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# Fractional Factorial Designs

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How to Design Fractional Factorial

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• The purpose of injection molding is not to get part strength. Part strength is determined by raw material. The only thing that injection molding can do is hurt part strength. Therefore, I want to maximize/optimize the function of injection molding …. SHAPE. The strength will be improved when the function is improved. There is a big difference between theoretical experimenters and practical experimenters. Theoretical experimenters don’t care about the output measure, they are interested in all the affects/interactions with the output measure … all theory stuff. Practical experimenters want to optimize a process … they don’t want interactions and they want their output measure to be the function of the process (so that they can optimize it).
• Purpose of blocking, so that the block does not affect factor output. I can determine affect of the block and the affect of the factors independently.
• Interactions are spread throughout the matrix. Therefore, if a factor is significant it overpowers the interactions. For example, if you look at AB interaction, there are no other factors that confound with it (same with any other interaction).
• Ln(s) is actually a better measure than s because the distribution of variance is skewed (cant go below 0). If you use s you can actually have confidence intervals that give you negative s, which you cant get.
• ### Fractional Factorial Designs

1. 1. Fractional Factorial Designs (I want acknowledge the teachings on DOE by the subject experts : Mr. Jim Quinlan on Fraction/Full Factorial Designs and Dr. Ranjit K. Roy on Taguchi.)
2. 2. Fractional Factorial Designs <ul><li>Contents </li></ul><ul><li>What is Fractional Design </li></ul><ul><li>Why use a Fractional Design </li></ul><ul><li>Confounding and Aliasing </li></ul><ul><li>Design Resolution </li></ul><ul><li>2 - Level Fractional Factorials </li></ul><ul><li>Other Designs </li></ul><ul><li>Selection Strategy </li></ul>
3. 3. Fractional Factorial Designs – Learning Objectives <ul><li>After completing this module, you will be able to….. </li></ul><ul><li>1. Design, conduct and analyze fractional factorial experiments. </li></ul><ul><li>2. Select an experimental design that allows clear evaluation of factor effects and interaction effects (especially 2-factor interactions) thought to be strong. </li></ul>
4. 4. Fractional Factorials <ul><li>In fractional factorial designs, a fraction of the total number of combinations are run. </li></ul><ul><li> A B C </li></ul><ul><li> 1. -1 -1 +1 </li></ul><ul><li> 2. +1 -1 -1 </li></ul><ul><li> 3. -1 +1 -1 </li></ul><ul><li> 4. +1 +1 +1 </li></ul><ul><li>These 4 runs represent half of the total combinations that exist for 3 factors at 2 levels. </li></ul>
5. 5. Why Use Fractional Factorials <ul><li>Often, experimenters have a large number of </li></ul><ul><li>factors to investigate. </li></ul><ul><li>The size of full factorials limits their use to only </li></ul><ul><li>a few factors. </li></ul>The full factorial for 1 2-level factor and 7 3-level factors requires… 2 1 x 3 7 = 4374 test combinations. These same factors can be investigated using only 18 of the combinations. Example
6. 6. Obtaining the Critical Information <ul><li>When all test combinations are not run, information will be lost. </li></ul><ul><li>If the test combinations (fraction) to be run are carefully </li></ul><ul><li>selected, the information we care about the most can be </li></ul><ul><li>obtained: </li></ul><ul><li>Main effects </li></ul><ul><li>2 - factor interaction effects </li></ul><ul><li>This approach has been standardized with the </li></ul><ul><li>development of numerous fractional factorial designs. </li></ul>
7. 7. <ul><li>A major concern with fractional factorials is that effects are confounded (confused) with each other. To illustrate this consider the standard 1/2 fraction of a 2 3 design. </li></ul>Fractional Factorials - Confounding The interaction columns are generated, as before, by multiplying appropriate columns, and are used only in the statistical analysis.
8. 8. Confounding (continued) Notice the ABC column of all +1’s; the ABC effect can not be evaluated. Notice the A and BC columns are identical as are B and AC, C and AB. This means that when effects are computed the analysis will give: A effect = BC effect B effect = AC effect C effect = AB effect
9. 9. Confounding (continued) This does not mean that the effects are equal, it means the design cannot separate them. What is being measured is A + BC, B + AC, C + AB, the combined effects. We say A is confounded with BC and write it as: A  BC, B  AC, C  AB. The actual effects being measured is called the Alias pattern.
10. 10. Caution When Selecting Type of Experiment <ul><li>Whenever all possible combinations of factors and levels are </li></ul><ul><li>NOT run, certain factor and interaction effects will be confused </li></ul><ul><li>together. </li></ul><ul><li>Care should be taken to select an experimental design that </li></ul><ul><li>allows clear evaluation of main effects and interaction effects </li></ul><ul><li>(especially 2-factor interactions) thought to be strong. </li></ul>Hmmm…Is it the Factor C effect? Or, the A x B Interaction?
