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Simplified Design of Experiments Quality Day Orange Empire 0701 October 28, 2010

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Goals of Todays Session Understand why Design of Experiments is preferable. Learn how to perform basic experiments to optimize results and reduce variation.

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Approaches to Experimentation Build-test-fix (What experimentation?) One-factor-at-a-time (OFAT) Designed experiments (DOE)

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Build-Test=Fix The tinkerers approach Impossible to know if true optimum achieved –Quit when it works! Consistently slow –Requires intuition, luck, rework –Continual fire-fighting

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One Factor at a Time (OFAT) Approach: –Run all factors at one condition –Repeat, changing condition of one factor –Continuing to hold that factor at that condition, rerun with another factor at its second condition –repeat until all factors at their optimum conditions Slow and requires many tests Can miss interactions!

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One Factor at a Time is Like What we conclude may be determined by where we are looking! The Blind Man and the Elephant

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Design of Experiments A statistics-based approach to designed experiments. A methodology to achieve a predictive knowledge of a complex, multi-variable process with the fewest trials possible. An optimization of the experimental process itself.

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Key Concepts DOE is about better understanding of our processes

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Injection Molding Process

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Concrete Mixing Process

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Microwave Popcorn Making Process

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Three Key Principles Replication [DOEs version of Sample Size] –Replication of an experiment –Allows an estimate of experimental error –Allows for a more precise estimate of the sample mean value Randomization [Run experiments in random order] –Cornerstone of all statistical methods –Average out effects of extraneous factors –Reduce bias and systematic errors Blocking [What can influence my experiment?] –Increases precision of experiment –Factor out variables not studied

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Steps in a Designed Experiments Study 1.Brainstorm the problem / causes 2.Design the Experiment 3. Perform the experiment 4. Analyze the data 5. Validate your results

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FACTORIAL EXPERIMENT IMPROVEMENT OF THE MEAN Determining Input Factors to Optimize Output

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Case Study You have been assigned to look into a problem with your companys lamination process. Customers have been complaining about separation in your wood laminate, and you are leading a team to determine how to reduce the separation.

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Step 1. Brainstorm You met with your team, and through your investigation, you determine that the best approach would be to minimize the amount of CURL Curl Laminate

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Step 1. Brainstorm After brainstorming and list reduction, your team decided to look at 3 factors that they felt influenced the amount of curl: A Top Roll Tension, currently set at 22 B Bottom Roll Tension, currently set at 22 C Rewind Tension, currently set at 9

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Step 2. Design the Experiment The team decides to study 2 levels for each of the factors, and determine the effect on the response variable, curl: FactorLow (-) SettingHigh (+) Setting Top Roll Tension1628 Bottom Roll Tension1628 Rewind tension612 Note: Team will make sure that their selected values are FEASIBLE.

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Step 2. Design the Experiment This experiment will have 2 levels (high and low settings) and 3 factors (top roll tension, bottom roll tension, and rewind tension) Number of Experiments = 2 3 = 8

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Step 2. Design the Experiment RunTop Roll TensionBottom Roll TensionRewind Tension 1 --- 2 +-- 3 -+- 4 ++- 5 --+ 6 +-+ 7 -++ 8 +++ Alternate -,+Alternate - -,+ +Alternate - - - - + + + + Note: If another variable is added, there would be 2 4 =16 runs The next design generation would be - - - - - - - -, + + + + + + +

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Step 3. Perform the Experiment Run Top Roll Tension Bottom Roll Tension Rewind Tension Replication 1 Replication 2 Average Std. Deviation 1 16 6 878887.5 0.707 2 28166 767877.0 1.414 3 16286 909291.0 1.414 4 28 6 838081.5 2.121 5 16 12 1019698.5 3.535 6 281612 929191.5 0.707 7 162812 100104102.0 2.828 8 28 12 929191.5 0.707 Perform the runs in random order to ensure statistical validity

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Step 4. Analyze the Data We are going to calculate the MAIN EFFECTS for the 3 factors. We are also going to calculate the INTERACTION EFFECTS between each of the 2 factor combinations and the three factor combination. Lets start with the main effects…

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Step 4. Analyze the Data Visualization of Main Effects Top Roll Tension Bottom Roll Tension Rewind Tension - + - - + + 91.5 87.577.0 91.081.5 98.591.5 102.0 Note: lowest curl achieved when Top Roll Tension HIGH and Bottom Roll and Rewind Tensions set LOW

