed-level and fractional factorial designs was stated by Montgomery 1991: It is our belief that the two-level factorial and fractional factorial designs should be the corn

Table of

treatments for

the 3 3 design

These treatments may be displayed as follows:

The design can be represented pictorially by

5.3.3.9 Three-level full factorial designs

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Two types of

3 k designs

Two types of fractions of 3k designs are employed:

Box-Behnken designs whose purpose is to estimate a second-order modelfor quantitative factors (discussed earlier in section 5.3.3.6.2)

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5.3 Choosing an experimental design

5.3.3 How do you select an experimental design?

5.3.3.10 Three-level, mixed-level and

fractional factorial designs

was stated by Montgomery (1991): It is our belief that the two-level factorial and fractional factorial designs should be the cornerstone of industrial experimentation for product and process development and improvement He went on to say: There are, however, some situations in which it is necessary to include a factor (or a few factors) that have more than two levels.

This section will look at how to add three-level factors starting with

two-level designs, obtaining what is called a mixed-level design We

will also look at how to add a four-level factor to a two-level design The section will conclude with a listing of some useful orthogonal three-level and mixed-level designs (a few of the so-called Taguchi "L" orthogonal array designs), and a brief discussion of their benefits and disadvantages.

Generating a Mixed Three-Level and Two-Level Design

TABLE 3.38 Generating a Mixed Design Two-Level Three-Level

Similar to the 3k case, we observe that X has 2 degrees of freedom,

which can be broken out into a linear and a quadratic component To illustrate how the 2 3 design leads to the design with one factor at two levels and one factor at three levels, consider the following table, with particular attention focused on the column labels.

5.3.3.10 Three-level, mixed-level and fractional factorial designs

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of four two-level factors and one three-level factor This is accomplished by equating the second two-level factor to AB, the third

to AC and the fourth to ABC Column BC cannot be used in this manner because it contains the quadratic effect of the three-level factor X.

More than one three-level factor

Generating a Two- and Four-Level Mixed Design

a design with one four-level and two two-level factors.

Initially we wish to estimate all main effects and interactions It has been shown (see Montgomery, 1991) that this can be accomplished via

a 2 4 (16 runs) design, with columns A and B used to create the four

TABLE 3.39 A Single Four-level Factor and Two

Two-level Factors in 16 runs

L 9 - A 3 4-2 Fractional Factorial Design 4 Factors

at Three Levels (9 runs)

L 18 - A 2 x 3 7-5 Fractional Factorial (Mixed-Level) Design

1 Factor at Two Levels and Seven Factors at 3 Levels (18 Runs)

The good features of these designs are:

They are orthogonal arrays Some analysts believe this simplifies the analysis and interpretation of results while other analysts believe it does not.

●

They obtain a lot of information about the main effects in a relatively few number of runs.

●

You can test whether non-linear terms are needed in the model,

at least as far as the three-level factors are concerned.

● 5.3.3.10 Three-level, mixed-level and fractional factorial designs

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of the section are:

DOE analysis steps

● Plotting DOE data

● Modeling DOE data

● Testing and revising DOE models

● Interpreting DOE results

● Confirming DOE results

● DOE examples

Full factorial example

❍ Fractional factorial example

❍ Response surface example

● Residual analysis

● Model Lack of Fit tests

● Data transformations for normality and linearity

●

5.4 Analysis of DOE data

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5.4 Analysis of DOE data

5.4.1 What are the steps in a DOE analysis?

Flowchart of DOE Analysis Steps

DOE Analysis Steps

The following are the basic steps in a DOE analysis.

Look at the data Examine it for outliers, typos and obvious problems Construct as many graphs as you can to get the big picture.

Response distributions (histograms, box plots, etc.)

Interaction plots

■

Sometimes the right graphs and plots of the data lead to obvious answers for your experimental objective questions and you can skip to step 5 In most cases, however, you will want to continue by fitting and validating a model that can be used to answer your questions.

Test the model assumptions using residual graphs.

If none of the model assumptions were violated, examine the ANOVA.

Simplify the model further, if appropriate If reduction is appropriate, then return to step 3 with a new model.

■

❍

If model assumptions were violated, try to find a cause.

Are necessary terms missing from the model?

■

Will a transformation of the response help? If a transformation is used, return

to step 3 with a new model.