11. 11. Design Resolution <ul><li>In the 2 3 half fraction design, factor effects are </li></ul><ul><li>confounded with 2 - factor interactions ( 2-f.i.). </li></ul><ul><li>We say this is a Resolution III design. </li></ul><ul><li>The general relationship is: </li></ul><ul><li>Resolution III: Factor  2 f.i. </li></ul><ul><li>Resolution IV : Factor  3 f.i., 2 f.i.  2 f.i. </li></ul><ul><li>Resolution V: Factor  4 f.i., 2 f.i.  3 f.i. </li></ul>
12. 12. Selecting Design Resolution <ul><li>Resolution III if believe no strong 2 f.i.’s are present </li></ul><ul><li>Resolution IV if suspect 2 f.i.’s, so don’t want main effects confused with 2 f.i. Not interested in evaluating 2 f.i.’s. </li></ul><ul><li>Resolution V if need to evaluate main effects and 2 f.i.’s. </li></ul><ul><li>The problem is that for a fixed number of factors, higher resolution designs require more runs! </li></ul>The standard approach for selecting which resolution design to use is:
13. 13. Minitab & Confounding <ul><li>Minitab will generate the 1/2 fraction, and produce the alias structure. </li></ul><ul><li>Select: Stat > DOE > Create factorial design </li></ul><ul><li>Click on 2 - level factorial (default generators) </li></ul><ul><li>Set number of factors = 3 </li></ul>Click on Designs Select 1/2 fraction Click OK then OK again. The design will be in columns 5, 6, 7 of the worksheet. The alias pattern will be in the session window.
14. 14. Minitab Output – Alias Structure Fractional Factorial Design Factors: 3 Base Design: 3, 4 Resolution: III Runs: 4 Replicates: 1 Fraction: 1/2 Blocks: none Center pts (total): 0 Some main effects are confounded with two-way interactions Design Generators: C = AB Alias Structure I + ABC A + BC B + AC C + AB *** NOTE ***
15. 15. Example – Injection Molding <ul><li>Small plastic wheels are produced by an injection molding process. An experiment is to be run. The objective is to maximize part strength. </li></ul><ul><li>Levels </li></ul><ul><li> Factors -1 +1 </li></ul><ul><li>A: Die Temp 130  170  </li></ul><ul><li>B: Nozzle Temp 350  375  </li></ul><ul><li>C: Shot Size 6.7 10 </li></ul><ul><li>D: Injection Press 700 900 </li></ul><ul><li>The team strongly suspects an AxB, and AxD interaction. </li></ul><ul><li>No other interactions are judged likely to exist. </li></ul>
16. 16. Comments About the Output Measure <ul><li>In this molding experiment, the team decided to measure part strength (a “Maximum Value is Best” characteristic). </li></ul><ul><li>Recall, the output measures that provide the greatest optimization power are those that relate to the basic function of the manufacturing process. (These are usually “Target Value is Best” characteristics). </li></ul><ul><li>Is the function of the molding process to create part strength ? </li></ul><ul><li>NO! </li></ul><ul><li>The function of the process is to create parts of specific dimensions. Factor combinations that minimize variation in dimensions , are combinations that most uniformly distribute particles and stress. </li></ul>What happens to “part strength” when stresses are uniformly distributed throughout the part? Often, the best way to improve strength and other problem-related measures is not to measure the problem directly. Instead, measure and optimize the process’ intended function. Remember
17. 17. <ul><li>A Full Factorial design will allow evaluation of the four factors </li></ul><ul><li>and all interactions, at the cost of 16 runs . </li></ul><ul><li>If the team is correct in their judgement that only two interactions </li></ul><ul><li>are likely to exist, then a Resolution IV will work, needing only 8 runs : </li></ul>Injection Molding – Selecting the Design Factors: 4 Base Design: 4, 8 Resolution: IV Runs: 16 Replicates: 2 Fraction: 1/2 Blocks: none Center pts (total): 0 Design Generators: D = ABC Alias Structure I + ABCD A + BCD B + ACD C + ABD D + ABC AB + CD AC + BD AD + BC
18. 18. Injection Molding – Layout and Data <ul><li>Two replications were run for each test combination as shown below: </li></ul><ul><li> A B C D Replicates </li></ul>Data is part strength.