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Step 4 Analyze the Data Main Effect Calculations Average the High Settings and subtract the average of the Low Settings:

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Step 4. Analyze the Data Interaction Effects Run Top Roll Tension (A) Bottom Roll Tension (B) Rewind Tension (C)ABACBCABCAvg Curl 1 ---+++- 87.5 2 +----++ 77.0 3 -+--+-+ 91.0 4 ++-+--- 81.5 5 --++--+ 98.5 6 +-+-+-- 91.5 7 -++--+- 102.0 8 +++++++ 91.5 Simply multiply the signs of the columns, i.e. + times + equals + - times - equals + + times - equals - and - times + equals -

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Step 4 Analyze the Data Interaction Effect Calculations Average the High Settings and subtract the average of the Low Settings:

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Step 4 Analyze the Data Pareto of Effect Size Pareto shows the absolute value of the effects for comparability of significance.

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Step 4. Analyze the Data Visualization of Interaction Effects Note: parallel lines indicate LACK OF INTERACTION

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Step 4. Analyze the Data Visualization of Interaction Effects Note: parallel lines indicate LACK OF INTERACTION

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Step 4. Analyze the Data Visualization of Interaction Effects Note: parallel lines indicate LACK OF INTERACTION

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Step 4. Analyze the Data Test of Significance We can test for the significance of the effects using the t-distribution and confidence intervals. The first step is to calculate the POOLED STANDARD DEVIATION for all of the observations….

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Step 4 Analyze the Data Pooled Standard Deviation We are studying 7 factors/ interactions (A, B, C, AB, AC, BC, ABC) in 8 runs We did 2 replications of each run. Note: if the number of replicates is the same for all runs, we can simply calculate the pooled standard deviation as the square root of the average of the variances

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Step 4 Analyze the Data Pooled Standard Deviation Run Replication 1 Replication 2 Average Std. DeviationVariance 1 878887.5 0.7070.4998 2 767877.0 1.4141.9994 3 909291.0 1.4141.9994 4 838081.5 2.1214.4986 5 1019698.5 3.53512.4962 6 929191.5 0.7070.4998 7 100104102.0 2.8287.9976 8 929191.5 0.7070.4998 Avg Variance=3.8113 Std Dev = 1.952 Simplified Calculation:

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Step 4. Analyze the Data t Statistic for Significance Key Information for calculation: =risk (Confidence = 1- ) [5%, 95%] p=number of + per effect column [4] r=number of replicates [2] S p =Pooled Standard Deviation [1.952] f=degree of fractionalization (in our case 0, since we are doing a full factorial) k=number of factors [3] Degrees of Freedom=(r-1)2 k-f [8]

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Step 4. Analyze the Data t Statistic for Significance The t statistic for 95% confidence (5% risk), 2 tailed test, 8 degrees of freedom is 2.306 The Error is calculated as: The confidence interval for each of the effects is: –Effect +/- Error t table for 2 tailed test

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Step 4. Analyze the Data t Statistic for Significance Since the Error = 2.25, any effect that is contained in the limits of: 0 +/- 2.25 Is considered NOT STATISTICALLY SIGNIFICANT FactorEffect Rewind Tension m(C)11.63 Top Roll Tension (A)-9.38 Bottom Roll Tension (B)2.88 BC Interaction-1.13 ABC Interaction-1.13 AB Interaction-0.63 AC Interaction0.63 Statistically Significant Not Statistically Significant

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Step 4. Analyze the Data Conclusion Based on this outcome, I will conclude that Top Roll tension, Bottom Roll Tension, and Rewind Tension are significant at the 95% significance level I will conclude that I should set: –Top Roll Tension HIGH (28) –Bottom Roll Tension LOW (16) –Rewind Tension LOW (6)

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Step 4. Analyze the Data Conclusion Run Top Roll Tension Bottom Roll Tension Rewind Tension Replication 1 Replication 2 Average Std. Deviation 1 16 6 878887.5 0.707 2 28166 767877.0 1.414 3 16286 909291.0 1.414 4 28 6 838081.5 2.121 5 16 12 1019698.5 3.535 6 281612 929191.5 0.707 7 162812 100104102.0 2.828 8 28 12 929191.5 0.707 Grand Average=90.0625

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Analysis of Data - Response Model The factor effects can be used to establish a model to predict responses.