Note: The above flowchart and sequence of steps should not be regarded as a "hard-and-fast rule"

for analyzing all DOE's Different analysts may prefer a different sequence of steps and not all

types of experiments can be analyzed with one set procedure There still remains some art in both

the design and the analysis of experiments, which can only be learned from experience In addition, the role of engineering judgment should not be underestimated.

5.4.1 What are the steps in a DOE analysis?

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5.4 Analysis of DOE data

5.4.2 How to "look" at DOE data

quantitative techniques and models are tools used to confirm and extendthe conclusions an analyst has already formulated after carefully

"looking" at the data

Most software packages have a selection of different kinds of plots fordisplaying DOE data Dataplot, in particular, has a wide range ofoptions for visualizing DOE (i.e., DEX) data Some of these usefulways of looking at data are mentioned below, with links to detailedexplanations in Chapter 1 (Exploratory Data Analysis or EDA) or toother places where they are illustrated and explained In addition,examples and detailed explanations of visual (EDA) DOE techniquescan be found in section 5.5.9

●

Scatter plot (for pairs of response variables)

● Lag plot

● Normal probability plot

● Autocorrelation plot

●

5.4.2 How to "look" at DOE data

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● Box plot

● Bi-histogram

● DEX scatter plot

● DEX mean plot

● DEX standard deviation plot

● DEX interaction plots

● Normal or half-normal probability plots for effects Note: theselinks show how to generate plots to test for normal (or

half-normal) data with points lining up along a straight line,approximately, if the plotted points were from the assumednormal (or half-normal) distribution For two-level full factorialand fractional factorial experiments, the points plotted are theestimates of all the model effects, including possible interactions.Those effects that are really negligible should have estimates thatresemble normally distributed noise, with mean zero and a

constant variance Significant effects can be picked out as theones that do not line up along the straight line Normal effectplots use the effect estimates directly, while half-normal plots usethe absolute values of the effect estimates

● Residuals vs independent variables

● Residuals lag plot

● Residuals histogram

● Normal probability plot of residuals

5.4.2 How to "look" at DOE data

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5.4 Analysis of DOE data

5.4.3 How to model DOE data

plots, Youden plots, p-value comparisons and stepwise regression

routines are used to reduce the model to the minimum number of neededterms A JMP example of model selection is shown later in this sectionand a Dataplot example is given as a case study

5.4.3 How to model DOE data

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5.4 Analysis of DOE data

5.4.4 How to test and revise DOE models

Outline of Model Testing and Revising: Tools and Procedures

❍ Residuals vs independent variables

❍ Residuals lag plot

❍ Residuals histogram

❍ Normal probability plot of residuals

❍

●

Overall numerical indicators for testing models and model terms

R Squared and R Squared adjusted

❍ Model Lack of Fit tests

❍ ANOVA tables (see Chapter 3 or Chapter 7)

❍

p-values

❍

●

Model selection tools or procedures

ANOVA tables (see Chapter 3 or Chapter 7)

❍

p-values

❍ Residual analysis

❍ Model Lack of Fit tests

❍ Data transformations for normality and linearity

❍ Stepwise regression procedures

❍

●

5.4.4 How to test and revise DOE models

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Normal or half-normal plots of effects (primarily fortwo-level full and fractional factorial experiments)

❍

Youden plots

❍ Other methods

❍

5.4.4 How to test and revise DOE models

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5.4 Analysis of DOE data

5.4.5 How to interpret DOE results

Checklist relating DOE conclusions or outputs to experimental goals or experimental purpose:

Do the responses differ significantly over the factor levels?

(comparative experiment goal)

●

Which are the significant effects or terms in the final model?(screening experiment goal)

●

What is the model for estimating responses?

Full factorial case (main effects plus significantinteractions)

❍

Fractional factorial case (main effects plus significantinteractions that are not confounded with other possiblyreal effects)

❍ Settings for confirmation runs and prediction intervals forresults

❍

●

5.4.5 How to interpret DOE results

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5.4 Analysis of DOE data

5.4.6 How to confirm DOE results

The interpretation and conclusions from an experiment may include a

"best" setting to use to meet the goals of the experiment Even if this

"best" setting were included in the design, you should run it again aspart of the confirmation runs to make sure nothing has changed andthat the response values are close to their predicted values would get

If the time between actually running the experiment and conducting theconfirmation runs is more than a few hours, the experimenter must becareful to ensure that nothing else has changed since the original datacollection

person/operator on the equipment, temperature, humidity, machineparameters, raw materials, etc

5.4.6 How to confirm DOE results (confirmatory runs)

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information gained from this experiment to design another follow-upexperiment.