19. 19. Injection Molding – Significant Effects <ul><li>From the MiniTab output we see that factors Die Temp, Shot Size and Injection Pressure and both interactions AxB, AxD are significant. </li></ul>Fractional Factorial Fit: Data versus Die Temp, Nozzle Temp, ... Estimated Effects and Coefficients for Data (coded units) Term Effect Coef SE Coef T P Constant 56.625 0.7235 78.27 0.000 Nozzle T -1.500 -0.750 0.7235 -1.04 0.330 Die Temp*Nozzle T 5.250 2.625 0.7235 3.63 0.007 -0.750 -0.375 0.7235 -0.52 0.618 Die Temp* Inj. Pre 10.750 5.375 0.7235 -7.43 0.000 Die Temp -10.000 -5.000 0.7235 -6.91 0.000 -19.500 -9.750 0.7235 -13.48 0.000 Inj. Pre 16.000 8.000 0.7235 11.06 0.000 Die Temp*Shot Siz Shot Siz
20. 20. Injection Molding – Response Table <ul><li>Minitab also gives the average output at each level of each factor and interaction: </li></ul>Least Squares Means for Data -1 -1 48.25 1.447 1 -1 49.00 1.447 -1 1 75.00 1.447 1 1 54.25 1.447 Mean SE Mean Die Temp (A) -1 61.63 1.023 1 51.63 1.023 Nozzle T (B) -1 57.38 1.023 1 55.88 1.023 Shot Siz (C) -1 66.38 1.023 1 46.88 1.023 Inj . Pre (D) -1 48.63 1.023 1 64.63 1.023 Die Temp* Inj . Pre Die Temp*Nozzle T -1 -1 65.00 1.447 1 -1 49.75 1.447 -1 1 58.25 1.447 1 1 53.50 1.447 Die Temp*Shot Siz -1 -1 71.00 1.447 1 -1 61.75 1.447 -1 1 52.25 1.447 1 1 41.50 1.447 Mean SE Mean
21. 21. Main Effects Graphs Data 1 - 1 1 - 1 1 - 1 1 - 1 6 6 6 1 5 6 5 1 4 6 Die Temp Nozzle Temp Shot Size Inj. Press Main Effects Plot (data means) for Data
22. 22. Interaction Graphs Nozzle Temp A, B A, D 1 1 - 1 - 1 6 5 6 0 5 5 5 0 Mean Interaction Plot (data means) for Data Die Temp -1 1 1 1 - 1 - 1 7 0 6 0 5 0 I n j . P r e s s Interaction Plot (data means) for Data Mean Die Temp -1 1
23. 23. Best Levels <ul><li>A -1 B -1 :highest from the AxB graph </li></ul><ul><li>A -1 D +1 :highest from the AxD graph </li></ul><ul><li>C -1 :highest from the C graph </li></ul>To maximize strength the levels to be used are: The analysis picks A -1 B -1 C -1 D +1 to be the combination to maximize strength. Notice, it was not one of the tested combinations.
24. 24. Prediction <ul><li>We will use A (Die Temp), C (Shot Size), D (Inj. Press.), AxB, AxD as the terms in our prediction (regression) equation. </li></ul>Regression Analysis: Data versus Die Temp, Shot Size, ... The regression equation is Data = 56.6 - 5.00 Die Temp - 9.75 Shot Size + 8.00 Inj . Press + 2.62 DTemp * NTemp - 5.38 DTemp * IPress Predicted Values for New Observations New Obs Fit SE Fit 95.0% CI 95.0% PI 1 87.375 1.713 ( 83.558, 91.192) ( 80.066, 94.684) Values of Predictors for New Observations New Obs Die Temp Shot Siz Inj . Pre DTemp *NT DTemp *IP 1 -1.00 -1.00 1.00 1.00 -1.00 The predicted average strength, 87.375, exceeds any in our test matrix.
25. 25. Factors that Affect Variation <ul><li>As was demonstrated in the previous module, the same data set can be analyzed to identify the factors that affect variation. </li></ul><ul><li>In this case, however, only two replications were used. At least three data values per test condition should be obtained in order to evaluate the effects on variation. </li></ul><ul><li>Therefore, the improvement strategy for this experimented is limited to maximizing the average. </li></ul>
26. 26. Blocking <ul><li>Blocking is a technique used to determine which test combinations </li></ul><ul><li>should be run, for example, within a given block of time. </li></ul>What is the purpose of running the tests in this manner? Question For example, we may decide to run the ABC – tests in the morning and the ABC + tests in the afternoon. The highest order interaction column is used to create the block. 1 5 6 2 7 3 4 8 Run order A B C AB AC BC ABC 1 – – – + + + – 2 + – – – – + + 3 – + – – + – + 4 + + – + – – – 5 – – + + – – + 6 + – + – + – – 7 – + + – – + – 8 + + + + + + +
27. 27. Fractional Factorials: 3-level factors <ul><li>Fractional designs for factors at 3-levels have also been developed. For example, with four 3-level factors, A, B, C, D, a full factorial consists of 3 4 = 81 test combinations. A 1/9 fraction, 1/9  3 4 , consisting of nine test combinations is available. </li></ul>The factors will be confounded with interactions as in the 2-level fractionals, but in a more complex fashion.