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Calculating Expected Response So, if we set A at low (-1), B high (+1) and C High (+1), we would predict: 90.0625 - 4.69 - 1.44 - 5.815 + 0.315 - 0.315 - 0.565 - 0.565 = 76.99 FactorEffectEffect/2SettingValue Grand Average90.0625 Top Roll Tension (A)-9.38-4.691 Bottom Roll Tension (B)2.881.44-1.44 Rewind Tension m(C)11.635.815-5.815 AB Interaction-0.63-0.315(1)(-1)0.315 AC Interaction0.630.315(1)(-1)-0.315 BC Interaction-1.13-0.565(-1)(-1)-0.565 ABC Interaction-1.13-0.565(1)(-1)(-1)-0.565

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Step 5. Validation of Model Run the process with the new settings for a trial period to collect data on the curl response. Compare the new data to the historical data to confirm improvement. A simple Test of Hypothesis of before and after data with a t test can be used.

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Step 5. Validation of Model We are trying to prove that the curl with the new process settings is significantly less than the curl with the old process settings at a 95% level of significance. Our historical standard deviation has been 3.5. –Ho: µ old µ new –Ha: µ old > µ new

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43 Commonly Used Z-Values 1-α (or 1-β)α (or β) Z Value.995.0052.575.990.0102.387.975.0251.960.950.0501.645.900.1001.282.800.2000.842

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Step 5. Validation of Model Sample Size Required: = 5% = 10% = 3.4 Change to detect = 2

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Step 5 Validation of Model Suppose our historical data (25 data points) is as follows for curl: –100, 95, 98, 102, 97, 90, 91, 98, 101, 95 –94, 105, 96, 104, 100, 96, 98, 96, 91, 99 –100, 102, 98, 95, 96 Average = 97.48 Sample Standard Deviation = 3.8419

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Step 5 Validation of Model Now, we run our new settings and collect 25 additional sets of data: –86, 77, 85, 81, 81, 83, 78, 79, 81, 80 –81, 78, 76, 75, 79, 83, 85, 81, 78, 78 –85, 81, 79, 83, 72 Average = 80.20 Sample Standard Deviation = 3.391

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t test – Assuming Equal Variances in Excel OldNewt-Test: Two-Sample Assuming Equal Variances 10086 9577 OldNew 9885Mean97.4880.2 10281Variance14.7611.5 9781Observations25 9083Pooled Variance13.13 9178Hypothesized Mean Difference0 9879df48 10181t Stat16.86034207 9580P(T<=t) one-tail4.21133E-22 9481t Critical one-tail1.677224197 10578P(T<=t) two-tail8.42266E-22 9676t Critical two-tail2.010634722 10475 10079 9683 9885 9681 9178 9978 10085 10281 9879 9583 9672 t critical (24 degrees of freedom) = 1.677 Calculated t value = 16.86 We can conclude with more than 95% confidence that the new parameter settings have significantly reduced the amount of curl in our process.

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FACTORIAL EXPERIMENT REDUCTION OF VARIATION Determining Input Factors to Reduce Variation

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Variation Reduction Standard deviations are not normally distributed, and therefore cannot be used directly as a response variable. Options include: –Use the natural or base 10 log to obtain normality –Use –log 10 (s) for normality –Use the F Statistic on the average variances for the high and low settings

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Data – Curl Reduction Run Top Roll Tension (A) Bottom Roll Tension (B) Rewind Tension (C)ABACBCABCAvg CurlStd DevVariance 1 ---+++- 87.50.707.004998 2 +----++ 77.01.4141.9994 3 -+--+-+ 91.01.4141.9994 4 ++-+--- 81.52.1214.4986 5 --++--+ 98.53.53512.4962 6 +-+-+-- 91.50.707004998 7 -++--+- 102.02.8287.9976 8 +++++++ 91.50.707004998

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Calculate an F Value for each Effect For Top Roll Tension r=# replicates (2) k=# factors (3) f=fractionalization (0) Any effects with a F value greater than 9.60 are significant at the 95% level. Significant for reduced variation???

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F Table for =.025

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Calculate an F Value for each Effect For Bottom Roll Tension r=# replicates (2) k=# factors (3) f=fractionalization (0) Any effects with a F value greater than 9.60 are significant at the 95% level. Significant for reduced variation???

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Calculate an F Value for each Effect For Rewind Tension r=# replicates (2) k=# factors (3) f=fractionalization (0) Any effects with a F value greater than 9.60 are significant at the 95% level. Significant for reduced variation???