5.4.6 How to confirm DOE results (confirmatory runs)

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by simply examining appropriate graphical displays of the experimentalresponses In other cases, a satisfactory model has to be fit in order todetermine the most significant factors or the optimal contours of theresponse surface In any case, the software will perform the appropriatecalculations as long as the analyst knows what to request and how tointerpret the program outputs.

Perhaps one of the best ways to learn how to use DOE analysis software

to analyze the results of an experiment is to go through several detailedexamples, explaining each step in the analysis This section will

illustrate the use of JMP 3.2.6 software to analyze three realexperiments Analysis using other software packages would generallyproceed along similar paths

The examples cover three basic types of DOE's:

A full factorial experiment

characterizing the effect of grinding parameters on sintered reaction-bonded silicon nitride, reaction bonded silicone nitride, and sintered silicon nitride.

Only modified data from the first of the 3 ceramic types (sintered reaction-bonded silicon nitride) will be discussed in this illustrative example of a full factorial data analysis.

The reader may want to download the data as a text file and try using other software packages to analyze the data.

Description of Experiment: Response and Factors

Response Variable Y = Mean (over 15 reps) of Ceramic Strength Factor 1 = Table Speed (2 levels: slow (.025 m/s) and fast (.125 m/s)) Factor 2 = Down Feed Rate (2 levels: slow (.05 mm) and fast (.125 mm)) Factor 3 = Wheel Grit (2 levels: 140/170 and 80/100)

Factor 4 = Direction (2 levels: longitudinal and transverse) Factor 5 = Batch (2 levels: 1 and 2)

Since two factors were qualitative (direction and batch) and it was reasonable to expect monotone effects from the quantitative factors, no centerpoint runs were included.

5.4.7.1 Full factorial example

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as a text file or as a JMP file )

Analysis of the Experiment

Step 1: Look at the data

5.4.7.1 Full factorial example

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First we look at the distribution of all the responses irrespective of factor levels.

The following plots were generared:

The first plot is a normal probability plot of the response variable The straight red line is the fitted nornal distribution and the curved red lines form a simultaneous 95% confidence region for the plotted points, based on the assumption of normality.

1

The second plot is a box plot of the response variable The "diamond" is called (in JMP) a

"means diamond" and is centered around the sample mean, with endpoints spanning a 95% normal confidence interval for the sample mean.

5.4.7.1 Full factorial example

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Plot of Response Vs Run Order

As hoped for, this plot does not indicate that time order had much to do with the response levels.

Box plots of

response by

factor

variables

Next, we look at plots of the responses sorted by factor columns.

5.4.7.1 Full factorial example

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Several factors, most notably "Direction" followed by "Batch" and possibly "Wheel Grit", appear

to change the average response level.

Step 2: Create the theoretical model

to be significant, and they are very difficult to interpret from an engineering viewpoint) That allows us to accumulate the sums of squares for these terms and use them to estimate an error term So we start out with a theoretical model with 26 unknown constants, hoping the data will clarify which of these are the significant main effects and interactions we need for a final model.

Step 3: Create the actual model from the data

After fitting the 26 parameter model, the following analysis table is displayed:

Output after Fitting Third Order Model to Response Data

Response: Y: Strength

Summary of Fit RSquare 0.995127 RSquare Adj 0.974821 Root Mean Square Error 17.81632 Mean of Response 546.8959 Observations 32

Effect Test Sum Source DF of Squares F Ratio Prob>F X1: Table Speed 1 894.33 2.8175 0.1442 X2: Feed Rate 1 3497.20 11.0175 0.0160 X1: Table Speed* 1 4872.57 15.3505 0.0078 X2: Feed Rate

X3: Wheel Grit 1 12663.96 39.8964 0.0007 X1: Table Speed* 1 1838.76 5.7928 0.0528 X3: Wheel Grit

5.4.7.1 Full factorial example

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