28. 28. Other Designs <ul><li>1. Plackett-Burman </li></ul><ul><li>2. Central Composite Designs (C.C.D.) </li></ul><ul><li>3. Taguchi Designs </li></ul>Note All of these designs are available in Minitab.
29. 29. Plackett-Burman Designs <ul><li>A series of Resolution III designs developed specifically for screening large numbers of 2-levels factors. </li></ul><ul><li>Only factor effects can be analyzed with these designs. </li></ul><ul><li>Factor effects are confounded with fractional portions of 2 f.i.’s. </li></ul>
30. 30. Plackett-Burman Designs <ul><li>Data Matrix (randomized) 12 runs:11 factors </li></ul><ul><li>Run A B C D E F G H J K L </li></ul><ul><li>1 + - + - - - + + + - + </li></ul><ul><li>2 + + + - + + - + - - - </li></ul><ul><li>3 - - - + + + - + + - + </li></ul><ul><li>4 - + + - + - - - + + + </li></ul><ul><li>5 + - - - + + + - + + - </li></ul><ul><li>6 - - - - - - - - - - - </li></ul><ul><li>7 - + + + - + + - + - - </li></ul><ul><li>8 + - + + - + - - - + + </li></ul><ul><li>9 - - + + + - + + - + - </li></ul><ul><li>10 + + - + + - + - - - + </li></ul><ul><li>11 - + - - - + + + - + + </li></ul><ul><li>12 + + - + - - - + + + - </li></ul>
31. 31. Central Composite Designs (C.C.D.) <ul><li>A series of designs that extend 2 K and fractional 2 K designs by adding additional design points. </li></ul><ul><li>Efficient designs for building quadratic models (prediction equations). </li></ul>
32. 32. Taguchi Designs <ul><li>G. Taguchi catalogued a large number of standard designs which he refers to as Orthogonal Arrays (O.A.’s). </li></ul><ul><li>The two most utilized are the L 12 2-level screening design which is a 12 run Plackett-Burman design, and the L 18 a mixed 2-level and 3-level screening design. </li></ul>Dr. Taguchi uses the L 12 and L 18 arrays to conduct optimization experiments. Classical Design of Experiment techniques utilize these same arrays for screening. Note
33. 33. L 12 O.A. Can handle up to eleven 2-level factors.
34. 34. L 18 O.A Can handle one 2-level factor, and up to seven 3-level factors.
35. 35. Variability Reduction Strategies <ul><li>Dr. Genichi Taguchi is credited for introducing DOE techniques to </li></ul><ul><li>reduce variation. His strategies include: </li></ul>1. Achieving robustness against noise using noise factor and Signal-to-Noise ratio. 2. Optimizing Ideal Function: achieving maximum sensitivity to an input signal and robustness against noise. (M is the input signal factor). A number of additional strategies accompany these experiments. Since, however, Dr. Taguchi’s techniques are beyond the scope of this course, we limit our discussion to the classical techniques described in the next slide. Note ABDC… Orthogonal Array ABCDE… Noise Level 1 Noise Level 2 S/N Orthogonal Array M 1 M 2 M 3 S/N N 1 N 2 N 1 N 2 N 1 N 2
36. 36. Variability Reduction Strategies (continued) <ul><li>Classical experimenters recognized the value of using DOE for variability reduction. The common classical approaches are: </li></ul>1. Find factors that affect variation by analyzing standard deviations computed from replications or repetitions. 2. Same as #1 except take natural log of standard deviation. 3. Use a noise factor to induce variation, but still analyze with s or In s. A B AxB Replications S Factorial Design A B AxB Replications In S Factorial Design A B AxB N 1 N 2 S Factorial Design
37. 37. Strategy for Design Selection <ul><li>All factors at 2-levels </li></ul>Full factorials 1/2 fraction of a 2 5 full factorial (16 runs) Screening design (12 runs) Number of factors K 2 K design 1/2 • 2 5 design L 12 K  4 K=5 6  K  11
38. 38. Strategy for Design Selection <ul><li>All factors at 3-levels - factors all quantitative </li></ul>All factors at 3-levels - some are qualitative Central composite design for model building Screening design Full factorials Screening design Number of factors K C.C.D. L 18 K  5 6  K  7 Number of factors K 3 K design L 18 K  3 4  K  7