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Calculate an F Value for each Effect For Interaction of Top Roll and Bottom Roll Tension r=# replicates (2) k=# factors (3) f=fractionalization (0) Any effects with a F value greater than 9.60 are significant at the 95% level. Significant for reduced variation???

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Calculate an F Value for each Effect For Interaction of Top Roll and Rewind Tension r=# replicates (2) k=# factors (3) f=fractionalization (0) Any effects with a F value greater than 9.60 are significant at the 95% level. Significant for reduced variation???

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Calculate an F Value for each Effect For Interaction of Bottom Roll and Rewind Tension r=# replicates (2) k=# factors (3) f=fractionalization (0) Any effects with a F value greater than 9.60 are significant at the 95% level. Significant for reduced variation???

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Calculate an F Value for each Effect For Interaction of All Factors r=# replicates (2) k=# factors (3) f=fractionalization (0) Any effects with a F value greater than 9.60 are significant at the 95% level. Significant for reduced variation???

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Lets Summarize… FactorF Value Top Roll Tension (A)3.460 Bottom Roll Tension (B)1.0000 Rewind Tension (C)2.412 AB Interaction1.4169 AC Interaction13.3994 BC Interaction1.8986 ABC Interaction1.31937 Note, the AC (Top Roll-Rewind Tension) is the only factor or interaction that exceeds the F Critical Value of 9.60.

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Conclusion To minimize variation, we find that variation is minimized when Top Roll Tension and Rewind Tension are set to the same levels (i.e. High-High or Low-Low). Our optimal settings to minimize curl were Top Roll High, Bottom Roll Low, and Rewind Low. Top Roll and Rewind influenced the curl in OPPOSITE DIRECTIONS! How can we decide?

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Case Studies Instructions: Work in teams to read the case study and then develop an experimental design. You should include: What factors to study How many and what levels to study How many replicates to take How many runs? Design your experiment.

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Case Study 1 You have been working late recently, and subsisting on microwave popcorn. As a result, you have decided to find the formula for the best popcorn. You are down to two brands, A and B. You also find that the time varies between 4 and 6 minutes and power between 75% and 100%. You judge the quality of the popcorn by taste and the number of unpopped kernels.

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Case 1 Design What factors to study How many and what levels to study How many replicates to take How many runs? Design your experiment

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Case 1 Design What factors to study How many and what levels to study How many replicates to take How many runs? Design your experiment

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Case Study 2 Your company has a high-pressure chemical reactor which filters impurities from your product. You have been asked to determine the proper settings to maximize the filtration rate. After a brainstorming session, your team decides that temperature, pressure, chemical concentration % and stir rate all influence the filtration rate (gallons per hour). Looking over the historical records, you see temperature has varied from 24 o C to 35 o C. Pressure can range from 10 PSIG to 15 PSIG, and chemical concentration from 2% to 4%. The lowest stir rate has been 15 RPM and the highest 30 RPM. Design an experiment to determine what the optimal levels would be to maximize the filtration rate.

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Case 2 Design What factors to study How many and what levels to study How many replicates to take How many runs? Design your experiment

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Case 2 Design What factors to study How many and what levels to study How many replicates to take How many runs? Design your experiment

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Case Study 3 A manufacturer of ice crease has hired you to assist them with achieving their target fill rate of 2.50 pounds. After discussing with them the relevant factors, it is concluded that there are two main variables impacting the final weight, fill temperature and the overfill %. Overfill is the percentage of air that is incorporated into the ice crease. The fill temperature is sensitive, and they have machines that vary from 20 o F to 25 o F. Overfill percentage is also tightly controlled. Studies have shown that overfill percentages greater than 120% are perceived as lower quality by customers and percentages less than 90% cause the ice crease to be difficult to scoop.

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Case 3 Design What factors to study How many and what levels to study How many replicates to take How many runs? Design your experiment

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Case 3 Design What factors to study How many and what levels to study How many replicates to take How many runs? Design your experiment

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Points to Remember 1.Get a clear understanding of the problem you intend to solve. 2.Conduct an exhaustive and detailed brainstorming session. 3.Teamwork – involve people involved in all aspects of the process being studied. 4.Randomize the experiment trial order 5.Replicate to understand and estimate variation. 6.Perform confirmatory runs and experiments to test your models validity.

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Your Assignment Find a problem and run a Designed Experiment using the methods we learned today within 1-2 weeks of completing this session. No problem – Optimize microwave popcorn or chocolate chip cookies!

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Questions?

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